fie is " it We lant bl i ENE be tt Wt ent 1 Huh HER oe Nt 1 a kik HE SRU you 5 3 lie ' in Oh nh 17 Le ‘i | Ge i inh Rie HH aA rt it Ihara ‘ Walt Ki (it ie Hea A ws Soi i a ie Lot) he HEEN ti Ait bj ae ; i a KEP Hi Mee Tin) UNA ; et iN it na vie ait 2. Sects =< ee Te Ener ee Bi es in eer En SS == | Ee i t= 3} AA ithe He Ki ia ie et HH ie ki ol we A ME ae ia iM eat an ait it . a Mat Pa AChE iit pare q Hi bit iad Dit i Ae a ae Bs a es Yi ae fe if iat} aid My Mae nit sits Wet nat nine je Ht rel th i ia ae AG gn a I i Ne ag Hart iv we aay et aes i i ee era aig its Nite! it te ‘ Ke WA in Che A ae eet ==> S- a Sere r= 3 —. SS en = ze = ee a Ht i : ti Gi +H, i EE Kn a. = a i My eats = = dz pe ien TT di en i wae i ete ie ital ni He en /l i i Gi) st i a ‘ Asia fas i faite | ca i oS i AE tantie ui HER re; 1 Tuas Atte att Ss
we tried to make probable in Suppl. N°, 80a $ 2.
The fact that in § 4 a satisfactory agreement with experimental
data was obtained, may, if the validity of the other hypotheses is
admitted as sufficiently approximate, be regarded as a confirmation
Ss)
of the above supposition concerning the magnitude of the energ
elements.
Astronomy. — “On Sernicer’s hypothesis about the anomalies in
the motion of the inner planets.” By J. Wo.tier Jr. (Com-
municated by Prof. W. DE Sitter).
‘(Communicated in the meeting of April 24, 1914).
To explain the differences between observation and calculation in
the secular perturbations of the elements of the four inner planets,
SEELIGER *) worked out the hypothesis that these are caused by masses
of matter, which by reflection of sunlight offer the aspect of the
zodiacal light. He imagines these masses to have the form of a flat
dise surrounding the sun and extending nearly in the direction of
the orbital planes of the planets and reaching outside the orbit of
the earth; the density of the matter within the dise has its greatest
value in the proximity of the sun, though it is very small even there.
For the calculation of the attraction of the mass of matter special
hypotheses on its constitution are introduced; we imagine a number
of very flattened ellipsoids of revolution with the sun at the centre,
the inclinations of the equatorial planes to the orbital planes of the
planets being small. It is evident that by the superposition of a number
of such ellipsoids we get a flat disc within which the density varies
1) This confirms at the same time the fact, that the introduction of the zero
point energy does not produce a change in the value of the entropy constant.
2) Das Zodiakallicht und die empirischen Glieder in der Bewegung der innern
Planeten. Sitzungsberichte der Bayerischen Akademie, XXXVI 1906.
24
after a certain law from the centre outwards. SERLIGER arrived at
the conclusion that two ellipsoids suffice, one of which is wholly
contained within the orbit of Mercury, the other reaching outside
the orbit of the earth. There appears to exist a certain liberty in
choosing the values of the ellipticities and the quantities determining
the position of the second ellipsoid. As quantities to be determined
so as to account for the differences which are to be explained
SEELIGER introduces the densities of both ellipsoids, the inclination
and the longitude of the ascending node of the equatorial plane of the
first ellipsoid with reference to the ecliptic, and a quantity not con-
nected with the attraction of the masses of matter, but relating to
the deviation of the system of coordinates used in astronomy from
a so called “inertial system”.
Last year Prof. pr SirrerR drew my attention to the necessity
of testing Srviicer’s hypothesis by calculating the influence of the
masses admitted by SEELIGER on the motion of the moon and the
perturbation of the obliquity of the ecliptic, which SeELIGER did not
consider '). I performed the calculations and arrived at the conclusion
that the perturbation of the ecliptic changes the sign of NeEwcoms’s’)
residual and makes its absolute value a little larger; further that
the perturbations of the motion of the moon are insensible. I may
be allowed to thank Prof. pr Sirrer for the introduction into this
subject and the interest shown in its further development. — One could
take the formulae required for the last mentioned purpose from
SEELIGHR’S publication; I did not do so, but developed them anew.
I give them here on account of small differences in derivation. First
I shall give this derivation and the results; after that I shall do
the same for the motion of the moon.
I. Perturbations of the ecliptic.
Let «,y,z be coordinates in a system the origin of which is at
the centre of the ellipsoid, while the axis of rotation is the axis of
z, k° the constant of attraction, g the density of the ellipsoid, a, a
and c its axes, then the potential V at the point z, y,z is given
by the expression :
ai wty?’ z du
Yes k'aga?e 1 — — ed
attu c+u/) (a? du)? tu
4
1) See DE Srrrer, the secular variations of the elements of the four inner
planets, Observatory, July 1913.
2) Astronomical Constants p. 110,
25
for a point outside the ellipsoid À is the positive root of the equation
vty? 2?
i Ed inte = 0; for a point inside A is zero.
Putting V = k*aqa’c2 and 2° + y’ + 2 =? we have
a _ 1 n° 2*(a*—c’) du
ts = pu HOH) HE pu
)
r 2*(a?—c’)
aA (HHA)
Perturbations caused by the first ellipsoid.
I develop in powers of 2? =6&, § being a small quantity ; for that
purpose we need (neglecting terms of the third order):
(=)=- af du
0s (a? 4u) (c* Hu)
& EA (a?—c’*)?
0g? J, rr are’) ,
I put r=a,(1+ 6) and develop the part of £ independent ot
5 besides the coefficients of the different powers of & in powers of §.
Introducing the quantities :
du du
alte Hue +u a= f- (a enn zel + u)? 5 Ju)“:
a,*—a? a,2—a? a,?—a?
a
a’?—a’t+e—p’* pee
we get:
4 We NE Eh
BO, 0, —2070,6+ @—pare)= +(+
7 AB 3 3 p
1 1 5 _v—e ris
Het grea? AAB ne ta 3)8°+
2 Pp ap
27 15 :
+6 + Sy + SE + (—5— G7 ear ls +
1 , (a@—e?)?
oF ee : 4,,5
a,"P
Let v be the true anomaly of the planet, p the angular distance
between the ascending node of the equatorial plane of the ellipsoid
on the orbital plane of the planet and the perihelion of the orbit,
26
J the inclination of the equatorial plane to the orbital plane, then
we have:
. z= — a, (1 + 5) sin (w + Wp) sin J
C= a,7(1 + §)? sin? (wv + Up) sin’ J.
For the calculation of the secular portion of the perturbative
function we thus need the secular portions of 57, 5” sin* (v + w) and
sin wp + wp) for different values of p. I get (denoting the secular
portion by the letter S) :
2
é
hf Ps it le
Sane 5
€ OE
3
8
gene
3 1
a(t BR eT
Se =
1 3
S sin? (v + w) = ae G e oF
=|
SE sin? (v + yee
tt 1
aye RA a rete) geene
3 1
S §* sin? (v + ww) = e* eo a 2 )
3 1
4 sin? zel — — — 2
S8* sin? wv + py) =e (= 3 cos2 yw
on
S sin’ (v + yw) = =
Substituting in the 2 coat for 2 we find:
done Ea ae ep eb) ee ee
Seay — ss a — a = me o/s) 7 E Ps A he
1 a, wee ge af Pp 16 ea
ae... 1 en nde 3 Se
+ a sin® J} —> C,a,*p* + & rea tiie eis
| NTR ORN
+ cos 2 W Te eI mea mi Tal T 30 RE
BOE nn ONE 13E ag
= 5 COSaW v—-—— —¥ ek,
64} TASTEN Te
Let 7, © and 9 be the inclination, the longitude of the perihelion
and the longitude of the ascending node of the orbital plane of the
planet, /, and ® the inclination and the longitude of the node of
the equatorial plane of the ellipsoid all with reference to a tixed
fundamental plane, e.g. the ecliptic of a certain epoch ; then we have:
sin J cos (W -— © + 94) = — cos J, sini -}- sin J, cos i cos (§ — P)
sin J sin (yw — & + QM) = sin (84 — D) sin J.
oe
27
0.7 AJ Oy Oy
09, 01 09," di
quantities required for the computation of the derivatives of 2 with
regard to these elements. In view of the calculation of the perturbation
of the obliquity of the ecliptic I do not use the elements # and ©,
but the elements p and q thus defined :
From these expressions we can determine he
p=tanisin $% gq = tani cos §%
I get:
aS ge 5 TEL 4 }
— = cosi | cos? — sin (Wp — @) + sin? — sin (W — @ + 20) |
p 2 2 (
On 2 ies 2
— = — COSÌ | cos” ~ cos (yw — w) — sin” — cos Www + 29)|
0g 2 2
Ow 2 i eet
sin J Te sin J tan cost cos Meos Jeosi cos”cos(tp-w)tsin” 5 608 (w-0+2Q)) |
Pp 2 2 a
2 ’ .
— cos“ sin(=@)-sin® sint 29%) | :
D ;
MOST sin cosJ cost
5 co eae
ME,
sin J a =sinJ tan
1
‘The differential equations for p and q are’):
dp 1 ov
din, na,2V 1—etcos*i 0g
dq — | OV
dere na,2V 1—e*cos*i Op
To verify these formulae I have used them for the computation
of some of the perturbations of {and Q, which are given by
SEELIGER *).
To compute the perturbation of the obliquity of the ecliptic I take:
Bite Dt ash dean,
(a*—c’) C,a,°.
V = — k’xqa’e
9
a
According to SeRLIGER’S data a= 0.2400, c = 0.0239, J=6°57'.0;
I get C,=0.426; taking as unit of mass the mass of the sun, as
unit of time the mean solar day I get log g = 0.7119 — 5 and I
find :
!) TisseRAND, Traité de’ Mécanique Céleste I p. 171.
di bee vr ze)
A = +4 01.573; SNL a = — 0”.049 ; SEELIGER gives:
2) For Mercury I get:
aen + 0/.091:
dt ed
SEELIGER: + 0.159 and +0”.088; the small difference is owing to the value |
get for Cy = 2.286, while from SEELIGER's data follows C; = 2.217.
di
107.574 and — 0”.049. For Venus I get: = —~ +0163; sini
28
ov Te OJ 0
— = — ktaga?e (a®—c’) C, sin J cos J — = — [0.5986—8] —
0 0 0
where the number within brackets is a logarithm.
Further:
0 OJ
— =—sn®; —=—cosh; D= 40°1'.8:
Op Òg
therefore
OS 0S
oe 108083 EI =O BS
Op 0g
therefore
OR OR
S= + [0.4069-—8]; —— + [0.48278];
Op dg
from which follows, taking as unit of time the century:
dp RS, dq
— = +. 0".065 ; — = — 0.054.
dt dt
Perturbations caused by the second ellipsoid.
Here the caleulation is much simpler. Introducing:
du : a du : = du
a =) =a ne Ey =| EE BL, =| 2 IE 3
(a? Hu) Ve? Hu (au)? Ven (4? + u)? (c? + uj"
0 0 0
we find:
3 ey ze 5
SQ = £,— ak, - oy Be (a?—c?)a,°E, sin? J | a Bee el cos2y
As a verification I have here also computed the perturbations of
the inclination and longitude of the node for some of the other
planets *).
To compute the perturbation of the obliquity of the ecliptic I take:
sin” J
V = — agate (ae) Bat
According to SEELIGER’s data a= 1.2235 and c= 0.2399; I get
di en SN }
‘) For Mercury I find: oA (DEP sam? ae = — 0”.013; SEELIGER gives:
¢
hi d
— 0/.057 and — 0.016. For Venus I find: 7 = -+ 07.007; 810% a = -+ 0.153;
SEELIGER: + 0”.009 and —+0”.144; the results differ somewhat; however, cal-
dS),
culating according to SEELIGER’s formulae, for Venus I find: sind si + 0”.154.
29
B, = 2.445 ; log q = 0.8582—9 ;
OV Ei 0S
p= [034011]; = 74°22! (1900.0), J— 7015;
therefore
OJ 0S
— — —[0.9836—1]; — = — [0.4305—1];
Op dg
therefore
ov OV
— = +. [0.8237—7]; — = + [0.7706—8];
Op og
from which, taking as unit of time the century, I get:
dp dq
== + 0".125 3» = = — 0" 447.
dt a : dt
Therefore the perturbation caused by both ellipsoids together is:
d. d
OO OT
dt dt
Let ¢ be the obliquity of the ecliptic for the time 4, «, the same
for the time ¢,, ¢ and Q, inclination and longitude of the node of
the ecliptic for ¢ with reference to the ecliptic for ¢,, then:
COS E == cos i cos &, — sinisin &, cos Sb,
from which, differentiating, we get:
de . di en:
— SNE — == — sinicos &, — — sin &, — (sin d cos Sb)
dt dt dt
therefore for t= t,:
de dq
ab Weds
_ ds
The perturbation of the obliquity of the ecliptic thus is a eae 0".501.
The difference between observation and theory given by Newcoms
is — 0".22 + 0.18 (probable error); this thus becomes + 0".28. The
addition to the planetary precession a is given by:
| da 1 dp.
o— <=: — == 0".478,
dt sin edt 2
li. Perturbations of the motion of the moon.
We shall now proceed to the formulae for the comprtitation of
the perturbation of the motion of the moon. As the petturbative
force in the motion of the moon we have to take the difference
between the attractions of the ellipsoid on the moon and on the
earth. Suppose a system of coordinates, the sun at the origin, the
axis of z perpendicular to the eliptic; let w, y,2 be the coordinates
30
of the earth in this system, «+ § y+, 248 those of the moon,
then the projections of the perturbative force on the three axes
are given by the expressions:
OV OV OV OV OV OV
raa de enn
The ratio of the distances sun-earth and earth-moon being very
large, I develop in powers of &, 1, 5, neglecting second and higher
powers. Then the expressions for the perturbative forces are:
adr te ave | OV Dr Van 6-0? ie DA : Vor, 0?V
Ox? 5 Dn a jnde” any i Oy” hi Oyo ef dede” + Sat gor Zi
and one can introdace as the perturbative function the function
aie Ee od NU rr NN 0 ONG dV
R=} 5 a? ef dy? +9 De 251 janes $5 = EE ani: | .
a
S
Here for x,y,z are to be substituted their expressions in elliptic
elements and then the secular portion of A is to be taken. Since
the powers and products of 5, 7,6, contain only the elements of the
orbit of the moon, the coefficients on the contrary only the elements
of the orbit of the earth we can take the secular portion of each
separately and multiply these together.
Besides the system just mentioned suppose another system 2’, 7, 2’,
the sun also being at the origin, but the axis of 2’ perpendicular to
the equatorial plane of the ellipsoid. Then we have
. z= «sin DsinJ, — yeos sin J, 4 zco0s J,
therefore
Dz! dz' dz!
gg in D sin Je; en = — cosBsinJ,; ie ee
Perturbations caused by the first ellipsoid
yv
From the expression given for & = a — we deduce, neglect-
p UGA C :
ing the terms having sz? J as a factor:
0 LQ AAN 5 a du Ag
Op: abv 2 | (a°+u)? (c?+u)'l2 Ne (a? 4-4)" (c° B Ayla
0? £2 4 U U
Òzòy (a? H2)(e* +2)"
0722 A wv z' A (at be du
ES C as — s
dede (a? +-a)?(c? + Aye 1 a*—c?) sin D sin Je Tu (Lah
@
v2 4y?
Oy? ve Je En aa Lule gs (a? +2)? (c? +a) 'b
de Aye!
yds (HIJ (+2)
du
pu (tue
=~ (a?— C°) 4-2 (a? —c’) cos mf,
la (a2 -
oe 2 du 8 r du
=-#fe >t ui)? (c? Luy'l2 eines J) (@ tu) (c° uy”
Substituting the elements of the orbit of the earth for.a, y,z and
neglecting the second and higher power of the excentricity I get:
de 2 HE LE
ken 2C, 4 Sesh ==
Ou? a,'p Oy? dady
aa " Oy PEBE Sr A =
= ~ sin Dan, — 2 (a°—c?) C, sin D sin J,
— = — —— ) cos D sin J, + 2(a?—c?) C,cos D sin J,
ET 2C, 2(a? —c*) We:
~-
~
Let © be the radius vector, v the true anomaly, © the longitude
of the perigee, §% the longitude of the node, 7 the inclination of the
orbit of the moon, then we have
§ = 9 [cos (v + W— M) cos 5), — sin (vt O— QQ) sin SV cos i]
rj = [cos (v-+ wW—§) sin SU + ai. SV) cos NM cos 7
S= osin(v+o—y)sini.
I write these expressions thus:
§=0 (Acosv + B sin v)
4 = 0 (Ccosv + D sin v)
S—o(Leosv 4+ Fsinv),
A, B, C, D, LE, F being expressions not containing the true anomaly.
For the formation of the required products we need the secular
portion of g° cos? v and g° sin? v; I get:
Sp cos? v—a',* (4 + 2e°) Sy? sin? v= ta’? (l1—e’?
being the semi-major axis of the lunar orbit.
Thus we get expressions as:
32
Neglecting terms like e? sin’ Dr e° sin — we get:
_
= r
tk L nti Aal ae 55
—— — = sn 1(1—eos2- e?| — + — cos 2
dre & 4 4
RAND nn
== si isin2 §% + — e* sin 20
a 4
+ re €
=e = is sind sin §), + e° sin — 1 = gin (20—§))— id sin \?
a Deen 9 ; a she = 9 9 x Ye 92 : £0
1 -_ ad -_
n l 1 nti(l 400820) Le 3.3) io ae
eN LL = CO8 S Ee | — —— COs 4H
eae ied ic 4 4
15 las : RR 5 8 3
—— —— sind cos {} + e° sin — { — —cos (2H —Q)) + — cos Q
ey: : 2 2 2
1 }
engl eee
ETE == — SUN "ee
a 2
Substituting in #& these expressions we get:
taupe Of bee eyes, Me ae A Bey l
ze —_| -2(,a,’+—+ 3e? | —-C,a,? }+4sin* —| -—=C,(a*-c*)a,?
Jan Aa - p Pp 2 2 Pp
ET dn ete at : Kier 1
+ 2(a?-—c?) sin J sin i cos (Ì,— P) Ga! ;
P
The only perturbations to be considered are those of the longitude
of the perigee and of the node.
The differential equations required are:
IO 10 d yi, 1° OR
e =e S-: sin 1 SSS,
dt na,'* de dt na, * di
One easily perceives that the last term in the expression for R
gives no sensible perturbation on account of the factor a*—c’, the
6 N
value of whieh is about aa and of the fact that §} has a period
of 18'/, years so that the coefficient we get by integration is about
thirty times as small as would have been the case if $} had been
absent. In the same way I omit the terni C,(a°—c’)a’, in the coeffi-
Rust on
cient of sim’ z and thus we have the following expression for P :
B ATEN de
— 8e? | ——C,a,* | — —sin? — |.
kaga?e 2 a," p p 2
l
I get C, = 0.678; — = 1.030 from which follows taking as unit
P
of time the century :
Perturbations caused by the second ellipsoid.
I find:
ee ten
Oi edu “ddy
ol = — 2 (a*—e?) E,sin ® sin J; er == 2(a?—c*) E, cos D sinJ :
Owdz j Aan Odier A uA
072
eaten 2E, —2(a— oc’) U,
from which follows:
R Ne
2 a? |
) Bie. 7 21,9 eb 7 ne 12 2 EA 2
Baa 2Ea, 3h ,a,*e? —E,(a?—c?)a,? sin® 1
Jt
+ 2(a’—c’) a,” E,sin J sin 1 cos (4, — D)|.
Although the term a’—c’ is not small, yet it is allowed to omit
the periodic term.
I get H, = 0.684, MW, = 2.445 from which follows taking as unit
of time the century :
da Rah, di
—== — 0".16 ; — == — 0" 28.
dt
Thus both ellipsoids together give :
both insensible amounts.
Astronomy. — “femarks on Mr. Wo.rtinr’s paper concerning
SEELIGER’s hypothesis.” By Prof. W. pe Srvrmr.
(Communicated in the meeting of April 24, 1914).
SEELIGER’S explanation of Nwwcoms’s anomalies in the secular
motions of the four inner planets consists of three parts, viz :
a. The attraction of an ellipsoid entirely within the orbit of Mercury
The light reflected by this ellipsoid is, on account of the neighbour-
hood of the sun, invisible to us.
b. The attraction of an ellipsoid which ineloses the earth’s orbit.
The light reflected by this ellipsoid appears to us as the zodiacal light.
c. A rotation of the empirical system of co-ordinates with reference
3
Proceedings Royal Acad. Amsterdam. Vol. XVII.
34
to the ‘“Inertialsystem”. This rotation is equivalent with a correction
to the constant of precession. The value of this constant which is
implied in Newcoms’s anomalies is that used in his first fundamental
catalogue (Astr. Papers Vol 1). In “The Observatory” for July 1913
I have shown that this constant requires a correction of — 1.24
(per century). Consequently, of SEELIGER’s rotation r only the part
7, =7r—i1".24 can be considered as a real rotation.
The position of the equatorial plane of the ellipsoid a was deter-
mined by SEELIGER from the equations of condition: he found it not
much different frem the sun’s equator. For the ellipsoid 6 the sun’s
equator was adopted as the equatorial plane.
It is important to consider the part which is contributed by each
of the three hypotheses towards the explanation of the anomalies.
By the way in which Srericer has published his results this is very
easy. It then appears that the ellipsoid a is practically only necessary
for the explanation of the anomaly in the motion of the perihelion
of Mercury, and has very little influence on the other elements.
Similarly the ellipsoid 5 affects almost exclusively the node of Venus.
The rotation # of course has the same effect on all perihelia and
nodes. In the following Table are given Newcoms’s anomalies together
with the residuals which are left unexplained by SREELIGER’s hypothesis.
In addition to SrRLIGER’S residuals [ also give residuals which are
derived: A. by rejecting the rotation 7,'), and C. by omitting the
second ellipsoid. The constants implied in the three sets of residuals
are thus
SEEMIGHR (9, = DAS p, > Pp. > Pe
if PLO + PbS form the stable phase pair, all the reactions
occur in the opposite sense and only the monovariant equilibria (2)
and (3) are stable, whereas then p, > ps > Pi > Pe
5. Starting from the mixture of PbS and PbSO, (for instance a
in fig. 1) we will, on withdrawal of SO, travel either through the
monovariant equilibria (1) and (4) (region PbS, PbSO,, Pb and region
PbSO,, Pb, PbO of Fig. 1) or the equilibria (2) and (3) (region
707
PbS, PbSO,, PbO and region PbS, PbO, Pb) to finally retain the
equilibrium Pb + PbS or Pb + PbO after eliminating the SO, as
much as possible.
Hence, the reactions (1) and (3), which are generally quoted as
taking place in the roasting reaction process cannot possibly indicate
both stable equilibria.
Of the p-T-lines which Scurnck and Rosspacn determined by addi-
tion of PbSO,, PbS and Pb and of PbS, PbO and Pb one at least
must, therefore, indicate an instable equilibrium or an equilibrium
between phases other than those which were brought together in
the reaction tube.
We will see later that both equilibria are metastable and that the
pressure lines recorded by them relate to the equilibrium between
other phases.
6. The supposition made in (2) sub a is not correct. Between
PbSO, and PbO there still arrive three basic salts as intermediate
phases, namely PbO. PbSO,, (PbO), PbSO, and (PbO), PbSO,. The
first of these can be in equilibrium with PbSO,.
The four monovariant equilibria mentioned in (3) now become:
Bushs Qe) Phe ASM le ja nent eas eer as dan
PbS + 7 PbSO,— 4 PbO. PbSO, +480, « .. . ()
2PEO .PbSO,-- BPS == 7Pb +E SOL 2) 4018)
Pira PSO, = 2 PHO: PLO, S052) 40, 8 @
and the alternative found must read:
either Pb + PbSO, stable and then p, >p, >> p, >> p‚ and only (1)
and (2) stable,
or PbS + PbO. PbSO, stable and then p, The tube was, therefore,
evacuated for the third time and then again a few times and each
time the equilibrium pressure was again measured. It now appeared
that the old pressure no longer set in, but that a lower pressure
was attained and the more so when more SO, had been withdrawn.
In succession were found 93, 75, 61, 54, 41, 34, and 28 m.m.
This different behaviour can be explained in two ways.
1. The equilibrium is no longer monovariant, but divariant. Instead
of three solid phases there are only two, one of which possesses a
variable composition. This phase might be a very basic sulphate
with a variable content in PbO. The fusion diagram PbO — PbSO,
of Scuenck and RassBacn gives, however, but little support to this
conception.
2. The pressures measured are not true equilibria pressures, but
indicate a stationary condition.
For if, on evacuating, the pressure falls below the equilibrium
pressure of equibrium (9) the basic sulfate (PbO), PbSO, can decom-
pose still further and give rise to the formation of (PbO), PbSO, .
Pb + PbO. PbSO, then strive, according to reaction (8) towards the
pressure p,, (PbO), PbSO, + SO, according to reaction (9), however,
in the direction — towards p,. And when finally both reactions
take place with equal velocity, we obtain an apparent equilibrium
at a pressure between p, and p, and dependent on the quantities
of the different phases.
It is even possible that PbO is also formed and that reaction (10)
thus takes place simultaneously.
17. The second assumption was the most probable one. In order
to test it more closely a mixture of Pb and (PbO), PbSO, was heated
in the pressure tube. From this mixture PbO only can be formed
as the third phase so that only one reaction, that of the monovariant
equilibrium (10), should be possible.
(PbO), . PbSO, was obtained by fusion of 1 PbSO, with more than
3 PbO. As porcelain is strongly attacked by PbO, the mixture was
heated in a magnesia boat previously heated and saturated with
lead oxide.
The result of the measurement at 780° was p = 23, after evacua-
tion at the same temperature again 22 m.m., then at 800°, 38 and
after evacuation successively 30, 22, 16 m.m. Thus no constant
equilibrium is attained.
On opening the pressure tube nearly all appeared to have been
715
fused and run through the boat although the temperature had not
got above 800°, whereas the eutecticum of PbO and (PbO), . PbSO,,
according to SCHENCK and Rasspacn is at 820°. Probably the MgO is
attacked by the PbSO,. The want of a suitable material which is
attacked neither by PbO nor by PbSO, or Pb at this high tempe-
rature renders a correct determination of dissociation pressures for
reaction (9) and (10) a matter of great difficulty.
If we accept the value of 20 mm. at 780° as the correct one for
the equilibrium (10), the p-7-line for this equilibrium would then
run as indicated by line V in fig. 2.
The p-7-line of equilibrium (9) then lies between III and V and
is indicated schematically in Fig. 2 by line IV.
18. Although from the preceding it is evident that PbS and PbO
cannot be.coexistent, a few experiments were made nevertheless in
order to confirm this opinion.
SCHENCK and RassBacH in all their publications consider the equili-
brium PbS and PbO to be stable although in their dissociation expe-
riments it had already been shown that with such a mixture repro-
duceable pressures were not always obtained, for instance if the
temperature had been raised to above 800°. They also noticed the
formation of sulphate, but assume that this can only be formed at
a high temperature and then remains intact on sudden cooling to
the dissociation temperatures.
We have now heated an intimate mixture of PbO + PbS for
some hours at 600—700° in an evacuated and sealed tube.
The reaction product perceptibly contained sulphate. This was
estimated quantitatively by boiling a weighed quantity of the product
with aqueous sodium hydroxide and then passing CO,. The PbSO,
present is then converted into PbCO,. After filtering and acidifying
the filtrate the sulphate was precipitated as BaSO,. A check analysis
was also made on a portion of the mixture that has not been sub-
jected to heating, under exactly the same conditions of boiling ete.
Mixture of 4 mols. PbO to 1 mols. PbS. Temperature 670—680’
Time of heating 0 ia 3 6 hours
Gram of BaSO, per gram of mixture 0,0498 0.0758 0,1000 0.1121
Additional sulphate formed on
heating at 680°
in gram of BaSO, per gram of mixture — 0.0260 0,0502 0.0623
If the mixture had been converted completely into basic sulphate
716
according to the equation 5 PbO-+ PbS = PbO PbSO,-+ 4 Pb, 1 gram
of the mixture should have yielded 154 mg. of BaSO,. Hence, a
large proportion of the PbO + PbS has been converted. *)
The pressures which Scuenck and RassBacn observed with a mix-
ture of Pb + PbO + PbS do, therefore, probably not relate to an
equilibrium of these three phases with SO,, but to another equilibrium.
By a comparison of their observations with our measurements it
appears that on heating at temperatures below 800° this -is the
equilibrium: Pb + PbO. PbSO, + (PbO), PbSO,, and on heating above
800° and then cooling, the equilibrium: Pb + PbS + PbO. PbSO,.
Also below 800° however, this latter equilibrium sets in, which
with a sufficient excess of PbS is the most stable, as shown from
the following experiment:
A mixture of + mols. of PbO to 1 mol. of PbS was heated in:
a pressure tube. The evolution of gas started at 660°. After evacua-
tion the following change in pressure was observed at 750°.
time in min. pressure
0 5
7 14
20 26
J+ 33
42 30
70 41
100 44
160 53
220 65
280 74
340 81
400 83
460 83
The pressure thus rises rapidly to + 38 m.m. and then increases
gradually to 83.
The first pressure falls on the p-7-line of Pb + PbO.PbSO, +
+ (PbO), PbSO,, the second on that of Pb + PbS + PbO.PbSO,.
Similarly was found with a fresh mixture on heating at 790° a
1) The high result of the sulphate content in the check experiment is very
striking, because both the PbO and the PbS employed were free from sulphate.
Evidently the conversion of PbS + PbO into sulphate takes already place at the
boiling heat in the aqueous solution, from which it follows that also at the
ordinary temperature PbS and PbO are nol stable in each other’s- presénee.
ae
AV
first halt at + 100 m.m. and then a slow rise to 236 m.m. Both
pressures are again situated on the above cited p-7-lines.
CONCLUSION.
19. Summarizing it thus appears that on abstraction of SO, from
a mixture of PbS and PbSO,, the subjoined monovariant equilibria
are successively met with, which are indicated in fig. 3 by the regions
a MEIN and. V.
PbS—PbSO,—PbO.PbSO, . . . I
jig (eee led Cr as ae er IE
Pb,—PbO.PbSO,—(PbO),.PbSO, . HI
Pb.—,PbO),.PbSO,—(PbQ),.PbSO, . IV
Bbg—(PbO), Pbs0.— PbO... 6250.40 V
The lead phase may contain a
little PbS in solution. As the con-
tent thereof varies in the different
equilibria, this difference is repre-
sented by the indices a, b etc.
Probably, however this sulphide
content is very small.
Fig. 2 indicates the pressures
in these monovariant equilibria
and the changes thereof with the
temperature.
Is)
Fig. 3.
Therein ce, A is the existential region of PbS + PbSO,
0) ER i ‘3 » 9 PbSO, + PbO. PbSO,
» EN Cr oe 5 é, oe hee OL Phew Pb
” ete eee a ii vane OO) PbSO, -E Pb
» ee, 5 see ys (E DO}; PbSO, + Pb
ire al Sl SS ¥ ordre DOSE Pb:
Hence, at the temperatures and pressures of region # all the
sulphur will have been expelled from the roasting material.
20. By substituting the values found in table 1 first series in
Q
4,571 T
thus obtained in pairs, Q, was calculated for the reaction:
PbS + 7 PbSO, = 4 PbO. PbSO, + 4 SO, +4 Q,
the equation log p= — + C and combining the equations
718
and as mean value was found — 38390 cals. Applying the samé
principle to the p-Z-values of table 2 we found as the mean value
for Q, in the reaction:
3 PbS + 2 PbO.PbSO, = 7 Pb + 5 SO, + 5 Q, -— 54324 cal.
In order to check these figures we eliminate the unknown heat
of formation of the basis sulphate from these equations:
PbS + 7 PbSO, = 4 PbO. PbSO, + 450, — 4 X 38390 cal.
6 PbS+ 4 PbO. PbSO, = 14 Pb +10 SO, —10 & 54324 cal.
7 PbS + 7 PbSO, = 14 Pb + 14 SO, — 696800 cal.
PbS + PbSO, = 2 Pb + 2 SO, — 99543 cal.
From the molecular heats *)
PbSO, = 216210 cal.
BbS: ==. 18420 5,
S0.==~ 71080 ;,
the calculation for the above reaction at 20° gives — 92470 cal.
The agreement is tolerable.
Delft. Inorg. and phys. chem. Laboratory
Technical University.
1) LANDOLT. BORNSTEIN, Phys. Chem. Tabelle 1912, 870 and 853.
(November 27, 1914).
KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,
PROCEEDINGS OF THE MEETING
of Saturday November 28, 1914.
Mons. AVD
DOG
President: Prof. H. A. Lorentz.
Secretary: Prof. P. Zeman.
(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 28 November 1914, Dl. XXIID.
NE B MNT ES.
G. J. Erras: “On the structure of the absorption lines D, and De”. (Communicated by
Prof. H. Lorentz), p. 720.
G. J. Erras: “On the lowering of the freezing point in consequence of an elastic deformation.”
(Communicated by Prof. H. A. Lorentz), p. 732,
G. J. Erras: “The effect of magnetisation of the electrodes on the electromotive force.”
(Communicated by Prof. H, A. Lorentz), p. 745.
H. KAMERLINGH Onnes and G. Horst: “Further experiments with liquid helium. M. Prelimie
nary determination of the specific heat and of the thermal conductivity of mercury at
temperatures obtainable with liquid helium, besides some measurements of thermoelectric
forces and resistances for the purpose of these investigations”. p. 760.
F. A. H. Scurersemakers: “Equilibria in ternary systems” XVII, p. 767.
F, A. H. SCHREINKMAKERS and Miss W.C. pe Baar: “On the quaternary system: KC1l—
CuCl,—BaCi,—H,0”, p. 781.
L. S. Ornstein: “On the theory of the string galvanometer of EINTHOVEN”. (Communicated
by Prof. H. A. Lorentz), p. 784.
L. S. Ornstein and F. Zernike: “Accidental deviations of density and opalescence at the
critical point of a single substance”. (Communicated by Prof. H. A. Lorentz), p. 793.
A. A. HiJMANs VAN DEN Berau and J. J. pe LA FONTAINE Scuuuirer: “The identitication
of traces of bilirubin in albuminous fluids” (Communicated by Prof. Ti. J. HAMBURGER),
p. 807. (With one plate).
M. W. BrIJRRINCK : “Gummosis in the fruit of the Almond and the Peachalmond as a process
of normal life”, p. 810.
Ernst Conen and W. D. HeLDperMaAN: “The allotropy of Lead” I. p. 822. (With one plate).
J. C. Kiuyver: On an integral formala of SrirLroms”’, p. 829.
F. E. C. Scurrrer: “On unmixing in a binary system for which the three-phase pressure
is greater than the sum of the vapour tension; of the two components’, (Communicated
by Prof, J. D. vaN per WaaLs), p. 834.
Mrs. 'T. EuRENFEST-AFANASSJEWA: “Contribution to the theory of corresponding states”,
(Communicated by Prof. H. A. Lorentz), p. 840.
A. F. Horieman: “The nitration of the mixed dihalogen benzenes”, p. 846.
J. BörsEKEN and W. D. Conen: “The reduction of aromatic ketones. III. Contribution to
the knowledge of the photochemical phenomena” (Communicated by Prof. A. F. Horrr-
MAN), p. 849.
P, Enrenrest and H. KAMERLINGH Onnes : “Simplified deduction of the formula from the theory
of combinations which Pranck uses as the basis of his radiation-theory, p. 870.
48
Proceedings Royal Acad. Amsterdam. Vol. XVIL.
720
Physics. — “On the structure of the absorption lines D, and D,”.
3y Dr. G. J. Erras. (Communicated by Prof. H. A. Lorentz).
(Communicated in the meeting of April 24, 1914).
Some time ago Prof. H. A. Lorentz drew my attention to the
results of an investigation by Miss G. v. Usiscn’). From phenomena
observed by the writer during the investigation of the polarisation
state of light emitted by a sodium flame in a magnetic field after
it had passed through a tube filled with absorbing sodium vapour,
she infers that the wave length for which the absorption of the
vapour is a maximum, depends on its temperature, and in such a
way too that on rise of temperature a displacement towards the
red takes place. The amount of this at 270° would be about 0.17 A.U.
with respect to the emission line. It seemed worth while to ascertain
this result by direct observation. During the summer months of last
year Dr. W. J. pr Haas and myself occupied ourselves with this question.
At first we intended to use an échelon-spectroscope for these researches,
observe by the aid of this the spectrum of a monochromatic source
of light, and then superpose the absorption lines of sodium vapour
on this. This vapour was in an iron tube, closed on both ends by
plates of selected plate glass, and provided with water cooling; in
the middle the tube, whieh contained there a vessel of metallic
sodium, could be heated. Such tubes were used by R. W. Woop
for the investigation of fluorescence of sodium vapour. First a blow-
flame served as monochromatic source of light, which was blown
by means of air in which a very finely divided solution (mist) of
soda was suspended. Afterwards the flame of a Méker-burner was
preferred, burning in an atmosphere in which a mist of soda was
also suspended. This was reached by placing the flame inside an iron
exit tube, at the bottom of which a reservoir was attached, which
was in communication with the air, and into which the soda-mist
was blown by means of an aspirator. This method appeared very
efficient to obtain a sodium flame of constant intensity, which is
moreover easy to regulate.
The lines obtained by means of this source of light, were too
broad for the investigation with the écheion-spectroscope when the
intensity of the light was sufficient for the observation, the self-
reversal moreover being very troublesome then. If on the other
hand, the light intensity was diminished till the lines were narrow
enough, the intensity was again too slight for the observation.
1) G. v. Usiscu. Inaug. Diss. Strassburg. 1911. Ann. d. Phys, 35. p. 790; 1911,
724
We have therefore then tried to see the phenomenon directly by
means of the spectrometer, which served for provisional dispersion
in the experiments with the echelon spectroscope. We were of
opinion that, the dissolving power of the prism system amounting
to 60.000, it must be possible to observe in this way a displacement
of the above given amount. Nothing was to be detected of this,
however. It is true that the at first narrow absorption line (which
has originated by self-reversal in the sodium flame) widened considerably
on the heating of the Woop tube up to about half the distance of
the two D-lines, but a displacement could not be perceived.
On account of Dr. pre Haas’ return to Berlin we had to stop our
joint observations at this point, and I continued the research alone.
First I tried to obtain comparatively narrow emission lines, which
should be intensive enough to superpose the absorption lines on
them in the investigation with the échelon spectroscope by electric
discharges in a heated evacuated Woop tube with sodium. This,
however, appeared impossible, the width remained considerable, and
the self-reversal troublesome. |
Then I took another course, and used an arclamp as source of
light. To obtain monochromatic light I used the above mentioned
spectrometer as monochromator, diminishing the widths of the slits
so that the issuing light comprised a range of only about 0,5 A. U.,
for some observations 0,4 A.U. In this case the adjacent spectra
will overlap only for a small part, as the distance between two
successive orders amounts to 0,39 A.U. The dissolving power of
the échelon spectroscope, which consists of 30 plates of 14,45 m.m.,
amounts to 450.000 *) for D-light.
The absorption lines of the are of light, which are’ caused by
the sodium vapour present in it, appear, observed in this way, to
be double. Both for D, and for D, there are two maxima of absorp-
tion, which are somewhat sharper for D, than for D,, and the
distance of which is smaller for D, than for D, under for the rest
the same circumstances. These distances are, however, variable. The
lamp burning normally, the distance generally amounted to 0.053
A. U. for D,, to 0.078 A.U. for D, for a point lying in the middle
of the are, when Sirmmens’ S A-carbons were used. It sometimes
occurred, however, for a certain pair of carbons that the distance
- was much smaller, down to half the value; sometimes too that it
was much larger, up to almost double the said amount. When new
1) The results of the observations made about the components of the mercury
lines by the aid of this échelon spectroscope, were in agreement with those of
most of the other observers.
48*
722
earbons are set burning, the distance is also much larger than the
normal one, when the arc hisses, the reverse takes place: the lines
grow fainter, and the distance grows smaller, in case of very decided
hissing they can even become entirely invisible. Also in different
places of the are the distance is different, for the negative carbon
the distance is much larger — about twice as large as a rule —
than for the positive carbon.
Between the two absorption maxima lies also a region of absorption,
which on the whole is of only little greater intensity than the
maxima of absorption. Now and then it makes the impression that
there are still more feeble maxima of absorption in this region; I
have, however, not been able to ascertain this with perfect certainty.
I could artificially modify the aspect of the absorption lines very
considerably by blowing a little soda mist into the are by means
of a tube placed parallel to the positive carbon, which lies horizontal ;
in order not to disturb the equilibrium of the are the blown in air
current had to be only very weak, while the quantity of sodium
could be modified by varying the concentration of the sodium
solution. It then appeared that always when soda was blown into
the are the distance of the components of the two D-lines increased,
these becoming vaguer at the same time. The greater the quantity
was of the soda that was blown in, the further the components
were split up, and the less sharp they became. This splitting up
could even reach an amount of about 0,3 A. U., in which ease they
were, however, very vague. The splitting was always perfectly
symmetrical with respect to the original double absorption line. The
maxima of absorption were — for so far as perceptible — of equal
intensity, the sharpness of the two components also seeming pretty
well equal. I have not undertaken further quantitative measurements
about this, since it would have been impossible to determine the
quantity of sodium in the arc, even when the velocity of supply
was known ; at most this quantity could be very roughly estimated;
nor was the phenomenon perfectly constant. Similar phenomena,
still less constant, however, were observed when carbons were used
which were soaked in a diluted solution of soda.
When the Woop tube is placed in the way of the rays of light
so that it follows the spectrometer, which cuts a small portion out
of the spectrum, so e.g. between object glass and eye-glass of the
reading glass, the absorption lines of the sodium vapour appear also
to be double, in which the distance of the components increases as
the temperature rises. At the same time, however, distinct phenomena
of anomalous dispersion are perceptible, as soon as the heating of
723
the tube takes place in a somewhat unsymmetrical way. For this
reason, and at the same time on account of the large differences of
temperature which must necessarily exist in the Woop tube, at
which there can be no question of saturate vapour, I did not under-
take quantitative measurements.
Finally in order to be able to carry out measurements which
should be liable to interpretation, I have generated the sodium vapour
in a vertical glass tube, which was first provided with some pieces
of sodium, then evacuated down to about 0.001 m.m. of mereury,
and sealed to. This tube was uniformly heated all over its length
by an electrical way, so that the temperature may be assumed to
be the same at all places, and accordingly the sodium vapour to be
saturate. In the enclosure there were made two apertures, through
which the light fell in horizontal direction. As the glass gradually
clouded somewhat at these places, | later on applied side tubes which
were also electrically heated, while a wider glass tube was also
used. The bore of the tube, with which I carried out my final
measurements, amounted to 28 mm. For a reason to be stated later,
this tube was placed between object glass and eyeglass of the reading
glass. Of course the image suffered by this, but nevertheless it was
possible to measure the distances of the components.
The pbenomena which I observed in this way were qualitatively
in perfect harmony with what I had seen by means of Woop’s tube,
and during the blowing in of the soda mist into the are. On tise of
temperature the distance of the components increases, while they
become less sharp at the same time. Up to almost 800° the distance
can be very well measured, the results of these measurements have
been represented in the curves D, and P,. At higher temperature
the width is too large to be investigated by means of the échelon
spectroscope, the phenomenon becoming very vague then, so that
the absorption maxima are clearly perceptible on slight magnitication
only, the light intensity is only little greater then between the absorption
maxima than in the maxima themselves. The greatest width measu-
red amounts to about 0,21 A.U., as is visible from the figure.
The resolution is always greater for D, than for D,; the curves
indicating the course in the two cases run perfectly parallel. On
the whole the components of D, are somewhat sharper than those
of D,; of D, the component lying to the side of the red is the
stronger and sharper, of D,, that which lies to the side of the violet.
I think IT have been able to observe with pretty great certainty that
the two components of D, are each double, so that the absorption
region would be bounded here by two absorption maxima on either
0.25
200°
side, which can be distinguished from each other with difficulty. I
have not been able to observe anything similar for D,, there the
absorption maxima seemed single to me. At 274° I found about
0.035 A.U. for the distance of the two absorption maxima, of which
each component of D, consists, at 290° about 0,045 AU. The dis-
tances of the components of D, indicated in the figure refer to the
extreme components. The region lying between the absorption maxima
situated on either side is on the whole of uniform intensity, which,
chiefly at tbe higher temperatures, is but very little greater than
that of the absorption maxima. Whether there are still more faint
maxima in this region, I have not been able to observe with certainty,
though I now and then got the impression that such was, indeed,
the case.
725
In the figure I have also indicated the mean amount of the width
of the region of the arc absorption by means of short dotted lines.
Further the curve indicated by p denotes the vapour tension of the
sodium vapour for the temperature in question. The scale of the
ordinates on the right hand side of the figure refers to this curve.
For the determination of this curve I made use of KRÖNER’s ') research
on the vapour tension of alkalimetals ; for this it was necessary to
extrapolate the values found by Kréner, for which purpose I used
Gritz’s*) formula, which is formed from Dvrré-Hertz’s *) formula
based on that of CLAPEYRON by assuming the validity of VAN DER
Waats’s law for the vapour instead of that of Boyny-Gay Lussac.
Grärz’s formula
containing four constants, | had to assume four points of the vapour
tension curve. I took three points for them, which had been directly
determined for sodium by Kroyer, viz. 7’= 693, p=2.00 ; T= 733,
p = 4.20; T=773, p= 8.64 (temperatures in absolute scale, pres-
sures in m.m. of mercury). I derived the fourth point, availing myself
of Ramsay and Youne’s rule, from Kréner’s determinations for pot-
assium and found for it 7'— 589, p=0.11. I found from this for
the constants using Brige’s logarithms, « = 28.877, log k = 164.88,
m = 48.748, n = 181438. By the aid of these constants I calculated
the values of p given in the figure.
One might be inclined to ascribe the observed phenomena to ano-
malous dispersion ; the observed dark lines would then be no absorp-
tion lines proper, but ‘dispersion lines”. If this were actually the
case, the light of the considered wave length would only have
changed its direction, without having undergone absorption. As to
the absorption lines in the light arc, taking the comparatively small
value of the anomalous dispersion at the densities in question into
_ consideration, the point of intersection of the rays of light coming
from the arc with the plane of the spectrometer slit could never be
far distant from the crater image. When this image is moved over
the plane of the slit we should therefore have to see light lines in
some positions instead of dark ones. As I have never observed
anything like this, not when I placed the are lamp in other positions
1) A. Kroner. Ann. der Phys. 40. p. 438. 1913.
2) Gratz. Zeitschr. f. Math. u. Phys. 49 p. 289. 1903.
8) Hertz Wied. Ann. 17. p. 177. 1882.
Dupré. Théorie mécanique de la chaleur. p. 69. Paris 1869.
726
either, so that the direction of the issuing beam of light with respect
to the light are was a quite different one, I think I may conclude
that the are lines are not to be attributed to anomalous dispersion.
Nor can for analogous reasons this be the case with the lines
which were observed after the light bad passed through sodium
vapour in a uniformly heated glass tube. Here too light lines would
have to be observed at some distance from the dark ones, of which
there was however, no question.
On the other hand — as I already remarked just now — when
the unsymmetrically heated Woop tube was used, I saw a sharp
light line by the side of the dark region, which latter became
blacker then at the same time; in fact besides the absorbed light,
also the anomalously dispersed light has vanished from this region.
Everything considered I am therefore of opinion that anomalous
dispersion has had no influence on my final results.
I will mention here another phenomenon, which at first made its
influence felt in a peculiar way. In my first experiments I had
placed the glass tubes in which the sodium vapour was generated,
before the entrance slit of the spectrometer, so that the whole beam
of white light passed through it. The measurements which I then
made of the distances of the components for different tubes, which
were distinguished by the thickness of the radiated layer of vapour,
were not in harmony; at the same temperature the distance of the
components was found larger as the radiated layer was thicker.
This peculiar phenomenon must undoubtedly be a consequence of
the presence of fluorescence light, which the sodium emits under
the influence of the incident white light. According to Woop’s
researches *) it is just the two D-lines which are very prominent
in the fluorescence light. This light will be the stronger as the
traversed layer is thicker. In this way it is explicable that the
absorption spectrum can be subjected to a modification which will
become greater with increasing thickness of layer.
When, however, the distance of the absorption maxima increases
in consequence of the superposition of the fluorescence light over
the absorption spectrum, which is greatly the case at higher tempe-
ratures (see the curves (D,) and (D,)'), it is easy to see that the
maximum, resp. the maxima, of the fluorescence light must be
situated between the absorption maxima so that the curve representing
the intensity of the fluorescence light, exhibits a rise at the place
of the absorption maxima, when we move to a point lying halfway
1) R. W. Woop. Phys. Opt. p. 444; 1905,
127
between the two absorption maxima. If the absorption maxima
coincided with maxima of fluorescence, the former wouid either not
shift their places, or they would split up. Hence we arrive at the
conclusion that at temperatures above about 260° the maxima of
absorption do not comcide with maxima of fluorescence, but that
the latter maxima, resp. maximum, lie between the maxima of
absorption.
I have indicated two curves in the figure for which the influence
of the fluorescence light is visible, the curves (D,)' and (D,). They
represent the distance of the components of D, and D, fora radiated
thickness of layer of 40 m.m., in which the tube of sodium vapour
was placed before the entrance slit of the spectrometer: the distance
from the tube to the slit was about 10 cm., the opening of the
incident beam being about 1 : 10. Under these circumstances it was
also possible to observe the fluorescence light by means of a spectroscope.
If the tube with sodium vapour was placed between object glass
and eye glass of the reading glass, the fluorescence can only be
brought about by the light that belongs to the narrow spectral range,
issuing from the spectrometer, instead of through the undivided
white light. It is easy to see that the part of the fluorescence
light, which in this case is already very faint, the part that
finally reaches the retina, will have to be exceedingly small
in comparison with the observed light; the influence of the
fluorescence light will, therefore, have to be imperceptible then. I
have actually convinced myself that when the tube with vapour is
placed between object glass and eye glass of the reading glass the
distance of the absorption maxima is independent of the thickness
of layer traversed by the rays. For this purpose I compared a tube
of 50 m.m. bore with the tube of 28 m.m. used for the measurements ;
in these two cases the distance of the components was the same at
the same temperature.
I think myself justified in drawing the conclusion from all that
has been observed that the distance of the absorption maxima of
the sodium lines is closely connected with the density of the vapour,
and that in this way that the splitting up increases with increasing
density. That what has been observed is chiefly an effect of density,
and not in the first place a temperature effect, is supported by the
fact that the influence of the soda mist blown into the are is for
the greater part the same as that of absorbing sodium vapour of
much lower temperature. The temperature at which the absorption
in the are takes place, will namely undoubtedly be much higher
than that of the vapour in the glass tube. On the other hand the
728
phenomenon in the are is dependent in a high degree on the quantity
of soda, the temperature varying very little as long as the equilibrium
in the light are is not disturbed.
In my opinion besides the density also the temperature can have
influence, though the latter will be slight. If the are lines were
exclusively dependent on the density of the vapour in the arc, the
horizontal dotted lines in the figure would have to cut the curves
for D, and D, in points for which the abscissae are equal. As this
seems to be almost the case, the influence of the temperature cannot
be very large.
Also with regard to the degree of the absorption — so the value
of the absorption index —, there can very well be difference between
higher and lower temperatures ; I have, however, not examined this.
And at last, the absorption lines of low temperature are somewhat
sharper than those in the light are.
Returning to what Miss v. Usiscu derived from her experiments,
viz. a displacement of the absorption maxima of sodium vapour on
change of temperature, we may question whether this result can be
brought into connection with the observations described just now.
In the experiments under consideration the main point was the
absorption to which the components of the two magnetically split
sodium lines (briefly called ZerMAN-components) were subjected in
the Woop tube filled with sodium vapour.
The measurements were made of the relative intensities of the
differently polarized beams of light both normal to the lines of force
of the magnetic field (transversal), and in the direction of the lines
of force (longitudinal) (in the transversal observations the beams
polarized parallel to and normal to the lines of force, in the longi-
tudinal observations both the circularly polarized ones.) These beams
of light were emitted by a sodium flame which was placed between
the poles of a magnet; the undispersed sodium light was subjected
to absorption in a Woop tube. This tube was every time heated to
a definite temperature, and the temperature being kept constant,
the magnetic field was varied till the difference of the intensities
of the differently polarized beams of light was a maximum; this
measurement took place by rotation of a glass plate, which served
as a compensator. For every temperature of the Woop tube the field
was determined, in which the difference of the intensities became a
maximum. Miss v. Ugisen makes the supposition that this difference
will be a maximum when one ZeemMaN-component coincides with the
maximum of absorption, and the other is not absorbed at all. By the
aid of this the writer deduces that at 270° the displacement of the
729
fe]
sodium lines would amount to 0.17 A.U.; in this ease the difference
of the intensities was therefore a maximum at a distance of the
ZEEMAN-components or on an average 0.34 AU.
Now on closer consideration it is clear that the correct interpreta-
tion of Miss v. Ubsiscn’s observations would be very intricate,
and many more data would have to be available for it. In the
first place we should have to know the correct distances and
intensities of the ZeEMAN-components, before they undergo absorption
in the Woop tube; further the accurate course of the curve that
denotes the connection between the intensity of the light transmitted
in this tube with the wave length, should be known. The absorption
maxima of sodium vapour not being sharply defined lines, much
will depend on the intensity and sharpness of these maxima; this
is the more obvious when it is borne in mind that the real maxi-
mum difference of intensity observed by Miss v. Ugiscn, constitutes
only a few percentages of the whole amount. With so small a
difference we should be sooner inclined to assume a difference
in absorption to that amount than as the author does, suppose
that one ZeRMAN-component is not weakened at all, the other
only a few percentages in the case of a thickness of layer which
is at any rate pretty considerable. It is easy to imagine cases in
which the absorption maxima are of equal intensity, but difference
of intensity of the ZrrMAN-components is a maximum, when they lie
outside the absorption maxima.
Everything considered the amount given by the author can only
represent the distance of the absorption maxima as far as the order
of magnitude is concerned; accordingly [I do not think that the
result of my observations (distance of the absorption maxima of
about 0.15 A.U. at 270°) is in contradiction with that of Miss v.
UBIScH.
Besides it is possible that the temperature has not been given
perfectly accurately. In this respect a Woop tube presents more
sources of errors than the uniformly heated tube which [ used.
Moreover the difference need not be very great, taking the very
rapid increase of the resolution in the neighbourhood of 300° into
consideration.
When seeking an explanation of the observed phenomena now that
it has appeared that in the first place there is here question of a
density effect, we are naturally led to look for a connection with
the widening of spectrum lines in general, and with the explanation
which Stark has given for it, which comes to this that this widening
730
would be the consequence of electrical resolutions of the spectrum
lines. It is easy to understand that the greater the density of an
(emitting or absorbing) gas, the more frequently it will occur that
the vibrating electron systems are in the neighbourhood of electric-
ally charged systems, and wil] therefore perform their vibrations
under the influence of the electric field of these charged systems ;
hence the spectrum lines brought about by these vibrations will be
the more perceptible by the side of those which arise from systems
which do not vibrate in an electrical field. In this way we shall be
able to obtain a great number of lines, which all being superposed,
can make the impression of a broad band. It is very well
possible that detinite groups of these vibrations can be predominant
which will give rise to the appearance of maxima of absorption
(resp. emission). If we wanted to give further particulars about this,
it would be necessary to enter into an examination of the mechanism
of the phenomena *).
As to the order of magnitude of the above described phenomena,
it is indeed interesting to compare it with the order of magnitude
of the electrical resolutions as STARK has observed them. When we
assume that a vibrating system is placed in an electrical field origi-
nating from an electrical elementary charge at such a distance as
the mean distance of the atoms in saturate sodium vapour of about
280° amounts to, the resolution of the D-lines, when taken as an
electrical resolution, would at this temperature agree — as far as
the order of magnitude is concerned — with the resolutions which
STARK found in this part of the spectrum, always on the supposition
of a linear course of the electrical resolution with the intensity of
the field.
Above 260° the observed resolution increases pretty accurately
with the power */, of the density, the increase being slower at the
lower temperatures; therefore what was observed just now about
the order of magnitude of the resolution at 280° cannot possibly
be of general validity, as this would require an increase not with
the power */,, but with the power */, of the density.
l also observed the D-lines in the solar spectrum, which also
exhibited two components each. On the whole the phenomenon was
in accordance with what is observed for saturate sodium vapour of
about 270°. The distance of the components was about 0.15 ALU. ;
1) Even without thinking of electrical resolutions, | pronounced the opinion
already before that the “own period” of a vibrating system might vary as
it was under the influence of neighbouring systems, and that widenings of spectrum
lines might be the consequence of this. Cf, G. J. Eras, Diss. Utrecht; p. 146 et seq.
731
I did not however carry out accurate measurements about this. They
further seemed to me slightly less sharp than those which were
observed for sodium vapour of low temperature, though this difference
was not very conspicuous. If the resolution were really only little
dependent on the temperature and possible other factors, we might
draw the conclusion from this that the density of the sodium vapour
in the chromosphere of the sun is as great as that of saturate sodium
vapour of about 270°.
I further made some observations on the emission lines of sodium
vapour. For this purpose I used a glass discharge tube which con-
tained some sodium, and which was heated to 200° or 200°.
It then appeared that the emission lines, both D, and D,, were
double, and that the distance of the two light lines increased with
increasing temperature. It is not impossible that self-reversal plays
a part in this; it was, however, peculiar in this that chiefly for D,,
the two light lines on either side of the dark core differed in intensity,
which would not have to be the case for self-reversal.
With regard to the emission lines D, and D, Mrenrrson ') has
pronounced the opinion that they would each consist of four com-
ponents, two intenser ones, and two very faint ones, the distance
of the intenser ones amounting to about 0.15 A.U. Fasry and Prot’)
are, however, of opinion, that reversal phenomena would play the
principal part in this.
In conclusion I will still state that already a long time ago I
observed for the emission lines of copper when this is in the light
are, resolutions of entirely the same order of magnitude as those
which I have now described for the absorption lines D, and D, of
the sodium, also witb the aid of the échelon spectroscope. I then
observed with a preity high degree of certainty that these resolutions
were greater as the density of the copper in the are increased. |
could not carry out measurements about this, however, as the amount
of the resolution was very variable, and besides I had no means {o
determine the density of the copper in the are.
Haarlem, February-April 1914. Physical Laboratory
of “TryreR’s Stichting.”
1) A. A. MicHELSoN and E. W. Morey. Amer. J. (3) 34. p. 427; 1887. Phil,
Mag. (5) 24 p. 463. 1887.
A. A. Micuetson. Rep. Brit, Ass. 1892 p. 170. Phil. Mag. (5) 34 p. 280. 1862.
2) Cu. Fasry and-A. P&érot. GC. R. 130 p. 653. 1900.
739
Physics. -— “On the lowering of the freezing poimt in consequence
of an elastic deformation.” By Dr. G. J. Erras. (Communicated
by Prof. H. A. Lorentz).
(Communicated in the meeting of May 30, 1914).
A number of years ago E. Riscke*) derived from thermodynamic
considerations that a solid body subjected to forces which bring
about an extension or compression, will in general exhibit a lowering
of the freezing point, also at those places of the surface where no
deformative forces are directly active.
This case may be extended to that of an arbitrarily deformed body.
1. Let the free energy per unity of mass be w, and the density g,
then the total free energy of a certain system will amount to
w= [owdr. og errs Se
in which the integration must be extended over all the material
elements @.dt. Further we make no suppositions at all on the state
of the system. ne
Let us suppose the system to undergo an infinitely small deform-
ation at constant temperature. We can always assume this deform-
ation to consist of the infinitely small dilatations 2,, y,, Zz, and the
distortions yz, Zz, vy, for which the well-known relations hold:
ds 07] _ 08
aan 49 Oy ES |
Pe Ase ee aes dE cond can
VTT Oy EER Wet “Seah eae”, a
when & ,$ denote the infinitely small displacements of the points
of the system.
In consequence of this deformation the free energy of the material
element odt will increase by the amount
Ow Ow Ow Ow Ow Ow
d = ag: an fi pe ae wae te ed =
o dr ee eee ay Yy + 5 Za + an Yz + ‘es = a zo) (3)
On the other hand work has been done by the external forces.
When the components of the joint volume forces which act on the
1) E. Rrecke, Wied, Ann. 54 p. 731. 1895. -
733
material element odt are oXdr, 9 Ydr, and oZdrt, and the components
of the joint external tensions which act on the surface elements do
of the surface that bounds the system: p-do, p,de, and p.do, the
total work of the external forces, the displacements being §&, y, ¢, will
amount to
JA = { OKEE Yn t Zo)de+ | (re$ + ry + p28) do
Now when the temperature is constant
OP == JA
holds generally as condition of equilibrium.
Hence we derive from (3), (4), and (5):
Op Ow Ow Ow Oy
felis LET EE (VC
= (e(XE + Fi + 2) dr + ie (paE + pi + PB) do
Ow dh:
ne
(4)
Making use of the relations (2) we get from this after partial
integration :
ow
DE es cos (Nu) + De (Ny) Lo (Nz | +
+ 4 ee cos (Ne) Ee —— cos (Ny) + = cos (Nz) | +
+6 se cos (Ne) te —— cos (Ny) me — cos (Nz)
|E | 0a Oy ; dz | |
d )
el (en | ; Pes) (eo)
T Pay ee a ie Oy aE
Jeu
Ou
eo)
pax Ow Ou Ow > |
(eon) ve me lo
ed Ls — L + 94 |—~
|
|
4-
ze | dr == (XE 4 Yn + Zour f (ned + ry pedo |
The quantities §, 47, and § for the different points of the system
being quite independent of each other, we obtain from (7) the relations:
734
Py 7 0
ow cos (Nw) +5 —— cos (Ny) an —— COS wal = |)
Uy
7 Ow \
MM
Òyz {4
|
0
Pz -- 0 Fea cos (Na) + op cos (Ny) gee en cos aaf =O
Ow Ow dw
em Oia ee ore
Q
ave dz x Oy | z
2 Yr ; \ Oy Yad ox at eae
ae Or 8 Oy Een )e zi
Ow )
(ea) Pens) Pea)
a = aah Me == 0 H
A Ow a dy | dz |
If we now introduce the internal tensions Xz, Y,, 7, Y:, Zo, Xy,
usual in the theory of elasticity, then hold for the components of
these tensions on an element of the surface:
Xv = X, cos(Na) + X, cos (Ny) + Xz cos (Nz) |
Y w = Yreos (Ne) + Fycos(Ny) + Ye eos(Nz) © . . (10)
ZN = Zi cos (Nx) + Zy cos (Ny) + Zz (cos (Nz)
Further in case of equilibrium:
pz + XN=O py + YN=0 Pz JANE 0 Feb
From (8), (10), and (11) follows:
Ow Ow Ow |
Sa SSeS TSS Vi ee
ik © Owe "y “dy, Se. |
0 0 0 he
en Ae ke p Kr en |
Oy: Ozx Oay |
The relations (12) introduced into (9) now yield the equations
OX; OA ROAS
TT
nn ER | or ES ei AEN
the known conditions of equilibrium for a deformed system.
735
2. If we now consider a material element which can be arbi-
trarily deformed, we can subject its state to an infinitely small
variation. With respect to the deformation this variation will be
determined by six mutually independent quantities, three dilatations,
which determine the change of volume, and three, which determine
the change of form. Hence speaking thermodynamically, the variation
of state of this element (which need not necessarily be infinitely
small, provided it is to be considered as homogeneously deformed)
is determined besides by the temperature, by six mutually independ-
ent quantities. It now follows from (3) and (12) that for a virtual
isothermal variation of state the following equation will hold for
the unity of mass
f
dy = — En (Xe + Y, Yy t+ 4e2z+ Yayjzt+ LZreat X pay) ENOR
If we now start from the unity of volume, and call the free
energy of it y’, the following form holds for it
dp = — (Karr + Yy Yy Leze} Pays + Zoer Xyty). « (15)
(In this it should be borne in mind that after the variation the
volume will in general be no longer equal to unity).
Now
en RG
holds generally for the free energy on change of temperature, when
in the expression for the external work with an infinitely small
variation no term with d7’ occurs as factor.
Hence:
]
dp = — PS (Kawa Vy yy + 2222+ Vy2+ Zoey + Aye) —ydT. (17)
holds for virtual variations of state, in which also the temperature
can undergo a change.
When we start from unity of volume, we have
dey! = — (Xyae+ VyYyt 2222 ig Fayed Zoet Xyay) — adt. (18)
where 7 represents the entropy of the unity of volume.
3. Let us now consider a system consisting of two phases, a
liquid and a solid state. We assume the system to be at rest. Let it
further as a whole be subjected to the hydrostatic pressure p, while
arbitrary deformative forces can be active on the surface of the
solid phase, with the exception of that part that is in contact with
the liquid phase; we exclude volume forces. Consequently the same
49
Proceedings Royal Acad. Amsterdam, Vol. XVII.
736
hydrostatie pressure will prevail everywhere in the liquid. We direct
our attention to a part of the system that contains a portion of the
boundary plane between the solid and the liquid phase. We assume
the surface that bounds the considered part of the system, for so
far as it falls inside the solid phase, to be invariable of position,
whereas we can subject it to variations of form for so far as it
falls inside the liquid phase. On this latter part acts then every where
the vertically directed hydrostatic pressure p. We take the part of
the solid phase that falls inside the considered part of the system,
as homogeneously deformed.
Let the considered part of the system contain m, unities of mass
of the solid phase, m, unities of mass of the liquid phase. The
direction of the normal to the boundary plane, which points from
the solid towards the liquid phase, may be called N.
For the part of the system in question are the free energy, the
mass, and the volume resp. :
p= mp tmp, |
M = m, + m, RN Ee Ll
V —= mv, + mv, |
when v, and v, represent resp. the volume of the unity of mass
of the solid and the tiquid phase.
We now subject this part of the system to a virtual change. For
this purpose we make a small quantity of one phase pass into the
other at constant temperature. This will be attended with a change
of the total volume of the considered part of the system. In virtue
of the suppositions made above this change of volume can only take
place through the change of position of that part of the surface
bounding the considered part of the-system, which lies in the liquid
phase. For the rest the state of the liquid phase will not change.
In order to keep also the solid phase in the same state, to leave
the quantities determining the deformation unchanged, it will be
necessary, to make the tensions of the part of the boundary surface
of the considered part of the system lying inside the solid phase
undergo infinitesimal variations. Since this part of the boundary
surface remains unchanged, no work will be required for this. The
only quantity of external work that we have to take into account,
will be that which is attended with the change of the part of the
boundary surface lying inside the liquid phase.
When dm, and dm, represent the changes of the quantities of the
two phases, then on account of (19), we shall have;
737
dp = yp, dm, + W, dm, Nen
0= dm, + dm, A os os GAG)
SV =v,dm, + v,dm,
tn connection with the above considerations the work done by
external forces amounts to:
dA = — pdV = — p(v,dm, + v,dm,) . .. . (21)
If we now apply the condition of equilibrium (5), we obtain,
making use of (20) and (21),
We Perret Pty se: ley ee on he pe)
This equation represents the condition of equilibrium for the two
phases in the case considered here.
4. Let us now imagine that the system consisting of the two
phases undergoes a real, infinitesimal change. The condition of
equilibrium (22) will then retain its validity. It is clear that it will
give us then a connection between the differentials of the variables.
As variables determining the state, we choose for the solid phase the
dilatations and distortions 22, %,, 2, Ye, 2x, Zy, besides the tempera-
ture 7’, for the liquid phase the volume v and the temperature 7. We
ascribe the value zero to the variables rz, yy, Zz, Yz, Zr, and w, in the state
from which we start (which, however, need not be without tension).
In order to be able to distinguish the difference between an eventually
ultimately reached final condition (which need not differ infinitely
little from the initial condition in mathematical sense) and the initial
condition from an actually infinitely small change of condition, we
shall represent the latter by de, dy,, dzz, dyz, de, dx, instead of by
Lx, Yip Zz) Yoo Zz) Ly, Which we shall use for the final condition that is
eventually to be reached. This does not alter the fact that the latter
quantities are always treated as if they were infinitely small.
Proceeding in this way we obtain by differentiation from (22):
oe ar Dae + ied dyy + 5 id ° dert ay: +e Hs dert
- . (23)
+ joi dey + pa, oan atten 28 * dv, + pdv, + v‚dp
In this ze dex denotes the increase of the free energy w‚, when
Ur
the initial state undergoes a dilatation dz: ete, just at this was the
case in (3) and the following formulae.
Now accotding to the theory of elasticity we have:
} 49*
738
1
dn = (da, + dyy + dzz). . Pe (84)
Sl
while further the well known relations:
Ow, ERE
OT = — Fe Ov, = =p . - . F : 8 (25)
hold for the liquid phase.
On introduction of (12), (24), and (25) we get from (23).
1 1 1 ;
(nsi) dT = dp (— oe =| ld Är—p) dez + |
9, 0, 9, ?
(26)
=p (Yy —p) dyy + (Zzp) dz. + Y.dyz + Zaden + X day]:
We can now put:
:
1, ah = (27)
In this we can call the “heat of melting” 7, by which that quantity
of heat is to be understood which must be added to convert the
unity of mass from the solid to the liquid phase, without the con-
dition of the two phases changing for the rest. We then get:
‘be 1 q.
Kl 5 dp + —— |(A,.—p) de Y,—p)d
pr & x POSTE le) te FLO (28)
+ (Z.—p) dz, + Y.dyz + Zeden + X,dzay|
When the only deformative force is the hydrostatic pressure, we
get the known formula of Tuomson and Cravsivus, since then the
following equations generally hold:
X;—p=0 Y,—p=0 Z,—p=—9 | (29) a
Y,=0 ae X=.)
Ff 1 1
am ene wan B (ek arne
NE VY;
If on the other hand dp =O, we get:
Ld Al
=
ar [(X:—p) dar + (Yy—p) dyy + (Zep) der + ey
+ Y.dyz + Zydz, + X,dzy|
Since the form between square brackets, provided with the negative
sign, represents the work performed in the deformative forces, with
the exception of the pressure p, a deformation will bring about a
lowering of the freezing point, when this work is positive.
5. We shall now assume that the initial state (for which we put
Was Yn Zes Ya, Za, ty equal to zero) is to be considered as without
739
tension. In this case (31) will also be applicable; we may then,
however, replace 7 and oe, by 7, and o,,, in which 7, denotes the
heat of melting, and 9,, the density of the solid phase in the tension-
-less state; then we have
TE. :
[(Xx—p) daz + (Yy—p) dyy + (Zep) dee +
Tro . (82)
+ Y.dyz + Zider + Xd]
If we disregard quantities of the second order, which we are
allowed to do when we consider the dilatations and distortions
as infinitely small, we can integrate (32), placing 7, mr. and @,,
outside the integral sign. We then get for the lowering of the freezing
point in the state determined by wz, yy. zz, ys, on, Ly,
be
Wands
f
ftp) dee + Wi) dy + Zp) des +
0
TQ 10
AP =
(33)
Yzdyz + Zaden + X,da,| é
The heat of melting in the state determined by z,, yy, ze, Yz. on, @y
will differ- from that in the tensionless state by an amount that is
of the same order as the dilatations and distortions themselves. For
an infinitely small change follows for the change of the free energy
from (14):
/
1
AES te (Ader + Yydy, + Z.dzz + Yidyz + Zaden + Xd)
Hence the difference in free energy between the deformative and
the tensionless state amounts to:
vy eee Uz «ae
|
Ay = en [Aude + Yydyy + Zeden + Vodye + Zaden + Xda].
0
For the difference in entropy between these states follows then
from (16)
By vee Yx oen
0 1
Ay at ala [Arda Y,dyy a 23022 + Y.dyz + Zrder +. Xda].
0
From (27) follows then for the difference in heat of melting:
Daens Uh, sas
ab
Ae =F, mila [Xadey-+ Y dy, + Zdz:+ V sdyz+ Zaden + X,da,| (34)
0
740
This will also apply to the case that the initial state is not
tensionless; only (34) does not represent then the difference jn
melting heat between the state z,...y-... and the tensionless state,
but the difference between the state a...y;... and the initial state,
which is not tensionless in this case.
6. Let us now suppose that forces act on the surface of the
solid phase which are exerted by solid bodies which rest on the
solid phase, and in consequence of the presence of which the sur-
face of the phase is not in contact with the liquid phase. We can
then imagine that a small part of the solid phase that is in contact
with the solid bodies which exert forces, is converted to the liquid
state, without the solid bodies changing their places. For this case
we can again draw up the condition of equilibrium.
We take the boundary plane of the solid phase as X Y-plane, and
suppose that the Z-axis is the normal to this plane which is directed
from the solid to the liquid phase. For the total free energy, the
mass, and the volume the following relations hold in this case:
¥—=m,y, +m, Y, |
M =m, + m,
V=m,v, + m,v, |
Let now an infinitely small quantity of the solid phase be con-
verted to the liquid pbase, then:
JSF = md, + W‚dm, + m, dp, + W‚dm, . . . (36)
During the conversion the volumes v, and v, will have to change,
as the total volume remains constant, as we supposed. The considered
change being a virtual change, we may assume that as far as the
solid phase is concerned, this change is brought about by variation
of z- alone. Then we get:
(35)
0
IY, = wheres 6 he BPN HS Sik nie Sachin en
Òz,
Further :
0
dw, = fe a ae en Vay ay he hae a
dv,
By introduction of (37) and (38) into (86) we get:
0 0
dF =m, et dein; vs dv, + w,dm, + pdm, . . (39)
ee Ov,
Just as before (see above under 3) the tensions at the surface that
bounds: the considered part of the system will have to vary now
too. We supposed, however, that this surface does not change its
741
position, so that no work will be done by external forces. The
condition of equilibrium is now:
bBo ee ee eee a)
We must use here the double sign seeing that there is only a
conversion possible in one direction. The sign = will hold for the
boundary equilibrium, i.e. the equilibrium at which a transition from
the solid to the liquid phase will just be possible. Now the equations
(35) give further
dm, + dm, = 0
m, dv, + v,dm, + m,dv, + vdm, = 0 |’ (41)
in addition we have
1
dss oe Pare. ua). cere TE
If we limit ourselves to the boundary equilibrium we get from
(12), (35), (39), (40), (41) and (42) making use of the equation
nd ROMO. ttle Moore mee Syst Ee
AN Wet, oe nt kee ee ee
Hence we get the same relation as condition of equilibrium between
solid and liquid phase as we had for the case that the two phases
were in contact. Therefore the conclusions about the lowering of
the freezing point will also be the same. Of course as pressure on
the solid phase must then be taken into account the hydrostatie
pressure, to which must be added that which is exerted by the
solid bodies which are on the solid phase.
7. We shall now consider more closely the amount of the lowering
of the freezing point, in which we shall make use of the expres-
sion (32). To calculate this amount it is necessary to know the
relation between the quantities v,... yz... and the tensions Y,... Y-...
In the most general case, the quantities 7,...y:... being considered
as infinitely small quantities, we shall be allowed to assume a
linear relation of the form:
rr — “ae Ae a iat iL as 35 nee + 522 + Gy gy
’ ; En AREN OEE (45)
Y.= iF patie + one 5 hed c Spas Hs > 46ty
in which
. Uik = Obi id Nee Ae (46)
will generally hold, because the tensions X;... Y,... according to
(8) may be considered as the partial differential quotients of the free
742
energy per unity of volume taken with the negative sign. Further
the coefficients « must be considered as functions of the temperature.
To this most general case, in which the number of coetficients amounts
to 21, answers a crystal of the lowest symmetry. This leads, of course,
to very intricate calculations.
We prefer, therefore, to consider the simplest case, viz. an isotropous
body. For this holds, if we use the prevalent notation °).
Ns 5 ee K [as ty 0 (zr a Yy + 22)| (47)
Y, == - Ky: \
from which equations can be Re
LT
a Ke
e= gel % ma +142)
. ° : ° = . ° e < : j (48)
Y,=—— Ys
K
In this the relation :
1436
ie
i= SERT (49)
exists between the coefficients AK and 4 on one side and the elasticity
coefficient Z on the other side.
Let us now consider a circular cylinder, the axis of which coin-
cides with the Z-axis. Let one end be rigidly fastened, while forces
resp. couples act on the other. Let the length in the direction of
the Zaxis be /, and the radius of the cylinder f. The conditions
of this problem may be satisfied by putting
KYO" Ro Ee eee
If P,, P,, P: are the components of the force, Qr, Q,, Q. the com-
ponents of the couple acting on the end plane, then for the other
tensions hold the expressions :
ei Pis AO vert A0 B ee ah hy
Takt ak. xR! R' ak
2Q.. P, (8486) (R?—2*)—y? Py 1440
x 20-9, Pe GHB + ee
nk' 2akR' 139 ak! 1,
pe Ps 1446 Py (8480) (R?—y?) —2?
ER RT 1436 |
Further :
1) Cf. among others G. Kircurorr, Vorlesungen über Mechanik,
1
Pee ee Se Pe
1426
E1+26
EE le ; . . . . (52)
/ 2 1439
H1+26
ee he. Sch Us
2 14-30 |
We shall now discuss some special cases.
1. Compression resp. extension.
In this only P‚5=0 is put, from which follows :
fps Ee
aR?
(in this the liquid pressure p is neglected).
Then the lowering of the freezing point is
E
VAW Ien fa
TQ
Making use of (52) and carrying out the integration, we get :
= ¥ = 0
AE EEE TE dn
1,0, 2E
which formula is in perfect concordance with RieckKeE's. *)
We apply this to ice, which we shall treat as an isotropous
substance.
RieckE assumes 0,7 kg. for the drawing-solidity of ice per mm?, and
calculates with this 0°.017 for AZT. As for most substances the
pressing-solidity is considerably greater than the drawing-solidity, this
diminution can probably be made larger in the case of compression,
so that it can be measured comparatively easily and with suitable
apparatus even the just mentioned lowering of the freezing point
would also be liable to be measured.
2. Sagging.
In this case we only assume that P,=0. When we consider a
point for which «= Rk, y = 0, then it follows from the formulae that:
5 P, 1420
As == 0 tn = YY, = —_ —_..
rR? 1430
The lowering of the freezing point of the considered point is:
df
[soy hee = fran
F0
1) E. Rrrcxe, loc, cit. p. 736 form. (20).
744
Making use of (52) we get after integration :
AT
"0, 1420 EF
In order to obtain a limiting value for Y., we make use of the
results of an investigation by Hess’) on the sagging of ice
erystals. He charged a crystal 2.9 cm long, 1.0 em broad, and 1,2 em
thick at its end with a weight of 5000 grams, without rupture taking
place. Let us assume by approximation that an ice cylinder of a
diameter of 1 em could bear the same load. We can then derive a
limiting value of Y, from (51).
If we introduce this into (52), we find finally, assuming that 6 = 4,
which is about correct for a great many substances, — 1.19 X 10-4
degree for A7, which quantity is probably not liable to measure-
ment. That this quantity is so small, is the consequence of the small
value of the maximum tangential tensions which ice can bear.
We considered the point on the circumference for which «= R,
y=O0. If on the other hand we take the point for whichw=0,
y= Rk, we get the formulae
Al
Zz = ——.FPy ja
RES
If as before, we again assume that an ice cylinder of a diameter
of 1 cm. can bear a load of 5000 grams at its end, we find for Z,
a value which appears to be greater than the value assumed by
Rircke. If we calculate the lowering of the freezing point by means
of this, we find A7’= — 0°.081, an amount that can be easily measured.
We see at the same time that the lowering of the freezing point
has different values at different points of the surface; a state of
equilibrium is therefore impossible. The rod of ice will diminish on
the upper surface and on the lower surface, and that much more
quickly than on the sides, which will also diminish a little. Further
this diminution will increase towards the end where the rod is loaded.
3. Torsion.
In this case only Q- 4-0. From the formulae (51) follows then
for the point 7=0, u &
ye 2s
mao meas
z ii
rR?
Taking the small amount of the tangential tensions which ice can
‘) H. Hess Ann. d. Phys. 8 p. 405. 1902,
745
bear into consideration, the lowering of the freezing point will again
become very small in this case. Since the tangential tension all along
the cylinder surface has the same value, equilibrium with the sur-
rounding liquid will now be possible.
Haarlem, May 1914 Physical Laboratory of
: “TryLer’s Stichting”.
Physics. — “The effect of magnetisation of the electrodes on the
electromotive force.’ By Dr. G. J. Eras. (Communicated by
Prof. H. A. Lorentz).
(Communicated in the meeting of June 27, 1914).
1. The question in how far magnetisation of the electrodes is of
influence on the electromotive force in a circuit in which there are
electrolytes, has already often been examined, without it being
possible to derive a definite answer to this question from the results
of these researches. Thus Gross’) found no definite direction of the
current in concentrated solutions of ferro salts, while in concentrated
solutions of ferri salts the magnetized electrode (both electrodes
consisted of iron) became the anode. ANnpREWs?) arrived at the same
result working with strong acids as electrolytes. NicHoLs and FRANKLIN *)
obtained results which were in concordance with those of Gross
and ANbREWs, in case a pole of a magnetized iron rod came in
contact with the electrolyte, which consisted of a solution of chromic
acid. In this case the electromotive force greatly increased with the
magnetisation, and reached the value of about 68 millivolts in a
field of 20000 Gauss. If on the other hand the neutral region of
the magnetized rod was in contact with the electrolyte, the sense of
the electromotive force was opposite. RowraNDp and Bei‘) found
that the magnetized electrode became cathode when acids that
attacked the iron, were used as electrolytes. Squirr*), who took
nitric acid as electrolyte, came to the same result. The maximum
electromotive force amounted to 36 millivolts, in a field of 10000
1) Ta. Gross. Sitz. Ber. d. kais. Ak. d. Wiss. 92. Dec. 1885.
2) ANDREWS. Proc. Roy. Soc. 42 p. 459, 1887; 44 p. 152, 1888.
8) E. L. Nrcnous and W. S. FRANKLIN. Am. Journ. of Science 31 p. 272. 1886;
34 p. 419, 1887; 35 p. 290, 1888.
4) H. A. RowLaNp and L. Bett. Am. Journ. of Science. 36 p. 39, 1888.
5) G. O. Squier. Am. Journ. of Science. 45 p. 443, 1893,
746
Gauss; on further strengthening of the field this amount did not
change. Also Hurmveeseu*) found the electromotive force in the same
sense, when diluted acetic acid or oxalic acid was used as electro-
lyte. In a field of 7300 Gauss the electromotive force amounted to
14 millivolts. Finally Bvcnrrer*) has occupied himself with this
question. His result is in so far entirely negative that he finds no
electromotive foree which would reach the value of 10—° Volts for
neutral solutions of ferro salts in the case of magnetisation of the
electrode in a field of 1200 Gauss. He further pronounces the opinion
that the electromotive forces found by Rowrarp are caused by
mechanical disturbances of the equilibrium (‘‘Erschiitterungen”), which
would be the consequence of the origin of the magnetic field. Then
BucHERER compares Hurmucrscu’s results with what has been theoreti-
cally derived by Dunem*), and concludes that no concordance exists
between them. DuHeEm arrives at the formula:
rhe
Sino? «acer og iz ot ae
2d.
i
in which / represents the magnetisation of the electrode, à the electro-
chemical equivalent of the iron, x the susceptibility, and d the
density, the electromotive force / being taken positive, when the
magnetized electrode is cathode. When we eliminate //;, /, and x by
the aid of the relations:
Bihar BSE Fa
we get instead of (1)
BE a 1
i= 1 — —
Sad. gn u
It has appeared to me that in consequence of an inaccurate expres-
sion for the energy of a magnetic field, this value of // is about u
times too small, so that we may write by approximation because u
has a large value:
Bk
Sr .d 6)
which expression, however, only holds when the electrolyte is a
neutral iron solution.
When the experiment is arranged in such a way that B may be
put equal to the external intensity of the field 7, we see from (3)
1) Hurmucescuv. Eclair. Electr. Nr. 6 and 7, 1895.
2) A. H. BucHerer. Wied. Ann. 58 p. 564, 1896 ; 59 p. 735, 1896 ; 61 p. 807, 1897,
5) P. DuHEmM. Ann. de la Fac. des Sciences de Toulouse, 188889. Wied.. Ann,
Beibl. 13 p. 101, 1889.
747
that the electromotive force would have to increase with the square
of the intensity of the field. On introduction of the values for iron
tA See 05, * 6.01.6; a= 29
we get for
H = 10000 Gauss = VAG SC 102 Volt,
In Bvenerer’s experiment the intensity of the field was 1200, if
the induction B had had the same value, the electromotive force would
have been 2.4 x 10— Volts. As this amount is much less than the
smallest value which Boererer could measure (10~° Volts), its negative
result cannot be considered in conflict with the theoretical result.
The results of the other investigators, who worked with acids as
electrolytes, are not at all in agreement with formula (3), in fact
they could hardly be so, as (8) rests on the supposition of a neutral
iron ‘solution.
As the case that the electrolyte is a dilute solution of the metal
of electrodes, which is assumed to be equal for the two electrodes,
is the only one that is liable to exact thermodynamic treatment, |
have calculated the value of the potential difference for this case in
what follows. Further I have communicated the results of experiments
made on this subject.
2. Let us now consider!) an arbitrary system in which also
electric currents and magnetic fields can be present. As variables in
this system we choose the temperature 7, further a number of
geometrical quantities «,,@,..., and finally the magnetic induction 8;
when the last quantity is known everywhere, then, besides the
magnetic field, the electrical current is also determined everywhere.
The external forces exerted by the system, are the components of
force A,, A,... corresponding to the geometrical quantities, besides
the external electromotive forces ©,. In order to be justified in leaving
JouLe’s heat out of consideration we shall assume that the conductors,
for so far as a current passes through them, possess no resistance.
We shall further assume that the system loses no energy by electro-
motive radiation and we exclude currents of displacement.
If the system undergoes an infinitely small virtual variation, we
first inquire into the work performed by the system on its surround-
ings. If. the variations of the geometrical quantities are de, da,...,
the corresponding work can be expressed by
1) The train of reasoning on which the general method. of treatment followed
here is based, was suggested to me by Prof. Dr. H. A. Lorentz, for which I will
express here my heartfelt thanks.
748
OW, = A, da, + Alda, oS el
Further the external electromotive forces will perform work per
unity of time equal to:
ae pee
(E,, €). dr
in which € denotes the electrical current. The work done by the
system amounts, therefore, per unity of time to:
W.
[CO
For this we may write:
dW,
= _fe + €,, €) dr Je €) dr,
in which € denotes the electrical force. Now in the conductors
€ En o (€ + Ser
From the supposition that in the conductors o will be infinitely
large, follows that here € + €, must be = 0, whereas outside the
ave
T
conductors ©—0. Hence the first term in the expression for
disappears. When we make use of the: expression :
€ =cecurl 5 *)
we get after partial integration
dW. 2
= = foo, curl €). ref Ld, Ely. do.
The second term disappears on account of the supposition that
no energy leaves the system through radiation. We finally get then
by the aid of the relation:
a
—cerl € =
Bf)
If the variation of B in the time dt is d%, we get:
ow, CO 0B). de . ERVEN
The total work performed by the system now amounts to the
sum of (4) and (5),
1) Here Lorentz’s system of unities is used.
749
JW — Ada, + A,da,+.. _fe POD dps a Seca)
If we now introduce the free energy of the system, the following
well known relation holds for it
w= E—T.H
when FE represents the internal energy, H the entropy. For au
infinitely small variaton we get from this:
oY — dJE—T . dH—H. OT.
Further
T. dH = dQ— dE + OW,
in which dQ is the quantity of heat added to the system. Making
use of (6) we get from this:
dW — — A,da, — A,da,.... + ,0%).dr—H.dT. . (7)
Let in a certain initial state, in which the variables “a,, a, ..
have the values a,,,a,,..., D being =O, the system have the free
energy W,. In the magnetic state, in which ¥ will have a certain
value everywhere, and the temperature and another quantity, e.g.
the external pressure have remained constant, the geometrical
variables will assume other values, which we shall denote by a, a...
We can now make this transition take place in two steps. We first
give the geometrical variables the values a,,a,, DB remaining — 0;
hence the free energy will increase by an amount A, W.
Further, while «,,@,... remains unchanged, we can bring the ~
magnetic induction B from zero to the final value; then the free
energy will increase by Ay. In this way the final state is reached,
in which the free energy will be:
Wel AP JAE ads ees eee (8)
Then according to (7) the following equation will hold:
Auw= {{(o.aB) ue , a ee var caren |)
3. Let us now return to the above discussed case, in which two
electrodes of the same metal are placed in the dilute solution of a
salt of this metal. The concentration of the solution can be different
at different places. We think the circuit closed by means of a wire
connecting the two electrodes. Let one electrode be in a magnetic
field, in consequence of which it is magnetized. We think the
magnetic field excited by an electromagnet, the leads of which
possess no resistance.
730
In the second circuit in which the electrolyte is found, we think
inserted an electromotive force -— M, which is in equilibrium with
the electromotive force / existing eventually in consequence of the
presence of the magnetic field; we shall assume that sense of circuit
which is directed inside the electrolyte towards the magnetised
electrode, to be positive. We shall assume also the resistance of this
second circuit to be zero.
We shall subject this whole system to an infinitesimal variation.
Let this variation consist in the passage of an infinitely small
quantity of electricity e through the second circuit, and that in that
sense that is directed inside the electrolyte towards the magnetized
electrode. We shall moreover assume that in this variation the
magnetic induction remains unchanged in all the points of the
system. We shall further assume that the surface that bounds the
second circuit, does not change its position.
In the first place we shall consider the work of the external
forces. These forces consist of: 1. the electromotive forces in the
first circuit (that of the electromagnet); 2. the electromotive force
— E in the second circuit; 3. the external pressure p. As we have
supposed DB to remain constant in all the points of the system, the
value of
fs N. do
which represents the flux of & through the first circuit will not
change either. It follows from this that no electromotive force is
active in this circuit, so that the work of this force is zero. The
electromotive force —E in the second circuit will perform work
equal to —/.e, when a quantity of electricity e passes. The whole
volume of the second circuit being supposed constant, the work of
the external pressure will amount to zero.
In all the work performed by the system is therefore
Wis Phe sept eN
Let us then consider the change of the free energy of the system.
For this purpose we shall examine what are the consequences of
the passage of the quantity of electricity e through the electrolyte
in the direction of the non-magnetized electrode towards the mag-
netized one. We shall call ‘the former the anode, the latter the
cathode.
If wv and » represent the absolute values of the velocities of cation
754
is the quantity whieh
and anion in the solution, then n =
u+v
Hirrorr has called “Ueberfiihrungszahl’ of the cation.
Of a current t the part „.t is carried by the cation, the part
(1—n).t by the anion. So the number of gram equivalents of the
cation in the unity of volume will increase per unity of time by:
Lae
aa (oli ll
~ div (ni) = — (1, yn),
as div.i=0O is; e represents the charge of a univalent gram ion.
In the same way the number of gram equivalents of the anion will
increase by the same amount per unity of time, so that the solution
will remain neutral. If £ is the valency of the molecule, and m the
molecular weight, the mass of the salt will increase per unity of
time by an amount:
m
ane Gi Vn).
If the quantity of electricity e, passes through the unity of surface,
and if i, represents the unity vector in the direction t, the increment is:
m.e, .
Bi, A) MEN on Geet)
In the volume elements which le on the surface, the increment
of the mass of salt will be per unity of surface:
Nt. €
dp :
k.g
when WN is the direction of the normal directed inward. The total
quantity of salt inside the solution will now increase by an amount:
fo ++ dmg) fa Coo + fan mi-nde =O
8 aye 5
when we apply Gauss's theorem and make use of the equation
divi=0. The quantity of salt, therefore, does not change.
The only change consists in this that the concentration in the
different volume elements is modified, and that a quantity of elec-
tricity e dissolved at the anode, has deposited at the cathode.
We shall examine what change the total free energy of the
system has undergone in consequence of what has taken place in
the electrolyte. Above we found the expression (8) for the free
energy, (9) holding for the “magnetic” part of it. We further chose
the variation so that the magnetic induction did not undergo any
change. In the first place we must now take into account that at
50
AN, 30 EREN
Proceedings Royal Acad Amsterdam. Vol. XVII.
752
the anode a certain volume of iron has been replaced by the solution,
whereas at the cathode the solution has been replaced by iron. The
volume of iron ee will correspond to the quantity of elec-
eo ee ‘
tricity e, when a denotes the atomic weight, £, the valency of the
atom, d the density of the iron. If we assume that at tbe anode
no magnetic field is present, the substitution of iron solution for
iron will not bring about any modification in the magnetic part of
the free energy. At the cathode, however, this substitution will give
rise to a change in the expression (9), which, when wu, represents
the permeability of the iron ssp u that a the iron, amounts to:
GMD — En fs ue Meee
when B means the absolute value of ® a B this value at the
cathode. When we speak of “at the cathode” or “at the anode”,
we mean by this that we must take the value of the considered
quantity at a plane that is at a very small distance from the
cathode resp. anode, this to evade the difficulties which the phe-
nomena taking place in the boundary layer might cause; we
shall examine this question more closely further on. On account of
the smallness of the considered volume we may assume that the
value of B is the same everywhere inside the volume. If we put
u, = 1+ 47x,, in which x,, the susceptibility per unity of volume,
is to be considered as a small quantity, we get about:
be DE - B
J (Ay W)= — —— | — (1 --4ax, — J—adB],. . (14)
: Gn Wam EA oe u
0
when xz, represents the susceptibility at the cathode. If we assume
l
u to be very great for iron, so that — is to be neglected with respect
u
to unity, and if we replace B by H, the absolute value of the inten-
sity of the field in air at the ee we get:
a.e
J (Au Wz TIE = “a —4xn %,) 2 ° e . (1 5)
Instead of (13), using the relation :
i
u
in which / is the absolute value of the magnetisation we may
further write:
753
B
4m .a.e {° ; N
d, (Au ¥)= DET AES EBi os ree GEN
ie
e
0
We must further take the changes of concentration in the different
elements of the electrolyte into account. If we introduce the concen-
tration c as the number of grammolecules of the dissolved substance
per unity of mass of the solution, this is modified by the passage of
the quantity of electricity e. If the density of the electrolyte is g,
then the variation of density, when the volume remains unchanged,
will amount to:
Ua UP ee en Se 9, a
in which dv is given by (11) and (412). By means of this we find
easily for the change of concentration :
1—me
de = dp ER ae ns kg has Pe ee
m. Q
Now for dilute solutions very nearly :
ep ee Ore en he Fa (GES
holds, in which y is the so-called absolute specific susceptibility per
unity of volume, which is considered as independent of 9 *). We
get from this by the aid of (17) and (18):
OR == ES
This then gives, as x, must be considered as small :
B
| §).d§} = — An B’.y . dv.
0
When we multiply this expression by the volume element dr, theti
introduce the value of dp from (11) and (12), and integrate with
respect to the whole volume of the solution, we get:
2ay.m Qary.m
é, (AuW) = — = | Bea agi.ar— vem | Be iy, edo.
V S
If we assume n to be constant, which is permissible, on account
of the relative smallness of this term, if we suppose further that
at the anode B= 0, at the cathode in air B— A, we get:
2nmyen —
FOE tales zer eld
d, (Àu WP) = Le
1) Relation (19) holds of course only as long as the specific magnetic properties
are independent of the concentration.
50*
754
as for the cathode holds:
= fatty, doe.
4. We must further consider the change of the “non-magnetic”
part of the free energy; as we saw above the values must be
assigned to the variables which they will have in the magnetic field.
The only change which is involved in this, is the change of con-
centration of the solution. If the free energy of the unity of mass
of this is w, then that of the volume of solution w of density @ is
t= pew,
when W, is the free energy of the solution. When we make use of
(17) and (18), and further of the relation :
vp, aes ae
which follows from (7), the variation of this will amount to:
; ow l—n
dd, — dp | + 2 + 4 | AL
o de m
For the free energy of the unity of mass of the solution we shall
use the well-known equation :
ww, tact Peo+....+KTeloge,. . . . (29)
in which y, means the free energy for c= 0.
In this we must give to the variables ce and wv the values which
they possess in the magnetic field in the state of equilibrium.
On the other hand we can, however, also imagine that different
concentrations can permanently exist in the different volume elements
of the solution, in such a way that no change can be brought about
in this by the magnetic field. Thus we shall obtain the potential
difference between the electrodes on arbitrary distribution of the
concentration. The supposition made is a fiction; the more so as
we have assumed also the resistance equal to zero; in general the
velocity of diffusion will namely increase as the resistance decreases,
with which permanent concentration differences are in contradiction.
This supposition, however, must always be made for such problems,
in order to be able to apply the laws of the reversible processes ;
hence we also make them here.
With the aid of (22) we get from (21)
NS
RI
dd WV, = dv E ee + — (1 + loge) — RTe + “| Wn
0) m m
When we use here the expressions (11) and (12), and integrate
155
over the whole volume of the solution, we get:
m / RE
ess En fo ayy A) |. +. = + — (1+ loge) — RTe + <| ‚dr
€ o m m
V
m dads c
+ he iN, oak nee + — (1 + loge) — RTe + “| ao.
ye 0? m
En dl
When we take n again as constant, we get:
m.n e 1 1 RE 6
ov, = er Wo. — Wor +P 0 = in OD as of
[ Cy =
k 0, m
when we denote the quantities referring to the anode by the index
1, those referring to the cathode by the index 2.
We may further write:
or, because
1 1
Wor = Wo, =F JZ ie aa =| :
ues es
There remains finally
mn e Rd C,
dw, SS re A — log — — (c,—¢,) (24)
€ m c
2
From the well known theorem for the free energy:
JWA dW =0
we get with the aid of (10), (16), (20), and (24):
_B \
An. (7 DREADS
K=—— d-I)dB ————_... H® —
d k.e
0 | aah Tks or)
EE
m.n 1 C,
oe, E log — — ee)
HAPE mi” ey
in which À is the electrochemical equivalent of iron.
Hence the potential difference consists of two parts, viz. one part
(the first two terms), which depends on the magnetic field, and a
second part, which depends on the concentrations at the electrodes.
The second term of the first part will increase proportional to the
second power of the externai intensity of the field, the first term,
too, in case of small intensities of the field, where we may replace
(16) by (15); at great intensities of the field, however, / will reach
756
the value of saturation, so that this term — which far exceeds the
other “magnetic” term — will increase only about linearly with
the external intensity of the field. The sense of the electromotive
force determined by the first term is directed inside the solution
from the non-magnetized towards the magnetized electrode. The
second part of the expression becomes equal to zero for c,=c,;
with neglect of the contraction which the solution undergoes on
concentration, this expression agrees with the potential difference
calculated by Hetmnontz'), between two electrodes which are in
solutions of different concentrations.
If we assume c,—c,, and neglect the terms which depend on
the susceptibility of the solution, the following form holds for not
too great intensities of the field (in which u is still to be considered
as very great)
a de
OT) eS el a ee (26)
If we use electromagnetic unities, this becomes:
oi ae
Bred:
which agrees with (3).
In order to simplify our considerations we have disregarded the
transition layers between iron and electrolyte; in them phenomena
will namely take place which cannot be examined in detail. It now
remains to prove that in the calculation of the free energy the in-
fluence of these transition layers may be neglected. For this purpose
it is necessary to assume that the thickness of the transition layers
is of the order of magnitude 7, when / represents so small a quantity
that we may assume that inside the thickness / the liquid is in
equilibrium with the electrode. We shall further assume the limits
of these transition layers on one side inside the iron, on the other
side inside the solution. Let the quantity of electricity e, which we
have passed through the solution, be of the order of magnitude §; the
same thing will be the case with the thickness of the iron layer,
which has dissolved at the anode, deposited at the cathode. This
iron layer may be infinitely thin with respect to the thickness of the
transition layer, and entirely fall within it.
We have already taken into account the change of the “magnetic
part” of the free energy, which is the consequence of the displacement
of the iron and the dissolved substance. Now we have still to take
into account the change of the state of the transition layers, which
1) H. Hermgorrz. Wied. Ann. 3 p. 201, 1878,
757
is the consequence of the conveyance of iron and dissolved substance.
The quantity of dissolved substance supplied resp. extracted in the
transition layers is of the order &, just as the quantity of electricity
e. As the volume of the transition layers is of the order of magnitude
/, the change of state inside these layer will be of the order A
Now there was equilibrium in the transition layers before the
variation; hence a variation of the free energy per unity of mass
2
of the order of magnitude (=) will correspond to a change of state
of the order 5 (the external work is zero). The variation of the
total free energy of the transition layers will therefore be of the
2
order = Thence we see that this variation may be neglected with
respect to the other variations of the free energy, which are of the order §.
5. We shall now still examine what will be the equilibrium
concentration in the magnetic field, i. e. that concentration which
will finally exist after the diffusion has been active between the
different volume elements. For this purpose we consider an infini-
tesimal variation of the total free energy W of the system. We
choose this so that all the parts of the system, with the exception
of the solution, remain unvaried ; moreover we leave the magnetic
induction B unvaried. We can, therefore, restrict ourselves to the
variation of W,, the free energy of the solution in the magnetic
state. For this free energy holds the expression according to (8)
and (9):
Piaf let [9.0 lar,
when we use the expression (23) for w.
As the susceptibility may be considered as small, we may put
for il:
B?
- = E W+ Di (1 —47 a Cac eee: Me (27)
We shall now let the variation consist in a change of the con-
centration, accompanied with a change of the specifie volume; in
this we leave the volume of every volume element unvaried, so
that the external work is equal to zero. We get the relation between
concentration variation and volume variation by eliminating do from
(17) and (18), by which we get:
758
1 — me
de =d ee er eD
mg
Now we get from (27), keeping in view that B remains unchanged:
JF, ale opty .do—20 B dx, | dt.
Making use of (19), (21), and (28) we get from this, when we
apply the thesis of the free energy :
1— me
fle + ap + i: een JR ‚|-do-dr=0
o m dc
Further exists the relation :
fear,
the total mass, from which follows:
foo.ar=o,
gee oe aaBtyondt. nt Bee
c
Mm
The formula
follows from the two relations as condition of equilibrium for the
solution in the magnetic field. For this we may put, just as above
in the expressions (22)
punt
tE (l + loge) — Red — —
from which el because at the anode | = 0, at the cathode B=H,
just as above for (24) :
Ted ee =
log? RT (eh aag BE) ee
m C, :
When we introduce this into (25),.we get for the potential differ-
ence in the state of equilibrium, at which also the solution is in
equilibrium
B
Am .a
E= (L=T JAB? rn ee ee
0
6. In order to test the obtained result by observation, I made a
number of experiments, in the first place with iron. The iron
used for this was electrolytic iron, which Prof. Franz FISCHER at
1) With neglect of the contraction which the solution undergoes, this result is
in accordance with the result derived by Vorer (Gétt. Nachr. Math. phys. Kl,
1910 p. 545),
759
Charlottenburg kindly procured me. The magnetized electrode con-
sisted of a circular plate, which was of the same size as the pole
plane of the electromagnet, and was rigidly fastened to it, a glass
plate serving as isolation. The other electrode was outside the field.
The concentration of the used solution of ferrosulphate was generally
5°/,. The results obtained with this may be summarized as follows.
On excitation of the magnetic field I always obtained a current in
the sense as the theory requires. The extent of the obtained effect
differed, however, greatly from the theoretical value; the measured
potential differences were, namely, between the strengths of the
field O and about 20.000 Gauss 10 or 20 times as great as the for-
mula would require. At first the course was about proportional to
the second power of the strength of the field, the effect reaching a
value of 6.3 Xx 10-4 Volts at about 16000 Gauss, which did not
change on further strengthening of the field. If the used solutions
were neutral, the effect remained pretty well constant after excita-
tion of the field. On the other hand for acid solutions (which con-
tained only very little free acid) a diminution and a reversal of the
effect soon took place, till a value was reached, about ten times as
great as the first effect after the excitation of the field. It is remark-
able that Rowranp and Beit also always found sueh a reversal,
whereas SQUumr found that above a certain strength of field the effect
no longer increased, which is in agreement with what I observed.
Another phenomenon that I regularly observed was the increase of
the resistance of the solution as it was longer in the tube. At last this
resistance can reach a value of some hundreds of thousands of ohms.
Besides I made experiments with nickel. The electrodes were of
so-called “Rein nickel” from the firm Kanipaum; as electrolyte
generally a 5°/, solution of nickel sulphate was used. No effect,
however, was observed with certainty, so long as the solution was
neutral. Probably there was an effect in the sense required by the
theory, but about five times smaller than for iron, which would
therefore harmonize better with the theory. It was, however, impos-
sible to obtain certainty in this respect, as the resistance of the
solution soon became exceedingly great, even up to more than
10° 2; moreover the zero position was very variable, much more
so than was the case for iron. It is peculiar that the large resistance
only consisted for very small electromotive forces; if on the other
hand the latter was a few volts, the resistance became only a few
thousands of ohms. For solutions of nickel sulphate greatly acidified
with sulphuric acid no other effect was found than in neutral
solutions ; there was no question of a reversal here,
760
Physics. — “Further experiments with liquid helium M. Prelimi-
nary determination of the specific heat and of the thermal
conductivity of mercury at temperatures obtainable with liquid
helium, besides some measurements of thermoelectric forces
and resistances for the purpose of these investigations”, By
H. KAMERLINGH Onnes and G. Hoist. Communication N°. 142¢
from the Physical Laboratory at Leiden.
(Communicated in the meeting of June 27, 1914).
§ 1. Zntroduction. Measurements of the specific heat and of the
thermal conductivity of mercury were considered to be of special
importance with a view to the discontinuity, found at 4°.19 K. in
the galvanic resistance of this metal. The preliminary results have been
already mentioned in Comm. N°. 133, for the measurements were
carried out as early as June 1912. We wished to repeat the experi-
ments, which we considered only as a first reconnoitring in this
region, because our opinion was, that, by some improvements in the
experiments, the accuracy of the results could be considerably
increased. Special circumstances frustrated this, and now, as there
seems to be no prospect of a repetition for the present, we com-
municate the details of our investigation.
§ 2. Thermoelectric forces. The tirst difficulty in these deter-
minations was the choice of a suitable thermometer. The measure-
ments already performed about the resistance of platinum, gold and
mercury did not give much hope, that there would be among the
metals a suitable material for resistance thermometers. We have
therefore investigated a series of thermoelements. The gold-silver
couple, a suitable thermometer at hydrogen temperatures *), showed
down to the higher helium temperatures a fairly large thermoelectric
power, at the lower helium temperatures, however, the thermoelectric
power diminishes rapidly, so that this couple did not satisfy the
requirements. Moreover, this couple was not at all free from disturbing
electromotive forces, which appeared at places of great fall of tem-
perature in the cryostat. Nearly all other elements were subject to the
same fault. But apart from this, none of the combinations was suitable.
Notwithstanding, we communicate the results of our determinations,
because they show clearly that according to the theoretical investi-
gations of Nernst and Kersom ®), the thermoelectric power of all
1) Compare H. KaAmerLiNGH Onnes and J. Cray, Comm, NO. 1075,
2) W. Nernst, Theor. Chem. 7e Aufl. 1913 p. 753. Berl. Sitz. Ber. 11 Dez.
1913p." 972.
W. H. Keesom, Leiden. Comm. Suppl. N°. 305 (Proceedings May 1913).
761
couples investigated approaches to zero at heliumtemperatures. The
different wires were measured against copper. After a preliminary
research, which included also the determination of the thermo-
10°v
+1300
+1200
400
Fig. 1.
electric forces of nickel and of six gold-silver alloys, the following
combinations were selected as most promising for investigations in
liquid helium.
Thermoelectric forces against copper.
if Ag Aw Aus Pt Pb | He Const. | VIA!)
er Sl Ke _ 28 | —129 | —257 | —298 | —457 | +1293 5 — 432
20° —28 | —282 | —326 | —68 | —553 | +1319 | —6280 | —819
4°26 —21 —375 | —328 | —58 | —559 | +1309 | —6630 | —990
3°20 | — | —383 — —59 — +1309 | —6630 | — 1002
zede. Cvet [07 Se ze = as, | ay | —1004
1) Au with 0,476 weight °/) Ag. EN
762
Figure 1 shows their thermoelectrie forces against copper, at the
absolute temperature 7’, given in the first column of the table. The
temperature of the second juncture was 16° C.
§ 3. Change of the resistance of alloys with temperature. As it
appeared impossible to find a suitable thermo-couple, our attention
was drawn to the change of resistance of constantin, which had
already been measured at hydrogen temperatures by KAMERLINGH
Onnes and Cray. This alloy shows bere a considerable decrease
of resistance at decreasing temperature; it was, therefore, probable
that constantin could be suitably used as a resistance thermometer at
helium temperatures. Experiments have shown that this was in fact
the case. Later measurements (see comm. N°. 142a § 4) proved that
also. manganin, whose resistance begins to diminish at decreasing
temperature and which has at oxygen temperatures a considerable
smaller resistance than at ordinary temperatures, is fit for tempera-
ture measurements in liquid helium.
§ 4. Specific heat of mercury. a. Experimental arrangements.
The method, used in the determination of the specific heat,
agrees most with the one used by Nernst in his investigations
about the specific heat. A little block of solid mercury hung freely in
a high vacuum and was heated electrically. The increase of temperature
was determined by means of a constantin resistance thermometer.
To obtain the little block of mercury the liquid metal (comp. fig. 2
with magnified fig. of details) was poured into the vessel C through
a capillary, provided with a funnel, which could be introduced
through m. First C was in the same way supplied by means of a
funnel with a small quantity of pure pentane, which, at the intro-
duction of the mercury, remains as a thin layer between the glass
and the mercury. A little hollow steel cylinder (thickness of the
wall ‘/,, m.m.), which contained the heating wire — a constantin
wire, insulated with silk and covered with a thin layer of celluloid
to avoid all electrical contact with the mercury — was immersed
in the mereury. Round this cylinder a second constantin wire was
wrapped, which was to be used as a thermometer. The little cy-
linder was, by means of silk wires (stiffened by celluloid) fixed to
a little glass rod, which could be moved up and down through
the tube B, and which was centred by constrictions in this tube.
This glass rod was connected to a silk wire, which could be
screwed up and down by turning the handle A. Now the mercury
was frozen by cooling down to the temperature of liquid air, The
763
thin layer of pentane, which is
spread over the glass, acts as a
viscous lubricant at this temperature
and prevents the sticking of the
mercury to the wall in freezing.
After having been frozen, the little
block of solid mercury was screwed
up by means of the handle K
and the temperature was increased
to about — 50° C. Thereupon a
high vacuum was established by
means of a GAEDE mercury pump
and the pentane was distilled off
into a tube, immersed in liquid air.
During the experiments the heat
insulation of the block appeared
to be so good, that the tem-
perature remained many degrees
above that of liquid helium, al-
though it was let down against
the glass wall. Therefore a little
gas had to be admitted in order
to ecol the block. This manipula-
tion succeeded perfectly, but the
gas could not be removed quickly
enough in the short time available
for the experiments. The loss of
heat of the mercury was thus
very considerable (decrease of the
temperature difference to half of
the original value in about 100
seconds) and therefore the cor-
rection, to be applied to the in-
crease of temperature while heat-
ing, remains the greatest source of
Fig. 2. Fig. 3. uncertainty. Nevertheless it seems
possible, that the results are accurate to about 10°/,.
The thermal capacity of the hollow steel cylinder with the thermo-
meter and the heating wires was determined afterwards by a separate
experiment, Fig. 8 shows the apparatus used for this purpose.
b. Results. Measurement at the boiling point of helium. The quantity
of heat supplied to the mercury amounted to 1,10 cal., the increase of
SAQA,
NEP
64
temperature, corrected for the loss of heat during the period of heating,
was 2,22 degrees K. whilst the quantity of heat, necessary to heat the
little steel cylinder with the thermometers 2,22 degrees, amounted to
0.11 cal. (result of a separate experiment). Control experiments showed
that the heating wire (used as a thermometer) and the thermometer
wire outside the steel cylinder had the same temperature. The mass
of the mercury was 314 grammes, so that 0,00142 cal./degree K. was
obtained for the mean specific heat between 4°,26 K. and 6°,48 K.
w fj
The relation of GRÜNEISEN ') ronte would *) have given c,‚=0.0037,
for 4°.27 K.
Measurement at 3°.5 K. Afterwards the experiment was repeated
at the temperature of liquid helium, boiling under a pressure of 6 em.
of mercury ; 0.000534 cal./degree K. was found for the mean specific
heat between 2°93 K. and 3°97 K.
Assuming this mean value of the specific heat, we shall calculate
now the value of this quantity for a definite temperature according
to Drsve’s formula, which holds for our very low temperatures
bie Tee
so that the mean specifie heat between two temperatures 7, and
Teas
CET)
A(T 7)
We obtain from the two experiments C= 0.0000088 and 0.0000127
respectively.
The agreement is not satisfactory; although, taking into account
that the absolute temperature occurs in the formula in the 4 power,
and that therefore small deviations in 7’ involve very large ones
in C, we may safely conclude from our experiments, that, weth
respect to the specific heat, nothing peculiar happens at the point of
discontinuity, and that we may content ourselves with a preliminary
mean value C= 0.0000110, when we assume for the moment that
the specific heat does not show any discontinuity at all.
We have then
C=
6501 ==. 0000TLO Te
or for a gramme-molecule
¢ = 0.00220 7™.
For the characteristic constant 6, introduced by Desir we find
1) E. Grinetsen, Verh. d. Deutschen Phys. Ges. 1913 p. 186.
*) Compare KamertincgH Onnes and Horst, Leiden Comm. NO. 142a Proceed.
June 1914.
765
with c, — 5.96 and
3 13
c= 0.00220 7 * — 77,938 — e — 464 —
O's 0e
| 060,
As a matter of fact, the specific heat has been determined here
at constant pressure and not at constant volume. In the foregoing
calculation, the difference of the specific heats ¢, and c,, given by
Gi Op Ae, I,
has been neglected. Indeed, A is about 7,2 .10—> and c, and T’are
both small.
Using Desir’s formula, we can compare our results with
those of PoLuirzeR') at somewhat higher temperatures. For this
purpose we calculate a value of 6 agreeing as well as possible
C
with Porrirzer’s figures of —, we find then 110. In fig. 4 c, is
On
plotted as a function of 7’ according to DrBrisr ; the values, deter-
mined experimentally, are indicated by circles.
The accordance at helium temperatures is bad, as could be
expected in consequence of the difference between the value of @
used in the calculation, and the one deduced from our experiments.
Ge
ce, |
go 09
oO Ww=s
Ie
3,75 | ae
425 | Ri
fe) kev he 0v So JO) _JID
Fig. 4.
1) See F. PoLLitzeR. Zeitschr. f. Elektrochem. 1911. p. 5.
766
Meanwhile we remark, tbat in Porrrrzer’s experiments too a distinct
deviation from DeBiE’s curve is to be noticed, in the sense of a
decrease of 6 (about 115—162) with decreasing temperature, which
would be, according to our experiments, very considerable down to
helium temperatures ; further that, according to LINDEMANN’s formula
and by comparison with lead (88), 6=61 is to be expected for
mercury.
§ 5. The thermal conductivity.
The thermal conductivity was determined by means of the appa-
ratus, represented in figure 5. A U-shaped tube, with double
walls, and closed at ene end, was provided
with mereury. The closed branch contained
a constantin wire S, insulated by means
of celluloid, which made contact with the
mercury at the free end. This wire’ was
used as a heating wire. The current return-
ed through the mercury itself by means
of a wire, in contact with the mercury at
the open end of the tube. The fall of tem-
perature was measured with 8 constantin thermometers 7, 7,
and 7, consisting of wire of '/,, mm. thickness, wrapped around
a small glass tube. The experimental arrangement is further ex-
plained by the diagrammatic figure. All wires were connected to
each other by two wires, insulated by thin layers of celluloid and
further running free through the liquid helium.
In consequence of a wrong manipulation during the preparation,
the tube had lost a little mercury, so that only the two lower
thermometer wires could be used. The heat developed in the heating
wire and the difference in temperature thus produced were measured
at two different temperatures, the one above and the other one
below the point of discontinuity in the electrical resistance. The
section of the cylinder of solid mercury amounted to 0.47 em’,
the distance of the thermometers to 5,0 em.
At the boiling point of helium the supplied energy was 0,633
watt/sec., the difference in temperature produced 0.58; at 3°7 K 0,0865
watt sec. and 0,23.
From this we find for the mean thermal conductivity between
4° 5 K and 5°,1 K. £=0.27 cal/em. sec. and between 3°.7 K and
9.9 K: £=0.40 cal/em. sec.
The thing, which immediately strikes us, is that there is here no
Fig. 5.
767
distinet discontinuity as was found at 4°.19 K in the electrical con-
ductivity, although the thermal conductivity becomes much larger,
when the temperature decreases. As there do not exist direct deter-
minations for solid mercury, we only can make a rough estimation
with the aid of Wirprmann and Franz's law. .
At the melting point, the electrical conductivity of liquid mercury
amounts to 1.10. 10‘ em. tand of solid mercury to about five times
as much, thus to 5.50. 10* emt 2-1, From this we tind by comparison
e.g. with lead about 0.075 for the thermal conductivity. The values
here obtained in liquid helium are 3.5 and 5.5 times as large.
Chemistry. “Aquwilibria in ternary systems”. XVII. By Prof.
SCHREINEMAKERS.
(Communicated in the meeting of Oct. 31, 1914).
Now we will consider the case, mentioned sub 3 (XVI), vizs
the solid substance is a binary compound of a volatile- and a
non-volatile component. A similar case occurs for instance in
the system Na,SO, + water + alcohol, when the solid phase is
Na,SO,. 10 H,O, or in the system FeCi, + HCI + H,O, when the
solid phase is one of the hydrates of ferric chloride, for instance
Fe‚Cl,. 12 H,0.
For fixing the ideas we shall assume that of the three compo-
nents A, B, and C' (fig. 1) only A and C are volatile so that all
vapours consist either of A or of C or of A+ C.
In fig. 1 CAde represents a heterogeneous region L—G'; ed is
the liquid curve, CA the corresponding straight vapour-line. The
liquid d, therefore, can be in equilibrium with the vapour A, the
liquid e with the vapour C and each liquid of eurve ed with a
definite vapour of AC.
Previously (XVI) we have seen that this heterogeneous region
L—G can arise in different ways on decrease of pressure, viz. either
in one of the angiepoints A and C or in a point of AC; also two
heterogeneous regions may occur, the one in A and the other in C,
which come together on further decrease of pressure somewhere in
a point of AC.
In fig. 1 we may imagine that the region /—G' has arisen in these
different ways; curve ed may of course also turn its convex side
towards AC. Besides this heterogeneous region L—G we also find
in fig. 1 the saturationcurve under constant pressure of the binary
51
Proceedings Royal Acad. Amsterdam. Vol. XVIL.
768
substance /’, represented by pq [we leave the curve rs, drawn in
the figure out of consideration for the present].
A
Fig. 1.
In the same way as we have acted in the general case | fig. 11 (1)|
or in the peculiar case (XI), we may deduce also now the different
diagrams.
T< Ty. At first we take a temperature 7’ lower than the
minimummeltingpoint 7’ of the binary compound #. Now we find
a diagram just as fig. 2 for the saturationcurve under its own vapour-
pressure of # and the corresponding straight vapour-line. In this
figure, in which the component-triangle is only partly drawn, Agn
is the saturationcurve under its own vapourpressure; we find the
corresponding straight vapour line Cg, on side CA.
When we assume, as is supposed at the deduction of fig. 2, that
neither a point of maximum-pressure, nor a point of minimum-
pressure occurs, the pressure increases from n towards /; conse-
quently it is lowest in m and highest in 4, without being, however,
a minimum in ” or a maximum in h. It follows from the deduction
that the sides solid-gas and solid-liquid of the threephasetriangles
must be situated with respect to one another and to the side CB
just as is drawn in fig. 2.
It is apparent from the figure that the binary liquids 4 and » can
be in equilibrium with the unary vapour C and that the ternary
liquids a, e and 6 can be in equilibrium with the binary vapours
a,, c, and 5. It is apparent that somewhere between the liquids c
and 6 a liquid g must be situated, the corresponding vapour g, of
which represents the extreme point of the straight vapour line Cq,.
When a liquid follows curve An, first from A towards g and after-
wards from g towards n, the corresponding vapour g, follows conse-
769
quently first Cy, from C towards g, and afterwards again this
same line, but in opposite direction, consequently from g, towards C.
Each vapour of this straight vapour line Cy, can, therefore, be in
equilibrium with two different liquids, the one of branch A and the
other of branch gi.
We may express this also in the following way: when we have
an equilibrium #4 L + G, then there exists under another pressure,
also an equilibrium /#’+ L, + G,, in which Z and ZL, have a
different composition; G and G,, however, have the same composition,
It is apparent from the deduction of fig. 2 that in curve hn also a
point of maximumpressure can occur. This case is drawn in fig. 3;
in represents again the saturationcurve under its own vapourpressure
and Cg, represents the corresponding straight vapourline; M/ is the
point of maximumpressure, J/, the corresponding vapour. The points
M,, M, and F must of course be situated on a straight line.
While under the pressure Pj, there occurs only one equilibrium,
viz. K+ Ly + Gy,, under each pressure, somewhat lower than
Py there exist two equilibria, for instance P+ Le + Ga, and
F+ L.G; we can imagine these to be represented by the
threephasetriangles Yaa, and Fee,, when we imagine both triangles
in the vicinity of the line FA/J/,. It follows from the deduction
of the diagram that both these triangles turn their sides solid-gas
towards one another, consequently also towards tbe line FMM.
Suppose, we want the curves ed and pg to move in fig. 1 with
respect to one another in such a way that a point of minimum-
pressure occurs on the saturationcurve under its own vapourpressure,
51*
770
then we see that this is impossible. Yet we can imagine a saturation-
curve with a point of maximum- and a point of minimumpressure.
When we trace curve An starting from n, we arrive first in the
point of maximum- afterwards in the point of minimumpressure.
We will refer to this later.
Tr < T. Now we take a temperature Ta little above the minimum
meltingpoint 7'p of the solid substance #. Then we must distinguish
two eases, according as the solid substance expands or contracts on
melting. We take the first case only.
Then we find a diagram like fig. 4 (XI); herein, however, the
same as in figs. 2 and 3, we must imagine that the vapourcurve
han, is replaced by a straight vapourline Cy, on side CA. We
will refer later to the possibility of the occurrence of a point of
maximum- and a point of minimumpressure.
We can, however, also get curves of a form as curve An and the
curves situated inside this in fig. 6 (XI); these curves show as well
a point of maximum- as a point of minimumpressure.
When we draw the saturationcurves under their own vapour-
pressure for different temperatures, we can distinguish two principal
types; we can imagine those to be represented by figs. 5 (XI) and
6 (XD. At temperatures below 7'p these curves are circumphased,
above P'p they are exphased. In tig. 5 (XI) they disappear in a point
H on side BC, in tig. 6 (XD) in a point & within the triangle. The
corresponding straight vapourlines disappear in fig. 5 (XI) at Tagan
the point C; in figure 6 (XI) they disappear at Tr in a point A,
the intersecting point of the line FR with the side CA.
771
Now we will consider some points more in detail. In order to
get the conditions of equilibrium for the system # + 4+ G, when
F is a binary compound of B and C and when the vapour consists
only of A and C, we must equate «=O and y,=0. The conditions
(1) (II) pass then into:
eZ 7
Phe da NAY aye met BD
ae ag Seg icy can ren
4,—#,—+P—=5
0a, dy
Now we put:
A= Uit ello a? and ZU, Aa loge of 7 (a)
Hence the conditions (1) pass into:
aU oe
verd ae Ae ia a i a RRA UE
a y
ae ae KT U, eo 4
AES Et ee +s — . . . . (4)
Le RTI 2g OU;
Oa ose te
elo Gi No Wer Hr vale OD)
1
When we keep the temperature constant, we may deduce from
(3)—(9):
[er + (y — B)s4+ RT] dz + [as + (y —B)t]dy = AdP . (6)
E tes > rr | de + [#,s — Bil dy =(A + OdP. . (7)
RT RT C7 or
r + —— | da + sdy — | r, + — | de, = —=—]|dP . (8)
v U, Ox, Oa
Here we must equate of course in A and C a=O and y, = 0.
In order to let the pressure be a maximum or a minimum, dP
must be =O. From (6) and (7) it follows that then must be satisfied :
a Bp Aly Dh On; verven a ee
This means that the point of maximum- or of minimumpressure
M (x,y) and the corresponding vapourpoint J/, (7,y,) are situated
with on a straight line (fig. 3).
In order to examine the change of pressure along a saturation-
curve under its own vapourpressure in its ends / and n (figs. 2 and 3)
we equate in (6) and (7) e=0O and x, =U. Then we find:
OV
[(y—B)s + RT] de + (y—p) tdy = | v- » + (B) | dP (10)
2, OV
| a+ Sar dea =| ¥,— 0+ as | ar AD)
Fy
772
The ratio 2,:2 has a definite value herein, as it follows from (5).
When we eliminate dy from (10) and (11), then we find:
E + (y—B) =| RT da =|8V + (y—8) V, — yr] dP. . (82)
The quantities in the coefficient of dP relate all to the binary
equilibrium F+ LG. When we call AV, the change of volume,
when between the three phases of this binary equilibrium a reaction
takes place, at which the unity of quantity of vapour arises, then is:
y—— AV, BFE GD Vg ee
Consequently we may write for (12):
ip — ite B RT ; id
LP = 7 JT Nae a teu sin we a
Now we introduce again, as in (XI) the perspective concentrations
of the substance A in liquid and gas; it is evident that the per-
spective concentration S, is equal to the real concentration 2, of A
in the vapour; we find for the perspective concentration of A in
the liquid:
Bu
py
=
(15)
so that we can write for (14):
dP at; SN RT
nr EER us, WEE See
de J ob a ER
When the vapour contains the three components, then, as we have
seen previously (14) (XI) is true; when we replace herein S, by 2,,
this passes into (16).
It follows from (16) that the sign of the change of pressure in
the ends A and n of a saturationcurve under its own vapourpressure,
depends on the sign of AV. Now AV, is almost always positive
for the binary equilibrium /’+2-++-G and it is only negative between
the points / and H [fig. 5 (XI) and fig. 6 (XI)|. Consequently AV,
is positive in the points 4 and n of figs. 2 and 2, also in the point
h of fig. 5 (XI) and 6(XI); AV, is negative in the point n of the
two last figures. Further it follows that the sign of the change of
pressure is not determined by the ratio #,:# (the partition of the
third substance between gas and liquid) but by the ratio S:, (the
perspective partition of the third substance between gas and liquid).
Let us take now a liquid of the saturationcurve under its
own vapourpressure in the vicinity of the point h of fig. 2, for this
we imagine triangle Faa, in the vicinity of the side BC. From the
position of Fa and Fa, with respect to one another, follows
773
S>a,. As AV, is positive in h, it follows from (16) that the
pressure must decrease on addition of a third substance. We see
that this is in accordance with the direction of the arrow in the
vicinity of h.
In the vicinity of point / of fig.3 is a, > S as follows from the
position of triangle Faa,. As AV, is positive, it follows from (16)
that the pressure must increase on addition of a third substance.
This is in accordance with the direction of the arrow in the vicinity
of h.
In the vicinity of point n of the figs. 2 and 3 S is negative (we
imagine for instance in fig. 2 triangle Fbb, in the vicinity of side
BC); as AV, is positive, it follows from (16) that in both figures
the pressure must increase, starting from 7.
Consequently we find: in a terminatingpoint of a saturationcurve
under its own vapourpressure, situated between Cand H, the pressure
decreases on addition of a third substance, when the threephase-
triangle turns its side solid-gas towards LC (fig. 2) and the pressure
increases when the threephasetriangle turns its side solid-liquid
towards BC.
As, therefore, at temperatures lower than 7'p (figs. 2 and 3) the
pressure always increases, starting from 7, and increases or decreases
starting from h, we find the following. When we trace curve nh,
the pressure increases continually starting from » towards h (fig. 2),
or we come starting from n first in a point of maximumpressure,
after which the pressure decreases as faras in h (fig. 3) or we come,
starting from n first in a point of maximum- and afterwards in a
point of minimumpressure, after which the pressure increases up to /.
As in point A of tig. 5 (XI) the pressure decreases starting from
h, consequent it is assumed here, that the threephasetriangle turns
its ‘side solid-gas towards BC. (Cf. fig. 2 and fig. 4 (XI); in this
last figure we imagine however curve /i,n, on side CA). In the point
h of fig. 6 (XI) is assumed that the threephasetriangle turns its side
solid-liquid towards BC.
Let us consider now the terminatingpoint » of the saturationcurve
in fig. 5 (XI) and fig. 6 (XI). As n is situated between F and H,
AV, is negative, when the threephasetriangle turns its side solid-
gas towards BC, then is S >>, and it follows from (16) that the
pressure increases on addition of a third substance. We then have
the case of fig. 5 (XI). When, however, the threephasetriangle turns
its side solid-liquid towards BC, then S<{w, and it follows from
(16) that the pressure decreases on addition of a third substance.
We then have the case represented in fig. 6 (XI).
774
When we consider the saturationcurve going through the point
Fin fig. 5 (XD and fig. 6 (XD, then for this point y= 8, conse-
quently, according to (15) S= a. From (13) follows also 4 V,=
Therefore we take (12); from this follows for y= 8
aP RT
lk A plies AEC es ORR
da az—0 V—v
As fig. 5 (XI) and fig. 6 (XI) are drawn for V >», the pressure
must increase starting from / along the saturationcurve going
through /.
As the pressure increases starting from /’ along the saturation-
curves under their own vapourpressure of fig. 6 (XI) and decreases
starting from a point m, situated in the vicinity of , somewhere
between J’ and n must consequently be situated a point, starting
from which the pressure neither increases nor decreases. This point
‚therefore, the point of maximum- or of minimumpressure of a
saturationeurve, and is not situated within the componenttriangle,
but accidentally it falls on side BC. It follows from the figure that
this point is a point of -minimumpressure; we shall call this the
point m.
The limiteurve (viz. the geometrical position of the points of
maximum- and minimumpressure) goes consequently through the
points m and R; it represents from m to # points of minimum-
pressure; starting from / further within the triangle, it represents
is
points of maximumpressure. This latter branch can end anywhere
between H and C on side BC.
The terminatingpoint of a limiteurve on side Bf’ can be situated
between Fand C, but cannot be situated between fand B. A similar
terminatingpoint is viz. a point of maximum- or a point of minimum-
pressure of the saturationcurve, going through this point. Consequently
in this point along this saturationeurve d?=0; from (16) it follows
that then must be satisfied :
S=. or Pe (ya, =O 2. on EE
Herein w and 2, are infinitely small; their limit-ratio is determined
by (5). As » and 2, are both positive, it follows from (18): y < 8.
The terminatingpoint of a limiteurve must, therefore, be situated
between /’ and C (fig. 6) and it cannot be situated between /’ and
B. In accordance with this we found above that one of the ends
of the limiteurve is situated in fig. 6 (XI) between n and FL.
Now we must still consider the case mentioned sub 4 (XIV), viz.
that the solid substance is one of the components. A similar case
115
occurs for instance in the systems: Z + water + alcohol, wherein
Z represents an anhydrie single, salt, which is not-volatile.
For fixing the ideas we assume that B is the component, which is
not-volatile (fig. 1), so that A and C represent the volatile components.
Now we imagine in fig. 1 curve pq to be omitted, so that the
curves ed and rs rest only, ed is the liquideurve of the region L—G,
rs is the saturationcurve under a constant pressure of the substance PB.
We can, in order to obtain the different diagrams, act in the same
way as we did before in the general case, or as in communication
XIII. For this we consider the movement of the curves ed and rs
with respect to one another on decrease of pressure.
As we assume that B is not volatile, these considerations are
not true, therefore, for points situated in the vicinity of B. Equilibria
situated in the immediate vicinity of 6 have viz. also always the
substance 6 in their vapour, so that the considerations of com-
munication XIII apply to these.
When we decrease the pressure, the liquideurve ed (fig. 1) shifts
further into the triangle towards the point 5, so that under a definite
pressure the curves ed and rs meet one another. Now we distinguish
three cases.
1. We assume that the curves ed and 7s meet one another first
in a point on one of the sides of the triangle; when this takes place
on side BC, then consequently the points e and r coincide in fig. 1,
while the two curves have no other point in common further. On
further decrease of /?, this intersecting point shifts within the triangle
and it disappears at last on the side A‘, when in fig. 1 the points
s and d coincide. Curve ed is situated then inside the sector Brs
and curve 7s inside the region CedA.
From this follows that the saturationcurve of B under its own
vapourpressure can be represented by curve habn in fig. 4, in which
the arrows indicate the direction, in which the vapourpressure increases.
The corresponding vapourcurve is the side CA; the liquid h viz. is
in equilibrium with the vapour C liquid with the vapour A and
with each liquid (a and 5) of An a definite vapour (a, and b,) of CA
is in equilibrium. It follows from the deduction that the threepbase-
triangles (Baa,, 6b),) turn their sides solid-gas towards the point /
and their sides solid-liquid towards the point 7.
This fig. 4 is a peculiar case of fig. 2 (XIII); when we suppose
viz. that the substance £ does not occur in the vapour, curve h‚a,b‚n,
of fig. 2 (XIII) must coincide with the side CA of the triangle and
fig. 4 arises,
~]
~
lep)
Fig. 4.
2. Now we assume again that the curves ed and rs (fig. 1) meet
one another first in a point of the side BC; this point of inter-
section shifts then on further decrease of P into the triangle. Under
a definite pressure we want a second point of intersection to be
formed by the coincidence of d and s (fig. 1). The two points of
B
en
mm, 6 A
Fig. 5.
intersection approach one another on further decrease of pressure,
in order to coincide at last in a point m. It is evident that m is a
point of minimumpressure of the saturationcurve under its own
vapourpressure; it is represented in fig. 5 by curve vambv, the
corresponding vapourcurve is the side CA. It is evident that the
vapour m,, which can be in equilibrium with the liquid m, is
situated on the line Bm.
3'¢. We can assume also that the curves ed and rs (fig. 1) meet
on decrease of pressure first in a point J/, which is situated within
the triangle. On further decrease of pressure then two points of
777
intersection arise; the one disappears on BC by the coincidence of
e and 7, the other on BA by the coincidence of d and s (fig. 1).
It is evident that J/ is then the point of maximum-pressure of the
saturationcurve of B under its own vapourpressure, the corresponding
vapourpoint M/, is situated of course on the line BM.
One can understand the occurring diagram with the aid of fig. 5 ;
‘herein we have to give an opposite direction to the arrows and we
have to replace the points of minimumpressure m and m, by the
points of maximumpressure J/ and JM, ; further the triangles Baa,
and £Lbb, are to be drawn, in such a way that they turn their
sides solid—gas towards the line MM.
We shall consider some points in another way now. In order to
find the conditions of equilibrium for the equilibrium B + L + G,
when the vapour consists of A and C only, we equate in the
relations (1)—8) @=1; in the general values of A and C (ID)
we put «c= 0, B=—1 and y, = 0. The condition for the occurrence
of a point of maximum- or of minimumpressure (dP = 0) becomes then:
Ves.) Sao Ns ea ea EE 1)
This relation also follows from (9), when we put g=1. This
means: the point of maximum: or of minimumpressure of the saturation-
curve of B under its own vapourpressure and the corresponding
vapourpoint are situated with point B on a straight line (fig. 5).
In order to determine the change of pressure along a saturation-
curve under, its own vapourpressure in its ends on the sides BC and
BA (figs. 4 and 5) we put in (16) g=1. We then find
dP a SND
en See hb ie a de Sa HN
OE aL ae A
In this S and AV, are determined by (13) and (15), when we
put herein 81. Consequently S is always positive. When we
apply (20) to the figures (4) and (5), then we see that the change
of pressure is in accordance with the position of the sides solid-gas
and solid-liquid of the threephasetriangles.
Now we have treated the case that either the binary compound
F (figs. 2 and 3) or the component B (figs. 4 and 5) occurs as solid
phase. When F and B occur both as solid phases, then the two
saturationcurves under their own vapourpressure can either intersect
one another or not. We only consider the case, drawn in fig. 6, that
the two curves intersect one another in a point; the vapour, being
in equilibrium with the liquid s, is represented by s, (s, or s,).
A similar case may occur for instance in the system Na,SO, +
water + alcohol, then curve cs is the saturationcurve under its own
778
vapourpressure of Na,S0,. 10 H,O (#), sa the saturationcurve of the
anhydric Na,SO, (B). Then there exists a series of solutions, saturated
under their own vapourpressure with Na,SO,.10H,O. (curve cs)
and one series saturated with Na,SO, (curve sa); the equilibrium
Na,SO,.10H,O + Na,S0, + Ls + Gs oceurs only under a definite
pressure P,. The solution ZL, has then a definite composition s and
the vapour, which consists only of water and alcohol has a definite
composition s, .
In the binary system Na,SO, + water, the equilibrium Na,SO, . 10
H,O + Na,SO, + vapour exists only under one definite pressure; we
shall call this pressure P,. In the ternary system Na,SO, + water +
alcohol the equilibrium Na,SO, .10 H,O+Na,S0,+L,+6, exists also
only under a definite vapourpressure /;. This pressure P, is influenced
B
Fig. 6.
by the watervapour and the alcohol-vapour together; now we may
show that the partialpressure of the watervapour herein is also equal
to P, and that the pressure of the alcohol vapour is consequently
PP
In order to show this, we consider the general case that in the
system A+ B+C (figs. 1—6) the substances A and C are volatile
and that a compound / of B and C occurs.
The binary equilibrium B + F+ G,, wherein the vapour consists
of C only, occurs under a single pressure P, only.
The ternary equilibrium B+ F+ G, wherein consequently the
vapour consists of A and C, can occur at a whole series of vapour-
pressures.
When we represent the § of B and # by § and &,, then the
condition of equilibrium is true:
OZ,
BDA) BREN
0a,
779
Hence follows:
OV.
E == by =. (P-— 8) (v. — @, =i) dP+(1—B)e,r,de, =0. (22)
el
When we assume that the gas-laws hold for the vapour G, then;
OV, ' RT ba
ae amd 2 7, a2 ee a a ee
Oz, Ten sie
From (22) now follows:
ees
ABP, HBP, SRT de, . . (24)
— a,
The coefficient of dP represents the change of volume when 1 Mol.
F is decomposed into 3 Mol B+ (1 — 8) quantities of G; this is
very nearly (1— 8) V,. As at the same time PV, = RT, we can
write for (24);
(Ue) dP =S Pao... NE
From this follows:
DE
Bas ae 2
le,
When we call the partial pressures of A and C in the vapour Py
fade, utene.) 4 it and Po — (1 — a) Po from 26) now
follows :
P
SP and) “Po 22, we 7 eee ee
a |
This means that in the ternary equilibrium B+-F+G the partial
pressure Po of the substance C is equal to the vapourpressure of
the binary equilibrium B+ F4 G.
When we bear in mind now that in a system the pressure and
the composition of the vapour do not change, when we add to this
system stili a liquid, which is in equilibrium with all phases of this
system, then follows:
In the ternary equilibria 6+ #4 G and 6+ + Ld G, the
partialpressure of the substance C in the vapour is equal to the
vapourpressure of the binary equilibrium 4+ F+ G,.
The first equilibrium (viz. BH F+ G') exists at a whole series
of pressures; both the others occur under a definite pressure only.
The binary equilibrium Na,SO .10H,O + Na,SO, + watervapour
has at 25° a vapourpressure of 18.1 m.m. when we add alcohol,
then, when the gas laws hold in the vapour, in the equilibrium
Na,SO,.10H,O + Na,SO,-+G and Na,SO,.10H,O + Na,S0, + LG
the partialpressure of the watervapour will also be equal to 18.1 m.m.
Now we will put the question, whether we can also deduce some-
780
thing about the change of pressure starting from s along the curves
sa and sc (fig. 6). In communication V we have deduced the following
rule. When the equilibrium solid + Z can be converted with increase
of volume into solid + LZ’ + G’ (in which L’ differs extremely little
from £) then of a threephasetriangle solid—liquid—gas the side
solid—liquid turns on increase of pressure towards the vapourpoint
and it turns away from the vapourpoint on decrease of pressure.
When we assume now that s (fig. 6) is not situated in the vicinity
of B or F (the equilibrium B + L and F+ L converts itself into
B+L’+G’ and F+ L/ + G’ with increase of volume) we can
apply the above-mentioned rule. We distinguish now according as
the vapour is represented by s,,s, or s,, three cases.
1. The vapour is represented by s,.
First we consider the threephasetriangle /’ss,. When the side Fs
turns towards c, then consequently it turns towards its vapourpoint
s,; the vapourpressure increases, therefore, starting from s along sc
towards c.
Let us consider now the threephasetriangle Bss,. When the side
Bs turns towards a, it turns, therefore, away from its vapourpoint
s,; consequently the vapourpressure decreases starting from s along
sa towards a.
Consequently we find that the vapourpressure starting from s
increases along sc and that it decreases along sa. It is evident that
this is only true for points in the vicinity of s; the occurrence at
a greater distance of s of a point of maximumpressure on sc and a
point of minimumpressure on sa, is viz. not excluded.
2. The vapour is represented by s,.
It follows from a consideration of the threephasetriangles /’ss,
and Bss, that the vapourpressure starting from s increases as well
along sc as along sa.
3. The vapour is represented by s,.
It follows from a consideration of threephasetriangles #ss, and
Bss, that the vapourpressure starting from s decreases along se and
increases along sq.
We can obtain the previous results also in the following way.
Between the four phases of the equilibrium 6 -+ /’+ ZL, + vapour
(s,, 8, Or s,) a phasereaction occurs on change of volume. We choose
this reaction in such a direction that vapour is formed, we call the
change of volume AY. ke
The point s (fig. 6) is a point of the quadruplecurve B + F+
+L+G; AV is positive for each point of this curve. When,
however, a point of maximumtemperature H occurs on this curve,
781
then AV is negative between this point H and the terminatingpoint
of the curve on side BC. It is apparent from the position of the
curves sc and sa (fig. 6) that point s is chosen on that part of the
quadruplecurve, where AV is positive. We distinguish now again
the same three cases as above.
1st. The vapour is represented by s,.
It is apparent from the position of the points #, B,s and s, with
respect to one another that the fourphase-reaction :
DR Ie NAV > 0)
F+L+G (Curve sc) | B+2L-+G (Curve sa)
i Bb | FIB+G
takes place; it proceeds from left to right with increase of volume
Hence it follows that the equilibria written at the right of tbe
vertical line occur under lower pressures, the equilibria at the left
occur under higher pressures. In accordance with the above we find,
therefore, that starting from s (tig. 6) the pressure increases along
se (equilibrium F4 2+ G) and decreases along sa (equilibrium
B+L+G).
2.4 and 3rd. Also in these cases we find agreement with the
previous considerations.
When a point of maximumtemperature MZ occurs on the quadruple-
curve B+ FH LG, then two points of intersection s occur at
temperatures a little below 77. When we consider now a point
of intersection s between /H/ and the terminatingpoint of the qua-
druplecurve on side BC, then AV is negative. This involves that
above in 1st—3'd increase of [ is replaced by decrease of P and
reversally. We find also the same when we consider the threephase-
triangles solid-liquid-vapour. To be continued.)
Chemistry. — “On the quaternary system: KCI—CuC1,— Ba Cl, H, O.”
By Prof. ScHREINEMAKERS and Miss W. C. pr Baar.
(Communicated in the meeting of October 31, 1914).
In a previous communication’) we have already discussed the
equilibria occurring in this system at 40° and at 60°; the results of
the analysis on which these considerations are based, we have hitherto
not yet communicated. Now we will communicate the results of the
analysis; all the points, curves etc. quoted in this communication
apply to the two figures of the previous communication (l.c). We
want to draw the attention to the fact that fig. 1 represents the
equilibria at 40° and tig. 2 the equilibria at 60°.
1) These Communications (1912 326.
789
TABLE 1.
Composition of the solutions in percentages by weight at 40° (fig. 1. l.c.).
Point) KCl | BaCl, | CuCh | H,0 | Solid phases
| |
a 0 0 | 44.67 | 55.33 | Cu Clo. 2 H2O
b 0 3.72 | 42.72 | 53.56 | BaClh.2H,O+CuCl.2H,0
5 0 28.98 0 11.02 Ba Cl. HO
d | 23.98 | 9.15 0 66.87 | BaCl.2H,0-+KCl
e | 28.63 0 0 71.36 | KCI
f 4 2188 0 22.85 | 55.62 K+ Doo
el tok ib 43.83 | 46.38 Cul 12 Hs ADS
b o | 3.72 | 42.72 | 53.56 Ba Cl, .2H,0 + Cu Cl,.2H,0
Es | 5.52 | 3.30 | 42.35 | 48.74 :
Sh | 9.88 | 2.99 | 42.07 | 45.06 | BaCl,. 2H,0-+ CuCl; . 2H,0 +Dj „5
gele 23008) |) 945 0 66.87 Ba Cl. 2 Hs O + KCI
» | 21.46 | 8.90 | 8.44 | 61.20 ;
ES
G | 20:61 | Aes | 14.31 | 57.45 ‘
i | 20.61 | 5.40 | 20.47 | 53.52 Ba Cl, 20 KG Bee
ns 0 22.85 | 55.62 cee bea
a 21.31 | 2.59 | 22.08 | 54.04 :
~; | 20.61 | 5.40 | 20.47 53.52 BaCl,. 2,0 4 KCl eee
es | 9.79 0 43.83 | 46.38 Cul 2E Dis
Ee 9 | 146 402 | 45.38 :
h | 9.88 | 2.99 | 42.07 | 45.06 | CuCl. 2H,0 + BaCl2H,0+ Dio.
i | 20.61 | 5.40 | 20:41 | 53.52 BaCly,.2H.0 KGS
o | 16.44 | 4.72 | 27.22 | 51.62 Ba Cls 2H,O + Dia.
8 | u.44 | 3.66 | 34.65 | 50.55 ;
h | 9.88 | 2.99 | 42.07 | 45.06 | CuCl.2H,O + BaCly. 2H,0 + Diaa
Composition of the solutions
TABLE I.
in percentages by weight at 60° (fig. 2 1.c.).
Point
KCI
BaCl,
CuCl,
41,42
43.57
H,0
Solid phases
CuCl, . 2H,0
CuCl, . 2H,0 + BaCi, . 2H,O
BaCl, . 2H,O
BaCl, . 2H,O + KCl
KCl
KCE DS
Die De,
CuCl, .2H,O + Di,
Boley 0 6.87 | 43.57 | 49.56 CuCl, . 2H,O + BaCl, . 2H,0
Vv
cs! 6.32 | 5.99 | 43.68 | 44.01 E
©,
1 | 12.45 | 4.93 | 44.09 | 38.53 | CuCl.2H,O + BaCly.2H,O+ Di,
eh BRO idle) 0 62.08 BaCl, . 2H,O + KCI
v
Ze | 23.15 | 10.01 | 12.01 | 54.83 ;
(5, |
l 23.78 5.97 24.61 45.64 BaCl» a 2H,O KCl + D, .2.9
ee eee
f t 2812 | 6 26.57 | 47.31 KCI + Di 5
vo
Ew, | 24.53 | 3.32 | 25.46 | 46.69 ;
©,
i | 23.78 | 5.97 | 24.61 | 45.64 KCl + BaCi» . 2H.0 + Di 25
eS iris ct 6 43.45 | 39.42 Das Dit
Vv
ES | 16.50 | 2.51 | 42.20 | 38.79 ;
©,
h | 15.75 | 4.75 | 40.84 | 38.66 BaCh, .2H,0- Die DR
bol ier Ie 6 46.40 | 39.93 CuCl, . 2H,O + Dis
8)
Ez | 13.04 | 2.52 | 45.24 | 39.20 :
5
1 | 12.45 | 4.93 | 44.09 | 38.53 | CuCl,.2H,O+BaCl,.2H,0+D,,,
keb)
ES | 19.58 | 5.40 | 32.37 | 42.70 BaCl, . 2H,O + D 4»
5 |
h | 15.75 | 4.75 | 40.84 | 38.66 BaCl,. 2H,O + Dy 22 + Di
h 1/15 4,15 40.84 38.66 BaCl, . 2H,O + D, 5.» + Di.
14.78 | 4.83 | 42,13 | 38.26 BaCl, .2H,O + D..,
1 | 12.45 | 4.93 | 44.09 CuCly. 2H,0 + BaCl,. 21,0 + Di,
38.53 |
52
Proceedings Royal Acad. Amsterdam. Vol. XVII.
784
Physics. — “On the theory of the string galvanometer of EINTHOvEN.”
By Dr. L. 5. ORNSTEIN. (Communicated by Prof. H. A. LORENTZ.)
(Communicated in the meeting of September 26, 1914).
§ 1. Mr. A. C. Crenorr has developed some considerations in
the Phil. Mag. of Aug. 1914), on the motion of the string galvano-
meter, which cause me to make some remarks on this subject.
For a string, immersed in a magnetic field //, and carrying a
current of the strength ./, the differential equation for the elongation
in the motion of the string is
04 Oy Oy HI
ke ee i oe
Pe Oe 5 0)
in which x is the constant damping factor, a? = —, 7’ is the tension
O
Q
and o is the density. The direction of the stretched string has been
chosen as the a-axis. For «=O and «—/ the string is fixed, so
y =. In deducing the equation the ponderomotive force is supposed
to be continually parellel to the elongation 4, which is only approxi-
mately true, since the force is at every moment perpendicular to
the elements of the string (perpendicular to / and H); but if y
may be taken small, then the equation (1) is valid. The approxi-
mation causes a parabola to be found for the state of equilibrium
with constant H and ./, instead of the arc of a circle, as it ought
to be; however, the parabola is identical with a circle to the degree
of approximation used. .
Dr. CRrrHORE now observes, that the equation (1) may be treated
after the method of normal codrdinates by putting
SITX
y = 2 Pein —— oy SS ke ROER
Besides the equation 1, he deduces a set of equations, the “circuit
equations”, which give a second relation between g‚ and J (from
(1) there originates in the well-known way an equation for every
coordinate y,). The obtained solutions will be independent, when
the circuit equation is true, and again their sum is a solution of the
problem. However, from the deduction of the circuit equation it cannot
well be seen whether this is the case, since not entirely exact
energetic considerations underlie this deduction. Now supposing the
string to be linked in a circuit with resistance /?, and self-induction
L, the circuit-equation may be easily found by applying MaxwEL1’s
1) Theory of the String Galvanometer of Einrmoven. Phil. Mag. Vol. 28, 1914, p. 207.
785
induction-equation. For in consequence of the motion of the string
in the magnetic field the number of lines of force passing through
the circuit changes to an amount proportional to
Expressed in the units used by Dr. Crrnorn, the induction-equation
now takes the form:
l
Bn rl g (3
Zee ST eo . . . . ° (3)
0
where / is an external electromotive force acting on the circuit.
§ 2. The problem of finding the vibrations governed by the
equations (1), (3) and the condition y=0O for e=0O and «=—/,
can be easily solved. First, let be 0, and so the question of free
(damped) vibrations may be put. Suppose that
Gap et = fb gat
where p is a function of wand J is a constant. Then the equations
change into
Om HI
Oa? ee.
(
0=AI+ Lio itt f pas.
9
— wv’ p + wap — a
0
Hence
l
dp iw
v4 ee ’ ; 2 — dx.
Ee Rr
0
Pulite ws at in ther first be
eN rst member n? and ————— =
utting w twx in the first m t, 2a o( + Lie) p we
have
: 1
2
n° pta? Laer [vee
Ox?
a 0
This equation may be satisfied by
n _n
g=Acs—a#+ Bane + C
a a
provided that
52*
786
a 3 nl a nl
=. (5 Asin + OB Les |C). .
n a nN a
whereas, because of the boundary conditions, we must have
A+ C=0
nl Amb
A eos — + Bsin— + C==0.
a a
This gives for the frequency the transcendental equation
x
nl?
eos
a nl a a
=P — sin — — +
or
BE, al a nl
n? sin — =p | l sin — — 2— 1 — cos — :
al a u a
From this it appears immediately that we must have
_ al
GN Oe a 2
2a
or
E nl nl. Za nl E
n° cos — = p | leos— — — UN — en rae (0
2a 2a n 2a
(5) ean be satisfied by
nl
eT 4 : (7)
2a
or, henee
ay (= =)
dm :
As is immediately to be seen, these are the damped vibrations of
even order, which the string can perform in the absence of the
current. It is evident that the presence of current and field have
no influence on tbe vibrations of even order. [f the resistance is
infinitely great, the constant p in the equation (6) is zero. In this
case the equations can be satisfied by n=O, or o =O, ie. the
string is at rest; and further by
nl
cos — = 0.
2a
Hence
nl dt
ze ie EE RETE vd ER (8)
of
187
we — iox = — Ra a
) e
The frequencies arrived at are those of odd order, altered by
current and field. For large values of R an approximate value of
n can easily be expressed in the form ng +s. From (6) follows
L
= Mei Aa
ek ok ns!
s being an odd number, / being taken zero, while for w and n, their
values for A= must be put. Taking x= 0, i.e. neglecting the
air-damping in comparison with the electrical damping, we find
sa 4H*il
iN : a. Uo Ste SEA Ae De eee
l Ee CW de @)
In the solution, therefore, there is a damping factor of the form
jn Apt,
Op ee
The influence of the damping is the less, the greater the value of
s is. This is directly evident, for if s is great, the string vibrates in
a great number of parts with opposite motion. The electromotive
force generated by those parts therefore is annulled.
In case A is small, the roots of the equation (6) are those of the
transcendental equation
nb 42a A nl
cos — gin — 0
2a n 2a
or
2a ; nl 1 foe
—_— td —_- = e e . e = e . e
nl 2a CON)
nl
The quantity — approaches to odd multiples of
of A an approximate form n,-+aF can be easily indicated. Taking
again L—=0 and «=O, we find
. For small values
bo| Q
where ms is an arbitrary root of (10). In case the resistance is small,
all vibrations suffer the same damping.
For fp we find
nl _ n(l—a) ~ na
sin — — Nn — sin —
2 a a
Ps =
Sid
sin —
a
1) Compare ‘for instance Riemann-Weser, Partielle-Differential Gleichungen, II,
p. 129.
hence for 4
_nl n (l—a) ns
sin — — sin — — sin —
: 5 a a a (i
ae Ee A . . .
d nl )
sin —
a
The real and imaginary part of this expression satisfy the equations
and the boundary conditions. A sum of solutions for different values
d
of w satisfies the equation. If and = are given for {= 0, we can
C
with the aid of the given functions find the solution. The found
proper functions are not orthogonal, but by an appropriate linear
substitution orthogonal functions can be obtained. If y is known, I
can be calculated from (3).
$ 3. It is useful to work out the problem. Using the assumption
(2) of CrrHORE, we obtain for p‚ the following set of equations
(taking k and / zero):
se : AHJ
Ps + Ns Ps = dele eee en (12)
STO
and
UH _ ps
RIA EES TA tn eN
nx s
where
SITA
Ns — ——
l
Here s is an odd number; for even values the second member
of (12) is zero, and the even vibrations are therefore unchanged.
Now putting
Ps = 0,6 ; Jd et,
and
AHI if
as — ’
STO Ns?—w?
we find
gr Sn
The frequency-equation therefore is
81 Hi 1
Ra 02. fae oe eae
ox, s? (ns? —@’)
This frequency-equation has the same roots as equation (6), which
if « and / have been taken O, takes the form
789
+
wl iH?! al 2a, al
Ee ia Ro CO EEEN
The identity of these frequency-equations can be easily shown.
2
Put
i=k, then (6a) takes the form
re == a
na s* (ns? —w’)
deus 1
Pe
Kk de 3?
: : a am
The sum of inverse squares of odd numbers is re Further,
O
n'a
ns’ = ——., therefore the first member amounts to
C
2
k a
1——+ 8&— &
w fe
For ig z we have
where s is again an odd number, therefore we obtain
Ne EN)
The equation (6) takes the oa
wl wl (i ' is
@ cos — = vS
: dl @ 2a 05 |= (15)
The equations (14) and (15) have the same roots, for the vectors
wl [ J
w and cos — do not contribute roots to (15).
atl .
Having found the roots of (14), we can determine y. Each root
; ; , srr
yields a Fourier series. In the case that (R = 0), sim —— must be
e
combined with one frequency only. For our case we have
= A, West . SAT
nn sin ——
s (ns? —w’) l
sre Ay 2 xt
3 in ee Sp e hpt ane Bad, OLON
Ss Ns —W,?
The Fourier series which is the vector of A Korg must be equal
to the function which in $ 2 appears as the vector of the same
exponential. This can be shown by direct development. It is apparent
that by a given frequency all the original normal coordinates are
790 6
set into motion. For very great and very small values of A, the
constants A in the expression (16) can easily be determined.
We can also use (9) and (11). Let us write (141) in the form
y _ gil I=
nl
COS =
2a
and let us introduce the value of n from (9), we then find
: 4FT*| ;
NE pn R i
vs Rosa’ (: ee =)
a J, a
ZH
where d; = a . Separating the real and imaginary parts, we find
vs ne
4H?
= 2734) / Os ae Net.
ek Resa IG (1 — cos =) cos ns t +- sin —— sin nt) A,
a a
ds Oeh ae st
+ { —| 1 — eos — | sin nst — sin — cos nst | Bs
R a a
For the time ¢= 0
J,
waz Ss cos) A, — sin" B, |
a
R
J, a ae)
ee ne in" (1 = ona °) B+ ae
Ros*ns
Y SUL
rig fi sin da =a, and f) 7 — de = bs; we have
and
0
20. l
be Se
R 2
l 2ns ARL
a5 = ts 5 As +R = d, Bs + =~
Ros”: ns
if l nsl l
For R=o we get b= — = Be ag 9 Ae Therefore Bs; = — 5 bs
2
and A; = — as. Putting
nsl
— 2 as
* Ins R
2 Bs
B ee Oi
s i + R
we have
ORDE des Sri
RO fs 2R .
4. 8D:
eN ae ze
R IR Rys'ns
These series are convergent, if the conditions for the ordinary
Fourier series are fulfilled. We can therefore calculate a, and 8, with
the help of the given formulae.
§ 4. In the case U is a given function of the time, our equation
can also easily be solved.
a. First if Z is constant, we have
E=RJ + H | dz.
0
The current / and y can be divided into two parts, the one
depending on ?¢, the other not; we indicate those parts by the indices
1 and 2. For the first part we have
Sa oY; ee
02? o
B RJ,
therefore
otha LE
O22 Ro’
from which y, can be determined if we take into account that y,
vanishes for «=O and «—/. The determining of the second part
leads to the problem treated in § 3. The solution can be used in
order to fulfill given initial conditions. If an initial value of / is
given, then y must fulfill at ¢=0 a condition following from (3).
6. Further, we can consider the case 4 = K cos pt.
Putting Z = 0, we can try the solution
y = p cos (pt + B)
J = 1 cos (pt + 8)
where p is a function of z. The first equation gives
EN ae
—pp—a ae a
This eqnation can be solved by
792
y= Acos x + Bsin® a+
or according to the above
( 4)
1— ¢os —
HI Pa a
es ars o u pl
sin —
a
sn tile AU
a :
Introducing this result into the second equation, we obtain
2
E cos pt =: RI cos (pt + B) + sin (pt +9)
Op
1 2
] (at)
Lin
a p ‚pl
sin —
a
Now take
l 2
1— cos —
U2 he | a . pl ( 4 ; :
Se SS —
opk p = a ‚pl T hie
sin —
r
then we find
2
(oac
A ap a a fe
Ecos pt=1 R? + (=S sin— | — — ~——__~_ + ) cos (pt+B-a)
uF ‘4
From this we find for the retardation of phase, =a; and for
the amplitude of /
E
I=
5
where r represents the square root in the second member. The
current / being found in this way, y can be determined from (17).
When Z does not vanish, we can suppose y and / to depend on
eit; and finally taking the real part, and following the above method
we find the values of y and J.
If we express y by (2), the solution can also easily be found. We
then have |
4HJ
Ps == sans” —p?) .
Substituting this into the second equation of § 3 (where zero has
been replaced by Ecos pt) we find
793
x 1
E cos pt = RI cos (pt + B) — ver (pt + B) XZ zj n=)
from which J can be found. The sum in the second member ean
be put in a way analogous to that of § 3, into a form identical
with (18). Our result does not agree with that of CREHORE (compare
p. 214). In our solution the retardation of phase is the same for all
vibrations, which is not the case in CREHORE’s paper.
It may be observed that in our problem we have to do with a
system of an infinite number of variables in which a dissipation-
function couples the variables; for eliminating / from (12) and (13),
we obtain
SESS Sp
Ds - Ns’ P= — =
sok s
The dissipation # takes the form
SH* (PN
Omen
moh s
Groningen, Sept. 1914.
Physics. — “Accidental deviations of density and opalescence at
the critical point of a single substance.” By Dr. L. S. ORNSTEIN
and F. ZeRNIKE. (Communicated by Prof. H. A. Lorentz.)
(Communicated in the mecting of September 26, 1914).
1. The accidental deviations for a single substance as well as
for mixtures have been treated by SmoLucnowski ') and EINSTEIN *)
with the aid of Borrzmann’s principle; by Ornstein *) with the aid
of statistical mechanics. It appears as if the considerations used and
the results obtained remain valid in the critical point. SMOLUCHOWSKI
has applied the formula found for the probability of a deviation
to the eritical point itself, and has found for the average deviation
of density
Lis
My
He has used this formula to express in terms of the mean density
Te
') M. Smorvenowskr, Theorie Cinétique de l’opalescence. Bull. Grac 1907 p. 1057.
Ann. der Phys. Bd. 25, 1908, p. 205. Phil. Mag. 1912. On opalescence of gases in
the critical state. W. H. Kreesom, Ann. der Phys. 1911 p. 591,
2) A. Einstein. Ann. der Phys. Bd. 33, 1910, p. 1276,
3) OrnsteIN, These Proc., 15, p. 54 (1912),
794
the accidental deviations in a cube, the side of which is equal to
the wave-length of the light used in the experiments on opalescence.
Now there is a difficulty with this formula, to which, indeed,
lead also the considerations of EiNsreiN as well as statistical mecha-
nics when worked out in an analogous way for the critical point.
In all these cases the mutual independence of the elements of
volume is presupposed. Now, let there be given for the element of
volume v the mean square of deviation viz. (n—n)*. Consider p
equal contiguous elements of volume »,,v,,etc., in which »,,n,, ete.
particles are situated, Ny, 1, ete. indicating the mean values of these
numbers.
Hence in the volume V=v, dv, +. there are V=n,+n, +...
particles.
For the mean value of N we have
N=n,+n,+...
subsequently
(N — Ny = {(n, —n,) + (a, — 2.) +-. f = p(n —n)?
since, the elements of volume being supposed independent of each
other, the means of the double products vanish. So we find for the
deviation of density that the product of volume and mean square
of deviation must be a constant.
Indeed the above-mentioned formula of probability for the devia-
tions of density is so far inexact, as the terms of higher order
appearing in it are at variance with the mutual independence of
the elements of volume, which underlies the deduction of the fre-
quency-law. In fact this deduction is only valid for such large elements
of volume that these terms are no more of any influence. It is
easily seen that this limit, above which the formula is valid, in-
creases indefinitely in approaching the critical point. This explains
also mathematically the wrong dependence on v found for the mean
deviation in the critical point itself.
Now one could try to deduce the formula to a farther approxi-
mation. However, also the supposition of independence of the ele-
ments of volume is inexact in case these are small, and it would
thus be impossible to ascertain how far the formula would yet differ
from reality. *)
1) A deduction of the inequalities in which the inexact terms of higher order
do not at all appear, is given by ZERNIKE in his ‘thesis, which will shortly
appear. As this deduction too uses the independence alluded to, the objection men-
tioned holds here also.
The remark of Einstein (lc. p. 1285) that there would be no principal difficulty
195
2. Now, in order to avoid the difficulties mentioned, it is necessary
to take into account the influence of deviatious in the one element
on the state in another. Let us divide the system into infinitely
small elements of volume. A molecule is considered to lie in the
element when its centre is situated in it. We consider an element
dv, in the origin of coordinates. Around this element we imagine
the sphere of attraction i.e. the region in which a molecule must
lie when it is to have any influence on the state in dv,. We determine
the numbers of moiecules for the elements of the sphere of attract-
ion in giving the deviations v,,r, etc. from the mean number of
molecules per unit of volume. f
We suppose the mean value of the density »,, when vp, etc. are
given, to be a linear function of the deviations », etc, i.e. we put?)
TE
Taking the mean value of », over all possible values of »,, it
appears immediately that C == 0, hence
Ha APE hr hal ee oee ale en ot QD)
The coefficients f denote the coupling of the elements, they only
depend on the relative coordinates, i.e. here, on «yz. That the in-
fluence of an element, when the density is given, must be propor-
tional to its size is immediately seen by considering the influence
of uniting two elements in (2).
We shall now write the sum (2) as an integral. For the density
in the element dv dy dz we put P,y2; further, we can dispose of f
in such a way that /(0,0,0)=0. Then for (2) we get
+
P, = (ff vreten) NE )
The integration may be extended here from —o to +o, f
being zero outside the sphere of action’).
in extending his deduction to a further approximation, is therefore mistaken. On
the contrary, the consideration of higher terms so long as the independence is
made use of, will not lead to anything.
1) Putting things more generally, we could write a series in ®, ete. instead of
(1). However, for the purpose we have in view, (1) is sufficient.
®) The quantity » can only take the values 1—adv and — adv, hence v is a
discontinuous function of the coordinates. One might be inclined therefore, to continue
writing a sum instead of the integral (3) and to solve the problem dealt with in
the text with the aid of this sum. In doing so one gets sum-formulae which are
wholly analogous to the integrals we used. However, we prefer introducing the
integral, as the discontinuous function v has entirely disappeared from formula (6)
only the function g appearing in it, which is continuous when the function f is
796
On the contrary, if vp, is given, » has another value for the
surrounding elements, than if », =O. Be in the element at xyz
Va == OE, Wat: Dale Ns. NE le (4)
and let us try to determine the function g, the function / being
given.
Now take the mean of formula (3), a fixed value v, being ascribed
to vp in a certain element dx, dy, dz
In «,y, 2, according to (4)
Peyz = YA ¥—Y,, 2—2Z,, D, da, dy,dz,). - - ~. fa)
For the first member we therefore get
Wy Yos Zo Vda, dy,dz,)
as f and g do not depend on the direction of the line joining the
elements. In the integral, (5) cannot be applied to the element
dx, dy, dz,; however, this element gives
VP, J (Yord du de,
Further taking g(0,0,0) zero, as it may arbitrarily be chosen,
we get
ge U 112510, de, dy, de) =| {fo (w-@,,y-Y ,.2-2,,9, da , dy ,dz,) f (wyz) dadydz +
af vj (1541571) de, dy ,dz,.
This is true for all values of vp, dx, dy, dz,, hence g must contain
this quantity as a factor, and we obtain
+a
He 41921) (fate YU zz) f (eye) dadydz = f(a,,y,-2,)
Now put 2z—a,=—& y—y,=1%, z—z,=5, and omit the index,
then for g we get the integral equation
+ a
Hey ,2) — { i he F(e@ +8 y+n, 2-5) (Ens) d§dyds = f(aye). . (6)
For g we have
Prijs = g(ay2) Dj do, ace) PS
from which it appears immediately that
Vryz Pye G(xYyzZ) V,? do, oo eh EP Re ea GEN
continuous. The integral-formulae obtained in this way are easier to deal with
mathematically, and besides the integral equation (6) has been solved, this being
not so easily found from the analogous sum-formula.
=
rey,
Now let us consider more closely the coefficient of g in (8).
Let « molecules be present in the unit of volume, then the mean
number of molecules in dv is equal to adv. If we take dv very small,
there may be no or one molecule in it. The chance for one molecule
is, therefore, adv; for none 1 — adv. In the first case » = pee a,
v
in the latter it is —a, thus
7 a
p= — a?
dv
or
de a u ED
Introducing this into (8), we find for the two elements w,y,z, and
Uelfsoe
Voy =S aga, — rs Ve Yrs 2¢—2r) . . . . . (10)
This result can be used to indicate the values of (W—N)? — A N?
for any volume.
AN = fra:
We have
DES af if v,? dv, dv, + fo. de, dye dz, dx, dy, dee
VV VV
from which applying (9) and (10)
AN’? =aV-+a fy (zeer Ye — Yrs Zl) dx, dy, dee der dye der.
VV
This holds for every size and form of V. Elaborating it for a
cube with side / the dependence on V is seen more clearly. Putting
Le — &, = Ye — Yr =, 2 — 2- —$, and integrating only. for Sns
positive, by which '/, of the integral in question is found then, we get
Mat hs at) | (5
AN? = Na sof (fen ff [a dy, dz,
000 Ent
ue
HN 4 8a ff (Oe Hn) + Gib nb +88) — Erb) 9 dE a
000 je
Hence
798
+1 El
AN? 6 “(la
Fe =|: +f {fo (ayz) dedydz — off Lel g dadydz
N l
ze} —l
Aaj 4-3
+3 {I = g da dy dz — 7 gde dydz.
ay ey
Every integral in this formula is always smaller than the prece-
ding one. If / is large with respect to the distance for which g has
an appreciable value, there remains only the first integral. For
any great volume we have
+a
AN?
— a14f[f ode dy a: te eres
N
3. In trying to determine the function / by means of statistical
mechanics, we meet with difficulties. Still something may be found
about the quantities »»- by applying the statistic-mechanical method
to our problem. Indeed statistical mechanics permit to introduce a
mutual action of the elements of volume.
We will avail ourselves of a canonical ensemble. We suppose the
molecules to be spherical and rigid, and to attract each other for
distances which are great with respect to their size. Elements small
with respect to the sphere of attraction therefore may still contain a
great number of molecules. But now we drop the supposition of the
sphere of attraction being homogeneously filled for all systems (or
at least for by far the greater part of them) *).
In ealeulating the number of the various distributions, we
must, for the potential energy of attraction, take into account the
mutual action of the elements; whereas, in calculating the exclusion
of definite configurations of centres, we may neglect the fact that
there is some correlation on the borders of the elements. For the
dimensions of the elements have been supposed large with respect
to the molecular diameter.
The mutual potential energy of the »-+ tr molecules contained in
an element dv, will be represented by
,@+t)?
ae aS
in this formula » represents the number of molecules contained in
the volume dv for the most frequent system. In this system the
distribution is homogeneous.
Poo
*) Cf ORNsTeIN, Toepassing der Statistische mechanica van GiBBs op molekulair-
theoretische vraagstukken. Diss. Leiden 1908, p. 43 and p. 110.
799
Of course, the potential energies will not strictly be the same for
different configurations within the elements, but we shall neglect this
complication. Further we will represent the mutual potential energy
for the two elements o and o by
— (v + te) (v + To) pz
dv
all elements of volume being put equal.
For the total potential energy we find, in this way
aig (VY Ht) So (v + Tp) ge
2dv
For the frequency & of a system with the given distribution of.
molecules we find
! me Slot), oen
SC 5 —(, d Vv) EEn . (wd Ein 2de
(we rt,
Here w is the function defined in the quoted dissertation on p. 48.
Supposing r < v and developing, we get,
il LIES l 1d _dlogw Px Z
— na i 4
20 Pee 1) af p da ? da r Odv Eik
S=C wr a-“e
Ee
The number of molecules per unit of volume represented there
by n, has been put a in this paper. The function @ and the faculties
are developed in the same way as in the quoted dissertation. The
double sum in the exponent gives the forms >.»>'1,p,,and 2T,2rg.
These forms are identical, as they consist of the same terms differ-
ently arranged, further 2,, is the same for all molecules and
Xr, = 0, consequently both sums vanish.
The constant C contains the factor Ne’/® along with quantities which
do not depend on the volume by summing up (12) over all possible
values of 7 (and taking into account that +r; = 0) we get N, the
total number of systems in the ensemble. So we find
2
w n
— — !
C
Oa ga
=o.
Vn 2200 3
the quantity A being the discriminant of the quadratic form in the
exponent.
When we write Zep =a, we find for the pressure p=
53
Proceedings Royal Acad. Amsterdam. Vol. XVII.
800
2
pn dlog® an
med N nn =
oO I dy 20 V?
Al: being very small with respect to the other factors, we may
neglect its influence in y'). The equation of state has the same
form as vaN DER Waars’ equation. However, the correlation is sensible
in the accidental deviation; for it changes the value of t°; and
vr, which vanish if the correlation is neglected, obtain values
deviating from zero.
Denoting by A and by A, the minors of the discriminant,
we have
So
ed )
where J is the number of elements into which the volume is divided.
2 Ee ld: AEG
Ihe condition A =O is equivalent to the condition 7 = 0. For
av
if we write down the determinant in some arrangement, and if we
add all rows to the first row, we get a determinant of which all
terms of the first row have the form
l ib a ; dlgw 1
a a 3
da Od
: ( = Pap.
Yp r da r
Strictly speaking, this is not true for some terms at the end of the row,
but as we have neglected the action on the borders, we may neglect
this fact too. In reality our considerations are only true for an infini-
tely great volume, where this difficulty disappears, as 4 is then an
infinite determinant.
Now if
ae dfo 1 pn
Y Y da Odv i
then A=.
Or if A=0O
1 _dfw 1
a? J De a=0
d a da 0)
which therefore agrees with Sen
u
1) Cf l.c. p. 129.
2) Cf. ORNSTEIN, Accidental deviations in mixtures. ‘These Proceedings 15,
p. 54, (1912). |
801
The quantities zt. ete. here found are related to those mentioned
above. And though a statistical deduction of the function / enter-
ing into details may lead to difficulties, yet it is clear that
statistical mechanics yield a correlation analogous to that expressed
in g.
If we should wish to continue the deduction of the conditions of
the critical point, we should have to use higher powers of t,, which
can be done without difficulty; we then find for the second condition
If we drop the supposition that the sphere of attraction is large,
we can use the function &, defined in the quoted dissertation. In
order to take into account the correlation, we must suppose the
integrals
~— & /
fe 10 dv, den Er aaa (n,)
defining &, to depend on nz for the element in question and also
D ? | k |
on the numbers of molecules in the surrounding elements. Therefore,
in general, the numbers of molecules of all elements will appear in
09,
On;
9n, , but the influence of distant elements is so small that
can be put zero.
By considerations analogous to those used in the quoted disserta-
tion, we can show that %(n,) has the form
ny Ny
V, (wos, ny, Dy)
in which n,n, denote the densities (molecular), in the elements
with which V, is in mutual action. The values of all n, are equal
for the most frequent system.
Now we find for ¢
Be Ran nd et
where P is a quadratie form in the deviations for the various
elements, containing squares as well as double products. The form
might be easily indicated, but we will omit it, as it is only our
purpose to show how in general the statistic-mechanical considera-
tions, changed in the sense of a correlation of elements of volume,
lead to formulae analogous to those given in $ 2. Here too the
mean square of deviation and the means of double products are
represented by quotients of minors of the discriminant of P and
—Q(Q the discriminant vanishes.
53*
dp
this quantity itself. Here too for
Uv
802
4. The above considerations can be applied in calculating the
critical opalescence. For that purpose we use the simple method
indicated by Lorentz'), which consists in superposing the light:
vectors caused by the influence of every individual molecule in a
point at great distance.
Consider in the substance through which a beam of light passes,
a volume J” great with respect to the wave-length, and take a distant
point P, the direction VP forming an angle g with the incident
TAY.
All molecules lying in one plane perpendicular to the line which
bisects the angle gy, will cause equal phase in P. Take therefore a
system of axes with the Zaxis parallel to this line, then the con-
tribution of one molecule will be
AR ;
B sin — (ct + 22 cos} pp)
ud
where 8 depends only on the kind of molecules, on 4 and on the
distance VP, u being the index of refraction.
The number of molecules in dv dy dz amounts to
(a + v) du dy dz.
The total light-vector in P thus becomes
5 2
p fod p) sin AA (ct +22 cos Eep) da dy dz.
ua
b
and the intensity
ple
2
Eel (at »,) (a 4-r-) sin = (‘t+-22, cos } p)
u (a
‚ar
sin — (ct +22. cos 4 yo)
wa
da, dy, dz; du- dy; dze.
Integrating with resp. to ¢, we get
1s = An j
oh B? fa? Ha (pz-p-) -|- verd cos a 2,--¢:) cos by | da, dy,dz,da,dyzdz..
VV
The mean value of this must be calculated. The term with », + ve
vanishes, and that with a? yields no contribution proportional to V.
We introduce the value of ».»- from formula (10), and for c=t
from form. (9). This gives
IH. fe LORENTZ, On the scattering of light by molecules. These Proceedings 13
p. 92 (1910).
7
803
An
ud
a BaV + = za f fa (Wo Urn Ya Yer Zr —Zz) CO8 — (2e) dt; dyn
VV
For a great volume one integration over V can be performed
(compare the deduction of formula (11)); further we put aV=N
_ Ax
and for the sake of brevity re =C, then we get
ur
+a
i ae a)
Ee B? ik ff foo Cz 9 (x, y, 2) docdya | pied Ps) REESE
The integral appearing here will be represented by G, that of
formula (LI) by G. It will be seen that the deductions criticised in
§ 1 yield an opaleseence proportional to r?,a quantity which accord-
ing to the above is proportional to i + G, whereas the opalescence
is proportional to 1 + G.
With the aid of the integral-equation (6) we can express G and
G, in the corresponding integrals of the function f, which we will
indicate by F and F..
Integrating (6) with resp. to «yz from —o to +, we find
ae = — ie.) aes
[fo (wyz) de dy dz {ff (S75) ds dy aff Je + Su + 12 + 5) dudydez —
+o
bai f (wyz) dndydz
F
oy ae a Vr ae ee ee 14
Den (14)
Multiplying (6) by cos Cz and -again integrating, we get
+ Ge
G, — | {fae asanazf | feo C (z+$) cos CO4- sin C (z HE) sin CS}
—- 0
f(et-S, yn, ¢+$) dudydz = F,.
The integral with the sines disappears because f and g are even
functions; we find
or
Fe
In order to apply the results obtained and to test them experi-
mentally, one might try to deduce f from molecular theory. This
would at best be possible under very simplifying suppositions and
er ch zalk By
Te
804
even then only an approximation can be obtained. Therefore we
will take another way. As remarked in § 1, the exact value of rv?
for very great volumes was already known. In our notation we have
fhe eT. a
TER AT
Vv emd
dv
where MN is the number of AvoGrapo, v the molecular volume.
According to formula (4) we have
re V=a(lt+@)=;
ay,
Putting these results equal, we get
v dp
1—_Fo=———.
RT dv
In the critical point #— 1. *)
The formula of opalescence first arrived at by Knesom and EINSTEIN
1) There appears to exist a closer correspondence between the given statistic-
mechanical method and the method using general considerations of probability, than
perhaps might be expected. The elements of the discriminant (which is an infinite
determinant in the former) agree with the function f in the latter. The former finds
from this the value of ve ve as the quotient of a minor with that discriminant,
the latter deduces this value from an integral-equation. In the critical point the
discriminant vanishes, corresponding to this the FrepHotm determinant of the
integral-equation is likewise zero. That this is the case when #’=1, appears by
more closely studying the equation
9g (515) — i fo (SS) f (+8, y+, +) dgdnd5 = 0
tt Sys EN
which only permits appropriate solutions if à = ak (ie. this is the only proper
value). For F =1 this is therefore the case with the equation (6) without second
member.
From the formula (15) it will be seen that form. (6) can be solved by a FouRIER
integral, Putting
+a
[ iL | cos mx cos ny cos lz f (ayz) dadydz = p (m, n, U)
g(m, n, l)
ii cos ma cos ny cos lz dmdndl.
8? l—p(mn, n, 1)
we have
g (zyz) =
805
5 |
„wl
Top. 22°V RI dv ep 16
— = = in? Se oe Sn
Fis DEN ip a ee)
in which represent
D distance of observation
u index of refraction,
w angle of electric force in incident light with direction of
observation,
will likewise be found by using in (13) the value found for /’ instead
of f.. The exact formula then will result by multiplying by
Nn
ed
Developing the cosine in #, we find
, 136;
1 7 ce 2 £ 5
F—F.= [fez (wuz) dudydz.
Representing this integral by « and introducing the value of C,
we get
the factor
o
.
P— F,=4n*(1 + cos) (5)
ur
The formula of opalescence then will be:
ES : 5553) eR
is eee du : aa)
Vv = sin”
Fop. hee V Gs dv Sy Aaa
Paese Net (Vicon)
(18) ")
The greater exactness of form. (17) as compared with (16) is
confirmed by the measurements of one of us (Z.). According to these
measurements, which however bear upon a mixture of liquids the
1) According to this formula the proportionality of the opalescence to 4—4,
which holds for higher temperatures, changes continuously in the immediate
neighbourhood of the critical point, into proportionality with 4-2, This real “getting
whiter’ of the opalescence should not be confused with the apparent changing of
colour which is always observed much farther from the critical point. The latter
indeed is only a result of the method of observation, as is clearly proved by the
measurements of one of us (cf, ZERNIKE thesis).
806
reciprocal value of a quantity proportional to the opalescence changes
linearly witb the difference of temperature TT, but by extrapolation
does ‘not vanish for 7 = 4 but More? Tj =O 01255 Wien
therefore for this value of 7—7Z}, the denominator of (17) is equal
to zero, we can find from this, using VAN DER WAALS’ equation, an
estimation for ¢/,. The calculation yields:
= 0,0022 or ¢ = 1,2.10-7 om.
The quantity « is a measure for the size of the sphere of attraction. For
ad
i= Affe f (eye) dadydz
(9 distance to origin) whereas in the critical point A
+o
[freer ae ay de |,
If f were constant within a sphere with radius A, then &’ would
be */, R®, and the above estimation would give
PD WO em
SU MEM sA B S¥e
1. The known formulae of critical opalescence give an infinite
value at the critical point. Efforts to escape from this difficulty have
furnished formulae for the deviations of density with a dependence
upon the volume, at variance with the assumed mutual independence
of the elements of volume. :
2. In order to obtain formulae applicable in the critical point, it
is found necessary to take into account the mutual influence of the
elements of volume, it being shown that near the critical point this
influence is sensible for distances large in comparison with the radius
of the sphere of attraction.
3. Two functions are introduced, one relating to the direct inter-
action of molecules, the other to the mutual influence of two elements
of volume. An integral equation gives the relation between the two
functions.
4. Corrected values are found for the mean deviations, and in
the formula of opalescence a correction is introduced. The latter
depends upon the sphere of attraction which can thus be calculated
from observations.
>. Further it is shown that the same results may be arrived at
by taking into account the mutual influence of the elements of
volume in the deductions of statistical mechanics,
Groningen, Sept. 1914.
A. A HIJMANS VAN DEN BERGH and J. J. DE LA FONTAINE SCHLUITER. „The identification of
traces of bilirubin in albuminous fluids.”
Bilirubin from human bloodserum (Chloroform-method).
Bilirubin from human ascites fluid Bilirubin from human ascites fluid
(Aether-method). (Chloroform-method)
Proceedings Roval Acad. Amsterdam, Vol. XVII.
807
Physiology. -- “The identification of traces of bilirubin in albu-
minous fluids.’ By Prof. A. A. HiJMANs VAN DEN BERGH
and J. J. pe LA Fontaine Scanvrrer. (Communicated by
Prof. H. J. HAMBURGER).
Several investigators have tried to demonstrate the presence of
slight quantities of bilirubin in albuminous substances, for instance
in normal human bloodserum. Most of them did this by adding
varions oxidizing substances, either directly to the serum or to an
alcoholic extract of the latter. The first oxidation-stages of bilirubin
having a green or a blue colour, the presence of bilirubin was
regarded as established if an addition of these oxidizing substances
gave rise to a green or a blue colour (OBERMAIJER and Popprr, STEIGER,
GarBerT*) and others). AvcuÉé®) employed a much more reliable
method based on the fact that bilirubin, in alkalic solution in the
presence of oxide of zinc, is changed, by careful oxidation with
iodine, into a substance with a characteristic spectrum. This reaction
had already been described by Sroxvis, but Avcné, who mentions
Srokvis’ work, owns the merit of having stated accurately the con-
ditions required if the reaction is to take place with absolute cer-
tainty, so that it may be used to demonstrate the presence of bilirubin.
Undoubtedly the reaction of Stoxvis-AucHE can be used with success.
Only the spectrum-line is very slight in the case of the small amounts
of bilirubin dealt with in this treatise: if the presence of bilirubin
is to be demonstrated in normal human serum by means of this
method, the layer of fluid intended for spectroscopic investigation is
to have a thickness of ten centimetres. And even then the result is
not always a positive one. For quantitative determinations this method
cannot be used.
Birrt extracted the serum at once with chloroform and carried
out his reactions with this *).
The reaction of ErnrricH has supplied us with an excellent means
of tracing bilirubin in bloodserum and other albuminous fluids and
of determining it quantitatively *). The characteristic difference in
colour between an alkaline and an acid medium increases its relia-
bility, whilst the reaction is an extremely sensitive one. It must,
1) OperMAER u. Popper. Wiener Klin. Wochenschr. 1908.
Sreicer. Dissert. Zürich 1911.
Gumpert. See for his werks the bibliography in: Clinique médicale 1910/1911.
2) Avené. Compt. rend. Acad. d. Sciences 1908.
8) Birri. Folia Haematolog. 1906 III. 189.
4) HijMANS van DEN Beren and Snapper. Deutsch. Arch. f. klin, Med. 1913,
808
however, be admitted that neither this nor any other colour-reaction
enables us to identify the presence of bilirubin with absolute certainty.
The possibility of other substances contained in the serum giving
the same reaction with the diazo-body may be esteemed less probable,
it cannot with absolute certainty be denied.
As far as we know it has hitherto been found impossible to
isolate bilirubin from normal human serum, which would have
afforded an incontestable proof of its presence. HAMMERSTEN has
attempted it?). But though in a great majority of cases he obtained
fine bilirubin-crystals from horse-serum, he never succeeded in ob-
taining them from normal human serum. From the terms used in
the latest edition of his Handbook of Physiological Chemistry we
eather that this investigator is not quite convinced yet of the presence
of biluribin in normal human serum.
With a view to researches on anhepatic bilirubin-formation we
needed a method which would enable us to identify with absolute
certainty the presence of small quantities of bilirubin in bloodserum,
exsudates and transsudates, if possible by obtaining the pigment in
the form of erystals. After some experiments we have succeeded in
this, starting from the property of bilirubin — which we have not
found mentioned anywhere — of dissolving readily in acetone.
To 10 em’. of bloodserum 20 em*. of pure colourless acetone are
added. An albumen precipitate is formed, which is centrifugalized.
The fluid at the top, coloured more or less intensely yellow, contains
all the bilirubin and only traces of albumen. This liquid is evaporated
in vacuo at the ordinary laboratory temperature.
If one has a good vacuum-pump at one’s disposal the liquid soon
begins to boil; after some minutes the acetone is evaporated. A watery
fluid remains in which, besides other serum-substances, all the bili-
rubin is dissolved. Then the fluid is shaken 2 times or more with
aether to remove the fatty bodies as much as possible.
These pass into the aether which is pipetted. The last traces of
aether are removed in vacuo. The aether may of course also be
removed by means of a separatory. Then a certain amount of chloro-
form e.g. 2 em*. are added, the fluid is slightly acidified with HCl
and shaken. The bilirubin then passes into the chloroform. By centri-
fugalization the watery fluid can easily be separated from the chloro-
form. The chloroform is washed thoroughly with water to get rid of
all the hydrochlorie acid and centrifugalized once more, the water being
removed by means of a separatory or by pipetting. Traces of water,
however, remain mixed with the chloroform, which sometimes renders
1) HAMMARSTEN. Maly’s Jahresber. 1878 II. 119.
809
the fluid slightly troubled. These traces of water are removed by
shaking with glowed sulphate of sodium. The latter is removed by
filtration. The result is a very pure solution of the yellow pigment
in chloroform (solution 4). It may be easily proved that this yellow
pigment is bilirubin.
1. If the chloroform solution is shaken with diluted KOH or NaOH
the pigment passes into the latter, while the chloroform loses its
colour (solution 5).
2. If now some acid is added till the fluid reacts distinctly as an
acid, then the fluid at tbe top loses its colour, the pigment passing
into the chloroform at the bottom.
3. If to the alkaline solution (see sub 1) HNO, containing some
HNO, is added, the result is the well-known colour-play of the reaction
of GMELIN.
4. If a slight quantity of a diluted iodine-solution in alcohol
(1: 100) is carefully poured on to the alkaline solution, a blue ring
is formed.
5. If to the alkaline solution first an equal volume of alcohol is
added, and then + of the original volume of the diazo-mixture ot
EnrricH, a red colour is the result. An addition of a few drops of
concentrated HCl changes the red colour into blue.
All these reactions together, prove conclusively that the pigment
obtained in the above way is indeed bilirubin.
Crystals of bilirubin can be easily obtained from the pure chloro-
form-solution (sol. A) in the following manner. The latter is poured
out into a watch-glass which is covered with another watch-glass
and placed in the ice-safe. The chloroform evaporates slowly and
on the watch-glass the microscopically visible, pretty, yellow bilirubin-
crystals are left. When HNO, containing HNO, is added, these
crystals present under the microscope the reaction of GMELIN.
We can also dissolve the yellow crystals again in some solvent
(chloroform, dil. NaOH ete.) and carry out the above-mentioned
reaction with them.
If one has no good vacuum-pump at one’s disposal the method
ean also be applied with the following modification suggested by
Dr. SNAPPER.
10 em* of bloodserum are precipitated with 20 cm’ of acetone.
The albumen-precipitate is centrifugalized. To the pipetted upper-
fluid some drops of water are added; then this fluid is washed
carefully with aether a few times, to remove the fatty substances
as much as possible. These volumes of aether are removed with
the pipette every time. Then some drops of ice-vinegar and 1 cm’
810
of aether are added to the fluid. All the bilirubin passes into the
mixture of ice-vinegar and aether, which separates entirely from
the fluid underneath. If this yellow-coloured aether is pipetted and
placed in an ice-safe in a loosely covered watch-glass, we likewise
observe that crystals are formed.
The accompanying picture is a micro-photograph of bilirubin
crystals which we obtained from the ascites-fluid of a heart-patient,
and from normal human serum.
Attempts to produce bilirubin-erystals by the above-mentioned
method from the intensely yellow-coloured serum of two icterus-
patients, led to a remarkable experience. If namely we placed the
chloroform-solution which, as appeared from various reactions, con-
tained much bilirubin, in the ice-safe, for the purpose of a slow
evaporation, the yellow colour at a certain moment when, owing to
the evaporation of the solvent the concentration had reached a certain
value, suddenly passed over into a green one, evidently by a change
of the bilirubin into biliverdin. The same phenomenon oceurred
when we evaporated the chloroform-solution in vacuo. It must be
distinctly understood that this occurred only with the solution obtained
from the serum of patients suffering from obstructive jaundice. We
cannot give an explanation of this phenomenon. Most likely the
ieterus-serum contains substances promoting the oxidation of bilirubin
into biliverdin.
Botany. — “Gummosis in the fruit of the Almond and the Peachal-
mond as a process of normal life.” By Prof. M. W. BrierINck.
(Communicated in the meeting of September 26, 1914).
lt has hitherto been generally accepted that the formation of gum
in the branches of the Amygdalaceae always is a process of patho-
logical nature. I have found that this opinion is erroneous, and that
gummosis occurs normally in the fruits of the Almond (Amygdalus
communis) and the Peachalmond (Amygdalus amygdalo-persica)
DunHamrL DuMONCRAU. ')
1) In some Dutch nurseries the peachalmond is simply called “Almondtree”.
The difference is in fact very slight as it consists only in the drying up of the
almond fruit before the epicarp opens, and the position of the flowers in pairs,
whereas the fruit of the peachalmond remains fleshy even at the dehiscence, and
its flowers are mostly single. Between leaves, flowers and branches no con-
stant differences are found.
GRENIER et Gopron (Flore de France T, 1, Pag. 512, 1848) call the peach-
EE
811
Contrary to what might be expeeted the phenomenon is the more
obvious as the trees are better fed and more vigorous. In specimens
on sandy grounds it can only be observed with the microscope.
As gummosis is the effect of a wound ‘stimulus, if is of import-
ance that this process also takes place in the normal development
of the healthy plant. The subject is moreover of practical interest.
All the chief facts relating to gum formation can almost unchanged
be applied to the production of gums in general, of gum resins,
and of resins, among which are substances of great medical and
technical value. As the study of the influence of parasitism has made
it possible to produce gum, and no doubt many of the other sub-
stances mentioned, in a more rational way than has been done till
now, a short review of the whole subject seems not superfluous.
Wound stimulus as cause of gummosis. Poisoning, and
parasitism also causes of this stimulus.
Gummosis in the Amygdalaceae is a process of cytolysis, whereby
young cells, freshly sprung from cambium or procambium, and
sometimes also young parenchyma, are more or less completely
dissolved and converted into canals or intercellular spaces, filled
with gum. In dissolved parenchymatous tissues usually remains of
not wholly disappeared cell walls are found; the gum of the phloem
bundles is more homogeneous, but always the microsomes of the
dissolved protoplasm are found. The nitrogen of the gum springs
from the dissolved protoplasm.
Formerly we proved *) that by such different causes as poisoning,
parasitism and mechanical wounding gummosis may be experimentally
almond Amygdalus communis var. amygdalo-persica. At present the name
Amygdalus persicoides (Koor, SERINGE, ZABEL) is also used, as in the Hortus
of the University of Leiden. The opinion that it is a hybrid is not sufficiently
founded. When grown from seed the tree seems constant (see MrlEr’s Conver-
sationslexikon, Articles “Mandel” Bd. 11, p. 853 and “Pfirsich” Bd. 13, p. 782, 1896)
and identie with the “English almond”, of which DARWIN reproduces a stone
(Domestication, 2nd Ed, Vol. 1, p. 858, 1875). The fruit is fleshy and bursts
open, the kernel is edible, not bitter. At Delft sowing experiments have been
going on a long time already, but under unfavourable circumstances. The root
cannot resist the winter temperature of the soil, hence, grafting on the plumtree
is required.
1) M. W. Beuerinck et A. Rant. Excitation par traumatisme et parasitisme, et
écoulement gommeux chez les Amygdalées. Archives Néerlandaises, Sér. 2, T. II,
Pag. 184, 1905. — Gentralblatt f. Bakteriologie, 2te Abt., Bd. 15, Pag. 366,
1905. — A. Rant: De Gummosis der Amygdalaceae. Dissertatie Amsterdam,
Bussy, 1906.
812
provoked in many Amygdalaceae, as almond, peachalmond, apricot,
peach, plum, cherry, and bird’s cherry.
But these three groups of causes may all be considered from one
single point of view, by accepting that gummosis is always the effect
of a wound stimulus, proceeding from the slowly dying cells, which
are found as well in every wound, as at poisoning and parasitism.
These dying cells may change into gum themselves, but besides, exert
their influence on cambium tissues to distances of some centimeters.
This distance-influence is the principal effect of the wound stimulus.
But poisoning by sublimate or oxalic acid, introduced under the bark,
can as well excite gummosis as an incision or a wound by burning
or pricking. Neither the dead cells nor the poison are the active
factors here; the stimulus proceeds from the slowly extinguishing cells,
so. that gummosts is essentially a necrobiotic process. Probably the
dying cells, after the death of the protoplasm, give off an enzyme
or enzyme-like substance, a lysine, fixed during active life, but, which
being freed by necrobiosis and absorbed by the young division produets
of the cambium causes their cytolysis. This reminds of the eytolysines
of the animal body, originating when foreign cells are introduced,
which liquefy the corresponding cells, for example the haemolysines
which dissolve the red blood-cells. Furthermore of the bacteriolysines
and of eytase, the enzyme of celiulose.
If the hypothesis of the existence of a “gumlysine” is right, — and
I think it is, — this substance must be of a very labile nature, for
when bark wounds are infected with gum, quite free from germs
of parasites, no more abundant gummosis is observed than at
mechanical wounding only. But a difference, however slight, will
certainly exist.
Gummosis produced by wound stimulus.
The influence of this cause is best studied in the following experiment.
A deep wound, penetrating into the cambium of a branch of
almond or peach, commonly soon heals completely, but it may be
that gum flows from the wound. This is the case when the trees
are in sap, thus in February or March at temperatures above 20° C.
and below 33° C. The experiment succeeds best with cut branches
in the laboratory. When the wounds are made in the open air in that
season no gummosis ensues, the temperature then being too low.*) In
') If the wounds are infected with Corynewm, an extremely copious gum production
follows in spring, as the parasite then finds abundant food in the branches. There
is, however, no season when wounds, infected with Corynewm, do not sooner or
later yield gum.
EN
815
summer the cambium of the still longitudinally growing part of.
young green branches may be caused to form gum by punctures or
incisions, but these wounds heal quickly, except when “kept open”
by Coryneum or other parasites.
As to thicker branches, wounded in spring, the micrescope shows
the following.
Around the wound a great number of gum canals are formed in
the cambium, about parallel with the axis of the branch, some centi-
meters long, which become the thinner and shorter as they are
more remote from the wound. The canals are separated by the
medullary rays, which are with more difficulty converted into gum than
the phloeoterma. All the gum canals together form a kind of net-
work, whose meshes are filled by the medullary rays. The whole
network has the shape of an ellipse, the “gum ellipse’, the wound
lies in the lower focus towards the base of the branch. The stimulus
extends over the ellipse, evidently farthest in the direction of the
branch, less far towards the base and sideways. So it may also be
said that the wound stimulus extends farthest opposite to the ‘“de-
scending” current of nutrient matter, following the phloem bundles,
or along with the “ascending” water-current, following the wood.
Evidently the gum canals are more easily formed in the better fed
ceils above the wound than in those beneath it, where the nutrition
must be worse. This is especially obvious in ringed branches. Wounds
in the cambium, directly above the ring produce much more gum
than those immediately below. ’)
Under ordinary circumstances the branches, after simple mechanical
wounding, are soon completely healed, and if the cambium at the
outside of the gum canals then again begins to produce normal
secondary wood, the gum canals may later be found back in the
wood itself.*) Evidently the healing takes place as soon as the
stimulus ceases, and so it is not strange that when it continues
by poisons or parasitism the gum production also continues.
1) The nature of the power, by which the food transmitting, ‘descending’
Sap current moves through the phloem bundles, is not known, It is thus not
impossible, that if the cause of gummosis is of a material nature, a lysine, moving
through the tissues, it is able to run in opposition to the “descending” current.
I think, however, that the extension of the stimulus does not go along the phloem
but along the xylem bundles and the young wood, with the “ascending” sap.
*) I have never seen distinct gum canals in the secondary wood, but accord*ng
to the descriptions they occur eventually.
814
Parasitism as cause of gummosis.
The connection between wounding and parasitism.
Wounds in peach branches treated with poisonous substances,
such as sublimate, produce gum much longer and more copiously
than the like wounds without sublimate. Other poisons have quite
the same effect. Now it is clear that the direct influence of para-
sitism on the organism must be sought in the action of some
poisonous substance. Hence it seems certain that what these three
causes have in common, namely necrobiose, or the slowly dying of
the cells surrounding the dead ones, is the base of gummosis, and
that parasitism, where necrobiose lasts as it were endlessly, must
be the most powerful instigator of the process.
That this simple view of the question has not yet taken root in
science is proved by the most recent treatise on our subject by
Mixoscu,') illustrated with beautiful anatomical figures. After the
publication of Dr. A. Rant and myself of 1905, he described the
relation of mechanical wounding to gummosis. But he did not think
of poisoning experiments, nor has he any belief in the influence of
parasitism on gum formation. Wiesner, in his recently published
paper on gums in the new edition of his “Rohstoffe des Pflanzen
reichs’”’, is also of the same opinion as Mikoscn.
For my object a short discussion of a few examples of parasitism
will suffice. |
The little caterpillar Grapholitha weberiana makes borings into the
bark of plum and apricot, and if the outermost corklayer is removed
by shaving it off, the butterfly finds so many fit places for deposing
its eggs, that the larvae creep in by hundreds and make new borings
from whieh later the gum flows out. These holes are coated with
a layer of slowly dying. cells, whence: the stimulus extends, which
produces the gum canals in the contiguous “cambium”. By cambium
I simply understand the not yet differentiated division products,
“young wood” and young phloeoterma. The necrobiotic cells, clothing
the continually extending holes in the bark, and the great numbers
of new individuals of the caterpillars, make the gum production a
chronical process. .
To explain the formation of the enormous quantities of gum
produced in this way, it seems only necessary to think of mechanical
wounding and not of any special excretion from the animal. But it
must be noted that the space, where the caterpillar lives during its
1) Untersuchungen über die Entstehung des Kirschgummi. Sitzungsber. d. Kais,
Akad. d. Wiss. in Wien. Mathem. naturw. Klasse. Bd. 115, Abt. 1. Pag. 912, 1906.
ek a a
815
growth, namely a vertical narrow canal in the innerbark, very neat’
to the cambium, could not possibly be imitated artificially.
s
Fig. 1. (Natural size). Gum producing peachalmond in September, whose
summit is cut off; the gum from the gum canals is afler drying, swollen by
moistening with cold water.
Much more common and interesting than the animal parasites are
the gum producing Fungi of the Amygdalaceae, five of which are
found in our country.’) The commonest and most vigorous is Cor-
1) Coryneum beijerinckii OupEMANS, Cytospora leucostoma PERsoon, Monilia
cinerea BoNoRDEN, Monilia fructigena BonorDEN and Botrytis cinerea PERSOON
(see Rant, l.c. p. 88). German authors also mention bacteria as instigators of
gummosis, I never found them.
a4
Proceedings Royal Acad. Amsterdam. Vol. XVII.
816
yneum beijerinckti Ovpemans (Clasterosporium carpophilum Apnrn.).”)
Pure cultures of Coryneum in bark wounds of almond, peachal-
mond, peach, cherry, plum, bird’s cherry, sloe, virginian plum, develop
with remarkable quickness and soon make the bark die off, evidently
in consequence of the secretion of a poison, Around the dead cells
the necrobiotie are found from which the stimulus issues, which,
penetrating into the cambium in the usual way, forms gum
canals in the young woud. Many mycelial threads of the parasite
itself are then. eytolised- and converted into gum. I think this fact
remarkable and a strong argument for the material nature of the
stimulus.
Undamaged branches are with difficulty infected by the parasite, but
it is easy, even by very slight -wounds and artificial infection, if
only the wounds be numerous, to obtain great quantities of gum.
This circumstance explains why nursery men dread wounds in the
trunks and branches of stone-fruit trees.
In the green shoots, especially of the peach, the formation of
anthocyan is observed in the enfeebled tissue around the wounds
infected with Coryneum when exposed to sunlight. *)
The supposition that secretion products of the parasitic caterpillar
or the Fungus could be the direct cause of the stimulus, is contrary
to the positively existing relation between mechanical wounding and
gunimosis.
Gum canals in the fruitflesh of almond and peachalmond.
To the preceding facts, long since stated, I wish to add the following.
Already in my first paper of 1883 I called attention to the circum-
1) BeEiERINCK, Onderzoekingen over de besmettelijkheid der gomziekte bij planten.
Versl. d. Akad. v. Wetensch. Amsterdam, 1883, — Contagiosité de la maladie de
gomme chez les plantes. Archives Néerlandaises, 1é Sér., T. 19, Pag. 1, 1886, —
CG. A. J. A. OUDEMANS, Hedwigia, 1883, NO. 8. — SACCARDO, Sylloge Fungorum,
Vol. 3, Pag. 774, 1884. — ApERHOLD, Ueber Clasterosporium carpophilum (Liv.)
ADERH. und dessen Beziehung zum Gummifluss des Steinobstes. Arbeiten der
Biolog. Abt. am Gesundheitsamte zu Berlin. Bd. 2, Pag. 515, 1902. ADERHOLD
has experimented with pure cultures of Corynewm, which | had made and sent
him. He himself has not executed any isolations of gum parasites. His determination
as Clasterosporium amygdalearum (LÉv.) is thus founded on the imperfect de-
scriptions from the older mycological literature, in which OupEMANS was no doubt
better at home than he. Like Linpau I reckon Clasterosporium to another family
than Coryneum.
*) The apperance of anthocyan in the light is commonly a token of diminished
vitality and often a consequence of necrobiose in the adjoining cells. Hence, wounds,
poisons and parasitism cause anthocyan production in the most different plants.
817
stance, that in the fruit-flesh of the peachalmond, and as I may add
now, also in that of the almond itself, there is a system of gum
canals, precisely corresponding to that of the vascular bundles. Of
these the phloem bundles are converted into gum canals by cytolysis,
either entirely or with the exception of the outer protophloem ; the gum
canal (gp Fig. 2 and 3) thus, is always immediately contiguous to
the woody bundle ei.
VALI
aud ult a |
Fig. 2 (3). Gum canals in the transversé
section of the fruit-flesh of a peachalmond :
ha hairs on epidermis; hw dermoidal tissue;
| bp chlorophyll-parenchyma; x] xylem bund-
les; ph phloem bundles; gp gum canals
sprung from phloem bundles.
Fa)
A
0 ad
Fig. 2 and 3 are reproductions from my
above mentioned treatises of 1883 and 1886.
7
%
of
SE aes |
The presence of gum in the canals of the fruit is easily shown.
In August or September the summit of a peachalmond fruit is cut
off and the fruit, or the branch with the fruit, is placed in water.
After some moments all over the section droplets of gum are seen
evidently issuing from the vascular bundles. As these bundles
are distributed through the fruit-flesh, running longitudinally and
transversely, and are partly reticulated, the number of droplets is
very great and they are of different size. In particular near the stone
they are big. If in August the gum is allowed to flow out in cold
water it dissolves completely or nearly so. In September the dissolving
is no more complete. By drying the gum, its solubility in cold
water gets almost lost, but it continues in hot water.
From lateral incisions also much gum flows out. In Fig. 1 the
drops are represented after drying, followed by swelling up in cold
water.
Although this gum does not only consist of dissolved wall material
54*
818
but also of cell contents, the microscope can only detect fine granules,
evidently corresponding to’ the microsomes of the protoplasm, which
are not dissolved during the eytolysis I could not find back the
cell nuclei in the gum, but in the cells of the not yet cytolised
phloem bundles, they are neither perceptible. As under normal cir-
cumstances the gum does not flow out, its volume must be about
as great as that of the phloem bundles which are cytolised. It is,
however, certain that the capability of the gum to swell up by
imbibition is much greater than that of the cell-tissue which gave
rise to its formation. It seems thus certain that imbibition with
Fig. 3 (360). Gum canal with surrounding;
gp gum; «xl xylum bundles, unchanged; ph non-
dissolved cells of the phloem bundles; cd thread-
shaped cells in a gum canal, originating from the
phloem bundles.
sufficient access of water must lead to a perceptible pressure and
also some thickening of the fruit-wall. This must promote the
opening of the fruit as well as the remarkable detaching of the
stone, although the required mechanical power for these processes
inust, no doubt, chiefly be the tension of the tissue of the paren-
ehyma of the fruit-wall existing independently of the gummosis.
Finally the stone is found quite loose within the fleshy shell, which
mostly opens like a bivalvate mollusk, but sometimes shows three
or four fractures. The vascular bundles, which pass from the fruit-
flesh into the stone, are thereby torn off clear from the stone. At
the base the separation seems provided for by an intercepting layer,
as at the fall of leaves.
819
The portion of the phloem bundles within the stone of the peachal-
mond is never converted into gum; in the almond itself such gum
is found in rare cases inside the shell.
Wound gum in the fruit-wall as a consequence of mechanical
stress of the tissue. Gumming almonds,
In many cases real wound gum is found in the fruits of the
almond and the peaehalmond, not proceeding from the gum canals
but from fractures in the parenchyma of the fruit-flesh. Its origin
must undoubtedly be sought in the tension or stress of the tissue,
which causes the opening of the fruit. An additional circumstance,
however, is required, namely a loss of vital strength, by which the
regenerative power of the tissue that coats the fracture is annihilated.
The therefrom resulting incapability of regeneration is associated
with the ripening of the fruit in a way not yet explained and
should rather be attributed to superfluous than to poor nutrition.
Parasitism is wholly absent in the production of wound gum from
the parenchyma of the fruit.
The fracture is mostly at the side where the two edges of the
carpels are grown together and the fruit later opens. Not seldom
in this case is wound gum seen to flow spontaneously from the
base of the fruit along the short peduncle. In other cases the wound
is at the side of the middle nerve of the carpel. Always the edges
of the fracture are coated with cells in a condition of necrobiose,
which is evident by their quickly colouring brown at the air, which
normal living cells do not. These necrobiotic cells and the adjoining
tissue produce gum. With the microscope not quite dissolved cell-
walls may be found in the gum, showing that the cells were about
full-grown when the process began.
In common almonds gum is sometimes found within the hard
shell, *) and eventually part of the kernel itself is then also changed
into real wound gum with still recognisable remains of the cell-
wall. In such almonds the phloem of the vascular bundles, which
run through the stone to the funiculus, is always changed into a
gum canal, so that the gum can reach the surface of the young
seed.
If we suppose that gummosis originates by the action of a cytolvsine,
it seems very well possible, that the lysine which has flowed inward
together with the “canal gum”, is able to attack the developing
1) The small quantity of gum found, especially in “hard almonds”, at the
surface of the shell, proceeds from the gum canals of the fruitflesh. The sugar
layer which covers the shell of the “soft” species is dextrose.
820
seed and is yet too labile to be demonstrated by infection of bark
wounds with gum. Experiments in this direction may perhaps be
effected with the peachalmond.
Wound stimulus as factor of development.
Formerly I thought that the presence of gum canals in the fruits
was accidental and should be explained by parasitism, although I
could not find any parasites.
In later years, with better knowledge, 1 again examined the gum
canals in the peachalmond and their surroundings repeatedly. Never
did I find a fruit without them, but they were not equally developed
in different trees from different gardens. In specimens of sandy
grounds they can sometimes only be found with the microscope.
Neither microscopically nor by experiments has it been possible
to detect gum parasites. This makes it quite certain that in the
formation of gum canals parasitism is excluded. *)
The great ease wherewith mechanical tension causes wounds in
the fruit-flesh of the peachalmond, gives rise to the supposition, that
the normal gum canals may be the product of some hidden wound
stimulus.
If this supposition is true, we cannot think of wounding in
the common sense of the word. When the flowers fall off, a
ring-shaped wound forms around the base of the young fruit,
but this is a normal process, taking place in an_ intercepting
layer and soon followed by complete healing. In the flowers of
peach, plum, apricot, cherry,, we observe the same without any
formation of gum canals in the fruit-flesh. Moreover, although the
peculiar structure of the layer between the woody peduncle and
the stone, along which the ripe fruit detaches, reminds of rent
tissue, no gum is formed at that spot and the layer also exists in
the other stone-fruits, where no gum canals occur.
So long as nothing else has been proved it must therefore
be accepted that in the phloem bundles of the fruit of the peachalmond,
where cytolysis takes place, the same factor of development is active
as that, which gives rise to the pathological gum canals in the cambium
of the branches. This leads to the conclusion, that the wound stimulus
belongs to the normal factors of development of this fruit, although
nothing is seen of external wounds. When considering, that the
phloem bundles are built up of extremely thin and soft-walled cells,
') The supposition, sometimes met with in literature that the gum of the Amyg-
dalaceae should consist of bacterial slime is quite erroneous. That parasitic bacteria
eventually occur as gum parasites, as is stated by some authors, I do not think
impossible, although till now I only found caterpillars and Fungi as active agents,
821
it is conceivable, that by great tension of the tissue in the surrounding
parenchyma, they undergo strain and pressure causing mechanical
rupture and necrobiose, centre and prey of the wound stimulus
being the phloem bundles themselves.
This conception is in accordance “with the fact that the gum
canals are broad in the fruits of well-fed trees on rich grounds,
which have a hard and solid flesh, wherein stress and strain are
certainly very great. Only here and there remains of the protophloem
along the gum canals are still to be found in such fruits. But in the
softer fruits of sandy soils, along the much narrower gum canals
not only the protophloem is still present, but also stripes of the
secondary phloem.
Summarising we come to the following conclusions.
Mechanical wounds in growing tissues of Amygdalaceae will some-
times heal directly, sometimes after previous gummosis.
The chief tissue, which is transformed into gum is the young
secondary wood newly sprung from the cambium and not yet
differentiated. By the wound stimulus a network of gum canals
is. formed around the wound. In thick branches, with a bark wound,
this network has an elliptical circumference, the wound being in the
lower focus of the ellipse,
If the stimulus is removed by the cure of the wound, the cam-
bium again continues to produce normal secondary wood, so that
afterwards the gum canals may be found in the wood itself.
If the stimulus continues the gum formation also becomes lasting.
The stimulus issues from the cells that die slowly by wounding,
poisoning or parasitism. Probably a eytolysine flows from these cells
into the young wood or the procambium; these bind the lysine and
liquefy to gum. Hence, gummosis is caused by necrobiose.
Young medullary rays and phloembundles are with move difficulty
converted into gum than the young secondary wood. But in the
fruit-flesh of the almond and the peachalmond it is the phloem
which changes into gum. The protophloem of the bundles often
remains unchanged.
Although gummosis in these fruits belongs to their normal develop-
ment, a wound stimulus is nevertheless active. This stimulus springs
from the strong tension in the parenchyma of the fruit-wall, which
gives rise to tearing, necrobiose and gum formation in the delicate
tissue of the phloem bundles. Consequently the wound stimulus is
here a normal factor of development.
It might also be said that the almond and the peachalmond are
pathological species, but thereby nothing would be explained.
822
Chemistry. — “The allotropy of Lead.” I. By Prof. Ernst COHEN
and W. D. HELDERMAN.
(Communicated in the meeting of Oct. 31, 1914.)
1. Indications concerning the existence of allotropic forms of
lead are found not only in the earlier chemical literature. Fourteen
years ago Ernst Conen’) pointed out in his studies on tin a clause
in Puurarcu’s (50—120 A.D.) Symposiaca (VI, 8) in which allusion
is made’) to the fact that lead is sometimes disintegrated spontaneously
at low temperatures.
This clause runs as follows: “No, the craving for food is not caused by the
cold, but in the body something takes place similar to that which happens with
metals in a very strong winter. There it is seen that cooling not only causes
congealing, but also melting, for in strong winters axovat boatiBdov (pieces of
lead) occasionally melt away, consequently something similar may be supposed to
PE
take place in the intestinal process, etc...
Moreover THEOPHRAST (390—285 B.C.) mentions such phenomena in his book
zegì args: “xattiteooy ydg padi zai woABdor Jy raxijvar eV TO
Iövro adyov zai yewdros ortog veavizot, yadxov dé Oayijvar.”
(It is told that tin and lead melled sometimes in the Pontos when it was very
cold in a strong winter, and that copper was disintegrated.
2. Sarnre-Crairm Devittn*) stated that the density of lead is a
function of its previous thermal history. He gives the following
figures (water at 4° = 1; Temp. ?) a
After quick cooling of molten lead 11.363.
u slow ys As ES » 11-254.
In a second experiment he found:
Density of lead electrolytically deposited 11.542.
After melting and rapid eooling 11.225.
About the value 11.542 he says:
“Mais telle est la rapidité avec laquelle se carbonate à l'air ce plomb extrême-
ment divisé, quil a fallu le transformer en sulfate pour en déduire ensuite le
poids de la matière employée. Cette complication introduil-elle quelque incertitude
sur le premier nombre, ou ne doit on pas plutôt l'admettre comme représentant
la densité de ce plomb parfaitement cristallisé ?”’
3. These values as well as others given in earlier literature have
to be accepted with reserve as generally no data are given about
1) Proceedings of the meeting of Jan. 26, 1901, p. 469. Zeitschr. f. physik.
Chemie 36, 513 (1901).
2) PLurarcHt Chaeronensis varia scripta quae moralia vulgo vocantur. Lipsiae,
ex officina Car. Tauchnitii 1820. Tomus IV, 339,
35) C. R. 40, 769 (1855).
Prof. ERNST COHEN and W. D. HELDERMAN, „The Allotropy of Lead I”.
(Natural size).
Proceedings Royal Acad. Amsterdam, Vol. XVII.
823
the purity of the material experimented on and as there often
exists some uncertainty concerning the method whereby the density
has been determined.
4. Kanipaum, Rota and Srepier') found the density of a pure
Oo
20
specimen of lead prepared by distillation in vacuo to be d En 11.341.
5. The values given by different authors for the specific heat of
lead vary within wide limits, as may be seen from the table given
in Arrae’s Handbuch der anorganischen Chemie.*) Moreover it may
be called to mind that Lr Verrier’) stated, that the specific heat
of lead is a function of its previous thermal history.
6. The facts mentioned above as well as the investigations of
Storsa, ‘) and those of Orro LEHMANN ®) render a new investigation
of the subject very desirable.
7. A year ago we carried out some experiments in this direction.
As the results were negative we experimented with other metals,
which yielded a more favourable result. Since a fresh investigation
on lead has given positive results, as will be proved below, we
give here also a short description of our earlier experiments, which
taken together with the new experiments furnish a confirmation of
the results obtained by us in the case of other metals (bismuth, cad-
mium, copper, zinc, antimony).
8. Our experiments have been carried out with lead which
contained only 0.001 per cent of copper and 0.0006 per cent of
iron (Blei-‘‘KAnLBAUM ’-Berlin). ®)
The metal was turned into shavings on a lathe and washed with
dilute nitric acid, water, alcohol and ether. After this it was dried
in vacuo over sulphuric acid.
9. The density of this material was found to be
25°
d Fe
Goble,
b. 11.330,
We put the metal into an aqueous solution of PbCl, and kept it
for 48 hours at 100°. After washing and drying it, we found:
1) Zeitschr. f anorg. Chemie 29, 177 (1902).
2) Bd. 8, 2te Abteilung, p. 633 (Leipzig 1909).
3) Comp. Ernst CoHEN, Proceedings 17, 200 (1914).
4) Journ. f. prakt. Chemie 94, 113 (1865); 96, 178 (1865).
5) Zeitschrift f. Kristallographie und Mineralogie 17, 274 (1890).
Ernst CoHEN and Kartsusi INouvr, Zeitschr. f. physik. Chemie 74, 202 (1910),
6) Mrumws, Zeitschr. f. anorg. Chemie 74, 407 (1912),
824
d —~ ne. A320.
de AL. S285
Another part of the original material was melted and chilled in
a mixture of alcohol and solid carbon dioxide. The determination
of the density gave the following result :
Ko
d ae ge. 11.330,
Beten on
10. As our determinations had been carried out with an accuracy
of 3 or 4 units in the third decimal place, it is evident from the
experiments described above that we had not been able to detect
any transformation in the lead experimented with.
11. However, some months ago Mr. Hans Herrer at Leipzic was
kind enough to call our attention to some phenomena which he
described in the letter which follows :
“Gelegentlich eines Vorlesungsversuches, der einen sogenannten ‘Bleibaum”
zur Darstellung bringen sollte, bereitete ich eine Lösung von 400 gr. Bleiazetat in
1000 ee. Wasser unter Zusatz von 100 cc. Salpetersäure (spez. Gew. 1.16), die
als Elektrolyt bei der Bleiabscheidung diente. Als Elektroden dienten bei dem
Versuch Stücke aus reinem Blei. Diese Bleistücke bliehen nach der Eiektrolyse
etwa 3 Wochen in der Lösung stehen. Als ich sie alsdann herausnehmen wollte,
bemerkte ich, dass sie ihre weiche, dehnbare Beschaffenheit völlig verloren hatten
und eine spröde, bröckelnde Masse geworden waren. Der Gedanke, es hier mit
emmer stabilen Modifikation zu tun zu haben, erschien mir um so wahrscheinlicher,
als das spröde Blei ganz dem grauen Zinn gleicht, beide Metalle zu der gleichen
Gruppe des periodischen Systems gehören und Metastabilität unserer Metalle nach
Ihren Forschungen nichts Befremdliches mehr ist.
Kurze Zeit darauf brachte mir ein Kollege ein Bleikabel, das an verschiedenen
Stellen eine weisse pulvrige Beschaffenheit zeigte von ganz ähnlicher Art, wie ich
sie an den vorhergenannten Bleistücken beschrieb. Wir machten darauf den Ver-
such reine Bleistücke unter konzentrierte Salpetersäure zu bringen und sie mit ein
wenig unseres spröden Bleies zu impfen. Der Erfolg blieb nicht aus: nach
wenigen Tagen hatten sich betriichtliche Teile der Bleistücke zu der bröckligen
Modifikation verwandelt,”
Mr. Heruer kindly invited us to continue these investigations ;
repeating his experiments with our pure lead we were able to corro-
borate his statements.
12. The lead was melted, chilled in water and cut into small
blocks (3,5 X 2 x 0,5 em). We put them into glass dishes which
were filled up with the solution mentioned by Herrer. The dishes
were covered with glass plates. The temperature of the solution was
15°—20°. The addition of some nitric acid has the effect that the
surface of the metal remains bright during the experiment.
825
In this way the electrolyte is in constant contact with the metal
and the inoculation which occurs can go on undisturbed.
The photographic reproductions (natural size) which accompany
this paper illustrate the development of the phenomenon. Fig. 1
shows a plate of pure lead in its original condition. Fig. 2 represents
the plates after having been in contact with the sotutión for some
days; there are to be seen deep cracks, which show that the material
has shrunk locally. In consequence of this an increase of the density
was to be expected which was proved by means of the pyenometer
(comp. § 19). Fig. 3 shows the plates after three weeks in the same
conditions: the metal has been disintegrated.
Repeating the experiment with 15 or 20 blocks we got in all
cases the same results.
13. We shall prove below that the phenomenon is not a chemical
one; the following experiment may give already an indication in
this direction. One of the blocks (+ 40-grams) was put into a cali-
brated tube which had been filled up with the solution mentioned.
This tube stood in a small dish containing the same solution. After
three weeks no evolution of gas had occurred, either at room tem-
perature or at higher temperatures. |
14. After this the phenomena described above were investigated
by means of both the pycnometer and the dilatometer.
A. Measurements with the Pycnometer.
15. We exclusively used the instrument (Fig. 4) described by
Apams and JonNstTon'), following the indications given by the authors.
Moreover we took the precaution of dipping the pycnometer into
water before weighing (empty) and wiping the
water off with a dry cloth. If this is omitted a
slight error occurs. as the surface of the pyenometer
is then not in the same condition as at the sub-
sequent weighings, after it has stood in a (water)
thermostat.
All determinations were carried out in duplicate
with two pycnometers (C and D) which contained
+ 25 cem.
16. Our investigations on bismuth, cadmium ete.
SS gy
NW
had shown that the pyenometer measurements have
to be carried out with special care. The volume
changes which accompany the transformation of the
Fig. 4,
1) Journ. Amevic. Chem. Soc. 34, 563 (1912).
826
different modifications are, it is true, not inconsiderable, but they
may be partially compensated in consequence of the simultaneous
presence of different forms. In order to detect the remaining volume
changes, very accurate determinations of the density must be made.
We shall see below that special precautions must also be taken
with lead. Evidently it is to be ascribed to such compensations that
these phenomena have escaped the attention of earlier authors.
17. We used toluene as a liquid in the pyenometer.
Its density was found to be:
25°
d ra 0.86013 by means of the pyenometer C.
0.86013 „ ORE Ee ee e D:
The quantity of lead used for each determination was 40 — 60
grams. The thermometers (divided into 0.05 degrees) had been checked
against a standard of the Phys. Techn. Reichsanstalt at Charlotten-
burg-Berlin. The weighings were carried out on a Bunrer-balance
with telescope. The weights had been checked by the method
described by TH. W. RicHarps ').
18. In the first place we determined (at 25°.0) the density ot
the lead immediately after its preparation for the experiments.
It was melted, chilled in water and filed to powder. It was then
treated with a magnet in order to remove traces of iron from the
file. We washed the powder with dilute nitric acid, water, alcohol
and ether, and dried it in vacuo over sulphuric acid. Its density
was now:
go
4)
dre 11.325, (Pycnometer C).
The metal was then washed and dried again in the same way;
KO
de 11.322, (Pycnometer D). After treating again in this way we
ane
found d—— 11.324, (Pyenometer D).
4° 3
19. We brought the metal into the solution of the acetate (temp.
15°). After standing for 3 weeks the material was washed and
o
25
dried. Its density was now d 5 11.340, (Pyenometer C)
11.342, (Pvenometer J).
1) Zeitschr. f. physik, Chemie 33, 605 (1900).
Bor
The figures show that there has occurred at 15° an increase of
17 units in the third decimal place.
20. We put the metal again into the solution which was kept at
50° (in a thermostat) for 120 hours. We found after washing and
drying
pas
le)
d ES 11.313, (Pycnometer C)
11.312, (Pycnometer J).
The density had decreased 28 units in the third decimal place.
21. The experiment was repeated again, this time at 25°.0 (in a
thermostat) for 144 hours.
ORO
We found: d/ : 11.327, (Pyenometer C°)
4 ;
11.329, (Pvenometer D).
An increase of 15 units in the third decimal place had occurred.
22. Our table I contains the results of these determinations:
BAAN ET ce oh
a eaf 25°
| LE
| EE
Without any previous treatment 11.324
After treatment at 15° or „341
5 je „ BO? ESL)
= a ae 11.328
B. Measurements with the dilatometer.
23. This investigation was carried out in the same way as has
been described in the case of cadmium *).
Some kilograms of lead were melted in a spoon and poured out
into an iron form. The metal cooled in contact with the air. After
filing it we treated it with a magnet and put it into the solution of
the acetate. Here it remained (at 15°) for 15 > 24 hours. After this
it was washed and dried in the way described above. We used
+ 600 grams in the dilatometer. (Bore of the capillary tube 1 mm.).
1) Proceedings 16, 485 (1913); Zeitschr. f. physik. Chemie 87, 409 (1914).
828
At 50°.8 the decrease of the level was 700 mm. (84 hours).
22 74°.4 EE) rise EE) 99 EE) EE) 275 EE) (2 EE) je
Whilst the first preparation ($ 20) had shown at 50° a decrease
of density, we now find an erease. From this result we may
conclude that there are more than two allotropic forms simultane-
ously present.
24. Special attention may be paid to a phenomenon which we
observed with all our preparations and which stands in close con-
nexion with the fact that lead as it has been known up to the
present, forms a metastable system containing simultaneously several
allotropie modifications of this metal.
It is generally known that when a bar of any metal which is
more electro-negative (resp. eleetro-positive) than lead is suspended
in a solution of a lead salt, the lead is thrown out of solution and
a lead tree is formed.
We found that the same phenomenon occurred when our pure
lead was placed in the solution mentioned above or in a (neutral)
solution of lead nitrate (30 grams of nitrate, 70 grams of water).
Soth at room temperature or at higher temperatures (50°) a lead
tree was formed in a few days.
25. We are in the case of lead in specially favourable circum-
stances for the observation of this phenomenon. The galvanic current
which is generated between the stable and metastable modification
decomposes the solution. The metal which is electrolytically deposited *),
shows in this case a characteristic form (lead tree) so that the
phenomenon is very striking ’).
26. We hope to report shortly on the different pure modifications
of lead and their limits of stability.
Utrecht, October 1914. VAN ‘rm Horr- Laboratory.
‘) That the phenomenon is not to be attributed to the presence of iron (0.0006
per cent) or copper (0.001 per cent) is proved by the investigations of OBERBECK
{Wied Ann. 31, 337 (1887)] and by those of KöNIGSBERGER and Mürrer [ Physik.
Zeitschr. 6, 847 and 849 (1905)].
>) We also carried out an experiment with tin: white and grey tin were put in
contact in a solution of SnCl, (Temp. 15°). After some time a great many
beautiful crystals of white tin were deposited by electrolysis upon the white metal.
(Comp. ERNsT COHEN and E. GoLpscHMip7, Zeitschr. für physik. Chemie 50, 225
(1905) |.
829
Mathematics. — “On an integral formula of Stmurses.” By Prof.
J. C. KrLuyver.
(Communicated in the meeting October 31, 1914),
In the Proceedings, and Communications, Physical Section, series
3, 2, 1886, p. 210, Srizites treats of definite integrals, referring
to the function
i= —=3
]—y" nl
In this function a stands for a positive odd integer without qua-
h |
dratic factors, and (- represents LEGENDRE’s symbol with the ex-
ha NA
: ( \m ave
= 1
hl \C ys (") ee
a
tension given to it by JACOBL.
Iik
‘As poles of the function f(y) only the points y=e are to be
taken into consideration, and for the residue, belonging to such a
pole, one finds
i a7 Zelk hal ie 2rihk
— —eé a Des ee e a
a h=1 a
From the well-known fundamental equation
Qrihk
h=a—1 /)j, 3 (= =F i k
Sef he at ah 8 — | Wa
n= a a
it follows, that a pole is only to be found in those points y=
in which 4 is prime to «. Consequently y= 1 is not a pole of the
function, and we have,
5 ] kzal / jh
» fMW=-—- 2 (1)
a hi a
from which it follows that —a/f(1) is equal to the sum of the
u
| ; h
numbers smaller than a, for which ( =-+1 (residues), diminished
. . . . U
With the sum of the numbers smaller than a, for which { — J—=—1
cl
is (non-residues).
In the paper quoted, Stipnrses considers the definite integrals
an
oa ‚att atx
fi (e—*) sin as da and Jr —2) cos — da,
al An
10
830
and he calculates the value of the first integral for the casé
a= 4w + 1, the value of the second integral for the case
a = 4w — 1.
In the following I give a shorter deduction for these results.
I suppose that the two positive, otherwise arbitrary numbers 3
and y have a for produet, that ¢ is a positive parameter and now
consider the integral er Shp
pen en
ey es E ) arie da.
In order to valeulate this integral, it is not necessary „as STIELTJES
does, to fall back on an integral formula treated by LrGENDRE and
by Aber. It need only be observed that in the upper balf of the
complex a-plane for increasing values of |t, the modulus of the
mitegrand approaches sufficiently rapidly to zero, to permit us to
equate the integral / to the sum of the residues in this upper half
plane, multiplied by 222.
TL
The poles of the integrand are the poles of f (cet that is to
. ki :
say the points «= —(4= 0,1, 2,...), where & is prime to a. The
P
residue of such a pole is
gee h=a-1 (hi Weds | (c= ) ran
ne Od pe jet ee (Say Ve dee
any A a Any a
hence
— a—1\? REDA : oxkt PE er? Dal
A | 5 EE al : ) = fe) Sree B al Z ) ern).
We ought to distinguish now between the two cases a = 4w +1
and a= 4w— 1. °
For a= 4w+1 we have
he —h — +5
El = + (=) and conse§uently f\e Pf J=—fle f J,
a
so that it follows from the result found for /, that
* oo mee /B _ ent
fil 7 ona ntede=yVe 7(. r). (a=4w dl) ..@
Y
On the other hand for a@—=4w—-1
nx
2nz
h —h ETE aps
6 — — (=) and consequently / (« Ê ) = df (- : ) ’
a
831
so that in this case it may be concluded from the integral formula that
2e 2x (Tt et ont
fel’ B ) cos 2nta dx — 4! (- 7 ) ae (a= tif Dees (EI)
y
0
As may be proved the equation (II) remains true if we suppose
t=O, and if the expansion in series
tol
|
~.
2rx 2amx
c as 5 m =o m = EPE
Wee bee OL eg
mater
is made use of, we get in this exceptional case
et m Liede ppd a i f
m=1\4 my Pe armas Ee 5 L. (== W— )
The results found by Srieurses have been derived with this, the
equations (1) and (II) may now, however, be used, to find other
results less known in the theory of numbers.
For real values of « the function / (e-“) has the property of approach-
ing rapidly to zero for positive and negative values of x of increasing
modulus. This leads to the conclusion that Fourier’s general sum-
mation-formula
fe +2
n—+n > 1 =o
= F(§-+n) =|F (y)dy +2 = | F'(y) cos Zan (y—8&) dy
n= 0 fi 0d
—1
may be applied, if we write
nr
i (7, (, 2 ia
and if we suppose OS §< 1.
Distinguishing again the cases a=4v+1 and a=4w—1, the
value of the integrals in the right-hand member may be determined
by means of the equations (I) and (II). It should be taken into
consideration in the summations in the left-hand member, that f (e¢—7)
changes its sign together with « or not, according as a is equal to
dw +1 or to 4w— |.
In this way the two following general equations are derived from
the summation formula.
= One Ed _ 2x(n+5) _ 2n(n—§) \
F(¢ ke = i(: Ê )-#(¢ Ê ie |
|
oie ; UD
| i p 1 f heh TREE
a= 2 = sncansf)é@ 4 |, (a= Aw + 1)
a
Proceedings Royal Acad. Amsterdam. Vol. XVII.
1 n= ore) ile
(EEA EE JN
la Mi (IV)
=/= ra) jn ok mijl 7 ) , (a=4w—1) |
n=
If in both members of these equations the functions fare expanded
into series, the summations indicated are to be executed still further.
1 shall, however, perform these reductions only for special values
of the parameter §, in consequence of which the general results are
simplified.
In the equation (III) I substitute therefore §—1, at the same
time I replace 3 by 5 and accordingly y by 2y. I further write
kK 7
e k= q; e v= q.
The numbers q and g’ are then positive and smaller than 1;
they satisfy the relation
9
2
En
log q X log q a
but are for the rest arbitrary.
In this way the equation (II) Sue into
= (—1)" f(g) = | = a (—1)" f(q'2"+),
n=0
and if the functions f are my into series, we shall find
r
y nd ii = (=) Gots | f | yea, 5 q ss 4
OY SS) SSS Og — < see | a= wl x
Gama aya +- go q m=1 1 sg En )
In the equation (IV) I substitute $—0. We have then in the
first place
n=
fa) + 2S 70% ig hi +2 570%},
and if again use is made of the expansion into series of the functions
i we find
y ise Up a O+ IS ie (=) nm alee ie uy
pa, Jg
RER i 1 M= '2m
SS an (1) he ere (a = 4w—1) . (VI)
q m=) lg
The equations (V) and (VI) completely symmetrical with regard
to gq and q are again conspicuous for the remarkable properties of
833
the arithmetical symbol Ee ) For the rest they show some similarity
a
with formulae in the theory of the functions, and point to a
' l
certain connection between the functions 9 (« -) and o(» ~).
f %
So it may be observed, that, from the equation
| Me (» *)
: : rey M= qm
v, (0 a v, (0. ) 5 — ( - 4 = Ten cos on an
; ov v Yn m1 1 +o
4 ’ B
in the case a= dw +1, ensues
9 R.. %
z 1 h=a-—1 h 3 is ’ B ; m= /m gr
v, 0, — va, U Dn, a — —~_—=4ya Zz ae —,
8 B h=t a IE 2 ) aN 1+ q2"
tg (lege
a
and the equation (V) proves that the expression
eet
5 . v, Te hie
1 t db em. ds G B
—%,(0,—]9,(0,-])- 2 [- =. (a = 4w + 1)
vg B B) nt \a G 5)
Bil =s
a
remains unchanged, if 8 is replaced by y.
In a similar way we conclude from
z
a, (« 5) oA
B M= ger à
—=zxcetvmt4ia XE — sin 2 mv
a (- =) m=1 gr
; B
for the case a = 4w —1 to
fe xt
=a—-1/p de 5) 1 is —i/h, ah en m gen
hen Vela ZOT te aleseee
as Vig a Ad & pda m=i \ a /1—q?"
a, a B
We can prove now, that
h=v—-1 /h
Pp G jee == Aaf (i)
fl
consequently it ensues from the equation (VI), that the expression
a 6 1 )
h=a—1 /] i ae
: = (= RUE (a = 4w — 1)
VB ia \a ae)
a’ B
holds its value, if 8 is changed into y.
834
Physies. — “On unmiving in a binary system for which the three-
phase pressure is greater than the sum of the vapour tensions
of the two components.” By Dr. F. B. C. ScuerreR. (Commu-
nicated by Prof. J. D. van DER WaaLs).
(Communicated in the meeting of Sept. 26, 1914).
1. In my investigation on the system hexane-water the remarkable
phenomenon presented itself that the three-phase tension of the two
liquid layers by the side of gas appeared to be greater than the
sum of the vapour tensions of pure hexane and pure water *). When
the tensions of the pure substances at a definite temperature are
denoted by P, and P,, the three-phase pressure by /, then for
temperatures which are not too far from the critical end-point
Pond
If we could speak of “partial pressures” for such an equilibrium,
this result would be impossible. A proof of this is found in VAN DER
W aars KoHNsTamM's “Thermodynamik”, which however is only
valid when the gas-laws*) hold for the saturate vapours. When the
gas-laws do not hold for the gas phases, in other words if the gases
possess surface layers, the proof is not valid, and the statement that
the three-phase pressure must always be smaller than the sum of
the vapour tensions of the components, holds therefore only for
rarefied saturate vapours.
In my paper on the system hexane-water I have shown that the
contradiction with the second law of thermodynamics, which at first
sight may be supposed to exist in the observed phenomenon, is only
an apparent one. We might, namely, be inclined to reason as
follows: If the three-phase mixture possesses such a high pressure,
the pressure exerted by the water and hexane molecules, or at least
that of one of them will have to be greater than the pressure of
water vapour, resp. hexane vapour over the pure components. If we
therefore bring the three-phase mixture by means of a semi-permeable
membrane into contact with pure hexane and with pure water
under their own vapour pressure, bexane or water will pass through
the semi-permeable membranes from the three-phase mixture towards
the pure liquid. We should then get splitting up of the three-phase
mixture, whereas just on the contrary the so high three-phase pressure
sets in of its own accord from pure water and pure hexane. This
is in conflict with the second law of thermodynamics.
" 1) These Proc. 16. 404. (1913).
*) Thermodynamik. IL. S. 476.
835
I have shown in the cited paper that the conclusion that the
three-phase mixture will expel water through a membrane permeable
to water, is really correct, and that probably mutatis mutandis the
opposite thing will apply to hexane. The error in the above reasoning
lies therefore only in the very last conelusion. I have pointed out
loe. eit. that it is, indeed, possible that two liquids, each under its
own vapour pressure, mix to a three-phase mixture that possesses
the property to get unmixed again into the pure components through
semi-permeable membranes; that this is not in contradiction with
the second law of thermodynamics, but that on the contrary this
phenomenon will be frequently met with in my opinion, also for
systems which do not present the special behaviour mentioned at the
head of this paper. Thus solutions of gases which are but sparingly
soluble in water will certainly expel water, when they are osmotic-
ally brought in contact with pure water of the saturate tension. The
observed phenomenon is therefore not in conflict with our theoretical
considerations.
2. To get an answer to the question whether the system hexane-
water presents an exceptional behaviour in the appearance of a
three-phase pressure which is greater than the sum of the vapour
tensions of the components, I have investigated a number of other
systems in the hope of findiug the remarkable phenomenon there too.
First of all I have chosen the system pentane-water. The pentane
which I had at my disposal, was however KaAnrBauMm’s “normal
pentane’, which is no pure normal pentane, but a mixture of normal
and isopentane, which can only be separated with great loss of
substance and time, as the boiling-points of the two substances lie
near room-temperature, and differ only little (slightly more than
8 degrees). This slight difference of boiling-point involves that the
pentane mixture behaves pretty well as a pure simple substance;
the isothermal pressure ranges for condensation are slight. I have
therefore given up the separation of the two pentanes, and compared
the vapour tensions of the pentane mixture and of pure water with
the three-phase tension of a pentane-water mixture. It is clear that
both the pentane mixture and the three-phase mixture must possess
a tension dependent on the volume at constant temperature, but
also the three-phase tension appeared to be only little dependent on
the volume. To execute this comparison of the pressures as exactly
as possible I have determined the pressures for final condensation
and for about equal volumes of gas and liquids both of the pentane
mixture and of the three-phase mixture. >
836
TABLE I. TABLE II.
5 5 if er ant Wee |
Pentane mixture. Threephase mixture.
Pressure (atm.) Pressure (atm.)
Tem: Tem-
Le arene 48 rg VG Re A VE ea VG
15123 17.63 150.15 22.45
151.9 Lies 150.7 22e
161.1 20.6 160.6 2.29
161.45 20.8 160.7 27.35
169.95 23.7 | 166.5 30.3
170.1 2a. 170.25 32.00
180.1 27.6 | 110.55 32.65
180.3 Ee | 180.3 39.0
190.25 32.15 | 180.5 39.25
190.3 32.05 171 | 44.1
19323 | 33.6
When the values of pressure and temperature indicated in the
above tables are graphically represented, it appears that the line
for the end-condensation coincides fairly well with that which holds
for equal liquid and vapour volume both for the pentane mixture
and for the three-phase mixture: the difference is nearly everywhere
smaller than 0.1 atmosphere, and is therefore of about the same
value as the errors of observations. When the pressure values are
read for definite temperatures from the erapen! representation, the
values of table 3 are found.
oe oe BA ENT
| Three-phase Pentane :
| Temperature pressure pressure Waterpressure Difference
150 22.4 Liss 4.7 0.4
160 27.0 | 20.3 6.05 0.65
170 32.4 23 7.8 0.9
180 38.8 27.6 9.8 1.4
187.1 44.1 30.7 11.6 Las
837
The values for the vapour tension of water have not been derived
from earlier observations, but determined by myself to prevent an
eventual deviation of the thermometer from vitiating the comparison.
All the observations have been carried out with a normal thermo-
meter, and with an Anschützthermometer verified by the boiling
point of pure aniline.
We draw the conclusion from the last column of table 3 that
the three-phase tension is again greater than the sum of the vapour
tensions of the pentane mixture and of pure water. The difference
appears again to be greatest at the critical endpoint — in all the
tables the critical values are printed in bold type —; with decrease
of temperature the difference decreases rapidly, and according to
the theory it must reverse its sign at temperatures where the saturate
vapours follow the gas laws.
The above described example shows therefore again a case of
very high three-phase pressure. Though these experiments would
have to be repeated with the pure substances to get perfect certainty
about the behaviour of the binary systems, the conclusion that the
pentanes and hexane behave analogously with respect to water,
seems yet sufficiently certain to me. Also the relative situation of
the critical end point with respect to the critical points of the com-
ponents is the same as for the hexane-water mixtures.
Finally I will still point out that the above only proves that there
exists a pentane-water mixture that possesses the repeatedly men-
tioned remarkable property, and this suffices also for my purpose;
other proportions of pentane mixture and water will probably give
rise to some change in the three-phase tensions because the pentane
mixture is not a simple substance; for the solubilities of the two
pentanes in water will probably not be in the same proportion as
the quantities of the pentanes in the pentane mixture; the difference
in the fifth column can therefore undergo some modification for
another ratio of the two “components”.
$ 3. The experiments of $ 2 confirming my supposition that the
abnormal value of the three-phase pressure would be a phenomenon
of frequent occurrence, | thought I had a great chance to find the
same peculiarity also for other binary systems. I have therefore
looked for binary systems of which it was known that for low
temperatures the threephase pressure lies higher than the vapour
tensions of the pure components separately and is about equal to
the sum. Dr. BicuNner drew my attention to the systems carbon
tetrachloride- water and benzene-water, which possess three-phase
838
tensions according to Rreraurr, which deviate little from the sum
of the vapour tensions of the pure substances. REGNAULT even asserted
that the tension of carbon tetrachloride-water mixtures is somewhat
higher than the sum, and thought he had to ascribe this to slight
contaminations; GrrNez has shown later that the three-phase tension
is really slightly smaller than the sum of the vapour tensions, which
is therefore in harmony with the theory. I have now tried to inves- —
tigate the two systems at higher pressure; I have, however, not
suceeeded in doing so with the system carbontetrachloride-water, as
the components act on each other at higher temperatures. The inves-
tigation is possible for the system benzene-water, and also this system
appeared really to furnish an example of the remarkable phenomenon.
Benzene free from thiophene (negative isatine reaction) was distilled
from phosphorus pentoxide; the boiling point under normal pressure
was 80°.2, and was therefore in perfect concordance with the value
given by Youre. The vapour tension line of this benzene was deter-
mined, and then the three-phase tensions of a benzene-water mixture
were measured and compared with the vapour pressure line of water,
which was also determined by the aid of the same thermometers.
To avoid corrections I have measured the three pressure values
TABLE IV.
| Temperature | en | Waterpressure | poe | Difference
150.0 | 10.6 | 4.7 5.9 . 0
160.0 13.2 6.05 14 0.05
170-0 Pets, | a6 a ata 1.8 8.5 DER
18020). a OA 9.8 | 10.2 | 0.1
| 190.0 fae 12.35 12.15 0.1
lice 750050. sees aOR ht 455 |. 14.8 0.2
210.0 eds | 18.75 | 16.7 0.45
220.0 42.9 | 22.8 19.45 0.65
230.0 | 50.9 ALPS 22.5 0.9
240.1 | 60.35 33.0 25.95 1.4
250.2 | 70.65 39.1 | 29.55 2.0
2001) ide 8215 allie eae 33.7 2.35
267.8 | 92.7 ‘calc.(52.4) calc.(37.35) 2.95
gS Yara ke ES [i Sees 37.5 —
839
always at the same temperatures; if a slight error should occur in
the absolute value of the given temperatures, this has no intluence
on the pressure differences. The thermometers which I used in this
investigation, have been tested by a resistance thermometer, the
resistance of which was determined for boiling water, naphthaline,
benzophenon, and sulphur. The obtained results are given in table 4,
the pressures are given in-atmospheres.
It appears from the last column of table 4 that the difference at
150° to 200° is only slightly greater than the errors of observation,
that the three-phase tension becomes appreciably greater than the
sum of the vapour tensions at 210°, and that this difference rapidly
increases with ascending temperature.
4. When we combine the results of the system hexane-water
and those of § 2 and 3, it appears that in the three systems the
three-phase tension is always greater than the sum of the vapour
tensions of the components in the neighbourhood of the critical
endpoint. Moreover these three systems present the same shape of
the plaitpointline in the 7-z-projection; the upper critical endpoint
always lies lower than the critical points of the two components ;
the plaitpoint line presents therefore a minimum temperature in the
7T-x-projection (homogeneous double plaitpoint). Though in my opinion
it is probable that the systems will behave perfectly analogously, a
further investigation would have to decide whether for all this
homogeneous double plaitpoint lies in the metastable region; Ll have
shown this for the system hexane-water in my cited paper. It is
remarkable that in the system ether-water the homogeneous double
plaitpoint appears in the immediate neighbourhood of the ether axis
or would perhaps lie outside the figure, so that the critical endpoint
in contrast with the above discussed systems lies between the critical
points of the components. In this system the said peculiarity does
not occur. Accordingly I think I have to conclude that the systems
which present critical endpoints which lie lower than the critical
temperatures of the two components possess three-phase pressures
which are higher at high temperatures than the sum of the vapour
tensions of the pure substances, whereas the opposite is the Case for
systems for which the critical endpoint lies between the critical
temperatures of the components. Perhaps this conclusion may contribute
to account for this remarkable phenomenon.
Anorg. Chem. Laboratory of the
June 25, 1914. f { j
University of Amsterdam,
840
Physics. — “Contribution to the theory of corresponding states.”
By Mrs. T. Exrenrest-Aranasssewa. D. Se. (Communicated by
Prof. H. A. Lorentz).
(Communicated in the meeting of September 26, 1914),
§ 1. Mersin") has tried to demonstrate that every equation of
state which contains the same number of material constants as
variables, is to be reduced to a universal shape (i.e. to such a form
that no parameters occur any more which vary with the substance),
if the variables are replaced by their relations to suitable special
values, which may be designated as “corresponding” for different
substances.
On closer investigation it appears, however, that the equality of
the number of the parameters and that of the variables is neither
necessary nor sufficient for the existence of corresponding states.
A method will be given here to decide whether a given equation
allows the existence of corresponding states. This method furnishes
at the same time the possibility to calculate the eventually corre-
sponding values of the variables for different substances.
§ 2. In the first place we shall define the term ‘corresponding
states” in a somewhat more general form. Let an equation be given
between a system of m variables: z,,,,... & and a number m of
such parameters: C\, C,,...Cn that they can vary with change of
definite cireumstances (for evample of the substance).
Let an arbitrary system of special values: 2,',2,',...%,' (we shall
briefly denote it by «;’) of the variables 2; be known, which satisfies
this equation for definite special values C, of the parameters Cj.
Let us introduce the following new variables:
et a, ep: vs me Un 1
Uy — Tj , Ys — 3 1 » Wee ed Yn — Sr . . . . ( )
L, 2, Tone
All the constants S; of the thus transformed equation can be
calculated as functions of the former constant coefficients, of the
values Ci’ and of the values aj’.
When the parameters C; assume other special values C;", other
systems of special values of the variables will satisfy the original
equation.
The case may occur that there is among them such a system of
values :
}) Mestin: Sur l’équation de vAN DER WAALS et la démonstration du théorème
des élals correspondants C.R. 1893, p. 135.
that on the substitution of
WB ee one ee ne Mea)
for wi, the constants of the transformed equation assume exactly the
same numerical values S; as in the first case. We call such values
Boden aoe corvesmonaent tot the palues ai! a la x,, and the
state defined by the values 2;", correspondent to that defined by the
values 2 (or corresponding to it).
The form to which the given equation is reduced in this ease
Ui Li
by the substitution y; = —, resp. i ae will be indicated by the
Ti wi
word universal.
$ 3. When for the system x; the system 2;" corresponding to it
has been given, the system zj” can be easily calculated, which cor-
responds with every other system zji’ of z; values, which satisfies
the equation in the first case, by the aid of the following equations:
| "
Lj ae
Vil ay" ;
Indeed the values 2;' resp. 2;" satisfy the original equation, when
the parameters C; assume in it the values C,' resp. Ci". When now
the substitution
Ui
kn veer ete an de onee ONE
Li
has been carried out, the constants S; which we have calculated,
assume other values, e. o. Si, and we must now find the values 21",
which keep the quantities S;; invariant on substitution of C;" for Ci’,
when the substitution:
Ge
vas (4)
"
; Uit
is carried out.
The values #;', however, satisfying the given equation,
1
Br
' —
Wilgen
Vit
satisfy the transformed equation. The constants of the transformed
equation do not change, when
! !
&; Li
is substituted for vj.
842
The fraction:
vi EE
vi ei
belongs therefore to the corresponding values w;", hence ej, corre-
sponds to wi.
Hence it is proved that in the case of a-system of values corre-
sponding to a system of solutions, there also exists a system corre-
sponding to every other system of solutions (when C;’ have been
replaced by Ci").
§ 4. To find a system 2;", if the system w,' has been given, we
take into account every product of powers of the variables:
Et ae Snilin <2) es,
which appears as separate argument in the given equation. We
shall therefore write the given equation as follows:
B(K PA IPs ae rs ee. Se =O ee
K; and Z; are constants with relation to 2;, L; are those constants
which do not oceur as factor of Pi, but in any other way. Among
the A; and L; are therefore also included the variable parameters
(for their functions).
Let us put that the constants A;, Z; in the first, resp. second case
have the special values:
Ki, Di, resp. Ki" pie"
(those among them which are independent of C;, have the same
values in both cases); they are to be considered as given. We can
write every variable also in the following way :
‚Ze !
Es = Ui = Ui Yi.
!
Ui
If we put them in this form in the equation (7), it assumes the
following form:
DQ Pis WEL 0 =
in which
Qt GE NEN EE EE
Pi = ag CR a oO. ee
PQ) = y fg Pea ee ee
Now it is evidently the question to find such values 2;" that when
Ci' is replaced by C;" and 2;' by 2;", all the constants Q; and Z£;
— eventually with the exception of one factor, by which all the
terms of the equation can be divided — assume the same values.
813
When we carry out this division — let the factor in question be
R (it can be both one of the ( and one of the Z;) in all 44/1
constants remain, which can have four different forms:
Aan B
Je R ’ ey
The required 2;" must now satisfy the following equations:
Qf' —= QQ
a i Os
Dt ver EL Tete GHS
Des i Bi
R' “ea R'
and besides the following equations must hold:
nn er ste Ua ea Ale SE ed Fo PLY
"
The number of equations (12), in which a," occurs, is quite
independent of the number of m of the variable parameters C;.
When all equations (13) are satisfied, and all those among the
equations (12) which do not contain 2;", the three following cases
can occur.
1. Equations (12) are in conflict with each other (a group of s
of the sought values is defined by more than s independent equations.
2. They have one, or a finite number of systems of solutions. (It
is required, though not sufficient for this that the number of independent
equations in which 2;" oecurs, is equal to n. Hence m must not be
greater than 7).
Which of the systems of solutions corresponds with the given
system w;', has to be decided by a further investigation in every
separate case.
This is the case in which we have corresponding states.
3. They have an infinite number of systems of solutions. (It is
required for this that ” is greater than the number of the equations
that are mutually independent). In this case we may speak of corre-
sponding states for the same conditions (e.g. for the same substance).
$ 5. We shall now examine how Mersin has come to another
conclusion. Mesrin starts from the conviction that all the constants
of an equation are independent of the choice of the unities, when
every variable in the equation bas been divided by a special value
of it. This is perfectly correct. [t is also true, as we have seen, that
every equation can be reduced to a form as meant here.
It is however not true that those constants that do not change
throuyh exchange of the unities, would also have to be unive. sal.
844
Mrsran seems to be not quite free from a confusion, which is indeed
pretty widely spread: between the change of a number occurring
in an equation through change of unities, (‘‘formal” change) and its
change through transition to other conditions (to other specimens of
the quantities which are measured by this number) (“material”
change).
In connection with this the assertion that in case of an equal
number of variables and parameters the latter can always be com-
pletely expressed in the former, is to be rejected.
$ 6. We shall illustrate what we have discussed by examples, |
which though fictitious, are as simple as possible. Their claim to
physical signification, can indeed always be vindicated in this way
that they are interpreted as equations for the geometric shape of
some physical system.
4; yaar datb (Nn ==rde 2);
a. Introduction of special values of the variables
y ‘hag x
oti ac,” a ae U, zie b
0
0 vo
v
b. Division by Q,=y,:
2 2
y EEN Eer b
=a mh Rakesh arene en
Yo Yio Sp Yin En Yo
c. Determination of the numerical values of the special values
of the variables satisfying the equation and of the coefficients :
1
LE 5 Vb
a
ax,” 1
y. ab
Ly 1
y ad
b
ee
y
d. Determination of the system of corresponding values:
Or eee
Yo ab Yo =d
ep
Ji ab : ab
b' aor. a.
a = —,
Yo oy a*b* ab
845
from which would follow that a'b'= ab, which would be possible
only when we have really but one independent parameter.
It follows, however, from the thesis of $ 3 that if for one system
of solutions there is not to be found a corresponding one, there
does not exist one for any other system of solutions.
Hence the given equation cannot be reduced to a universal form.
2. y—=ar + abe + b° (n= mass)
y ONS a
a. Yo =a,’ | — |] + abe, — + DB?
Yo vo vo
A y se as (=) | abs, a 4 b*
y 0 y 0 vo y 0 vo y 0
—b
C Yo = — b? ; U, rn
a
2m 3 a b?
A ae en? ey ae A ena
Yo Yo Yo
b'
d ¥, = — 6° ’ B =S
a
3 y=? Ja n= Ose = lh)
x 2
a „5 = U (2) SE U, i
Yo vy vy
; y =“ (=) ve
Yo Y vy Yo Xo
1 2
C A nn ’ Yo = T
a a
El el a:
OE LE
1 2
d ea : And
a a
41). pvz=AtBTtCT (n= 3, m == 0)
p v 7 In 2
a. Pov =A BT, — HCT | =
Poo is LS
See. eee Date mA ie ED ee ES Noe ere Dee ae
Bie yao |
As 3 8 independent of GC T°, the two last comparisons are contra-
dictory, so that even if A= A’, we should not have corresponding
states.
Leiden, August 1914.
1) This example fails in the Dutch text.
846 ;
Chemistry. — “The nitration of the mixed dihalogen benzenes”.
By Prof. A. F. HOLLEMAN.
(Communicated in the meeting of Oct. 31, 1914).
When in benzene are present two substituents and a third
is introduced, the Substitution velocity caused by the two groups
already present is unequal. From the data given in the literature
it may be deduced that those velocities for the substituents pointing
to the p-o-positions decrease in the subjoined order:
OH > NH, > halogens > CH,.
The question now arose how to express those velocities also in figures.
Dr. Wrisaur has done this for chlorine and methyl by determining
in what proportion the isomerides are formed in the nitration of
o-chlorotoluene. In this compound the positions 4 and 6 are occupied,
under the influence of methyl, by a nitro-group, the positions 3 and
cH, 5 under the influence of chlorine. If now we determine
esa the proportion in which the mononitro-chlorotoluenes 4 + 6
5 3/ are present in regard to the iscmerides 3-+ 5 in the
\YY nitration product, this is then also the proportion of the
substitution velocities caused by methyl and chlorine, because they
can exert their action in this o-chiorotoluene independently of each
other; for the positions which are substituted under the influence
of methyl are different from those that are substituted under the
influence of chlorine. For this proportion was found CH, : Cl=1:1.475.
Dr. vAN DEN AREND had previously determined the proportion in
which the nitro-p-chlorotoluenes are formed in the nitration of p-chloro-
toluene. If now, with the above mentioned ratio, we calculate the
relative quantities, those calculated figures appear to agree approxi-
mately with the observed ones.
These researches, carried out in my laboratory, have now been
continued, partly by Dr. HEINEKEN, so as to determine also the ratios
of the halogens. The method followed previously for the quantitative
determination of the isomerides, namely by means of the solidification
curves, could, however, not be applied here as the two nitro-p-
chlorobromobenzenes give a continuous series of mixed crystals and
because it was to be expected, on account of the fact that the
properties of the nitrodihalogen benzenes present a strong mutual
resemblance, that this would occur in other eases also.
Hence, for the quantitative determination of the isomerides present
in the nitration mixtures, we made use of the property that a halogen
in nitrohalogen benzenes is taken from ihe nucleus by Na-methoxide
847
only then when it is placed in the ortho- or the paraposition in
regard to a ee Of the isomerides
cl Cl Cl
a ed Br ZN Bete
7 ie
TORRE
NO.
I, IV and VI only B of IL, If and V only bromine will be
replaced by OCH,. Hence, if we determine the proportion in which
chlorine and bromine are split off from the nitration product of
p-chlovobromobenzene and from that of o-chlorobromobenzene, we
get at once the proportion wherein in the first nitration product
I and II are present in the second one IV + VI on one side, UI + V
on the other side. This method has also the considerable advantage
that all the isomerides for the construction of the fusion lines
now need not be prepared individually and that in the nitration
products the various isomerides need not be separated; this latter
attempt in particular would, presumably, have failed owing to
insuperable obstacles.
The results obtained are as follows:
Nitrationproduct of p-chlorobromobenzene contains 45.2°/, of the
Cl
ANNO:
compound , and 54.8°/, of the isomeride 1, 3, 4.
Noe
Br
Cl Cl
Br NO, \Br
Nutr. product of o-chlorobenzene consists of 55.5°/, of + k
| No, he
cl
jn caer
and of 44.5°/, of | 3 ; or in molecular proportion 1 :0.80,
ssi ey
The substitution velocity caused by chlorine and bromine when
present together in the benzene nucleus is therefore as 1: 0.80.
When calculating this proportion from the composition of the nitration
product of p-chlorobromobenzene it must be remembered that in the
nitration of chlorobenzene there is formed 30.1° , of the o-compound,
but in that of the bromobenzene 38.3°/, of the same. If we call
« the ratio of the velocities caused by chlorine and bromine we have
30.1: 38.317 —= 45.2 : 54.8,
or
(op
Proceedings Royal Acad. Amsterdam. Vol. XVIL.
848
from which a= 0.96. Hence, the result is here Cl: Br =1 0.96,
The mean result of these two experimental series is therefore:
Gl Br = 10:88:
The nitration of p-chloroiodobenzene caused the separation of large
quantities of iodine with formation of p-chloronitrobenzene. Hence,
for my purpose it was unsuitable.
In the nitration of o-chlorovodobenzene, there was also some separation
of iodine, but the formation of o-chloronitrobenzene did not amount
to more than about 3°/,. When determining the proportion in which
Cl and I were separated from the nitration product by NaOCH, a
correction for this must therefore, be applied. For the velocity ratio
Cl: I was thus found the mean value of 1:14.84.
It now became interesting to also investigate the nitration product
of o-bromoiodo benzene quantitatively as to its components. For, as
Cl: Br ‘was found :==4#:0:80 and..Cl-1l=4171:84 Bret ishoae
be = 1: 2.30, if indeed the two halogens present, act quite indepen-
dently of each other.
Also in this nitration a little separation of iodine took place; the
content of o-bromonitrobenzene in the nitration product was in
this case 4.4°/,. Applying a correction for this the mean ratio
Br: L=1:1.75 was found, which rather differs from the calculated
figure. If, however, we calculate the percentages of the isomerides
with the ratios 1.75 and 2.30 the theoretical value gives 69.7°/, of
the isomerides Br, 1, NO, = 1, 2,3-+ 1, 2,5, and the experimental
value 63.6"/, which may be considered as a sufficient approximation
if we bear in mind the difficulties of these quantitative determinations.
The conclusions from the above are obvious. Since it has
appeared that two substituents simultaneously present in ortho- and
in parapositions do not sensibly interfere with their respective
actions in regard to a third entering group, we shall be able to
calculate from the figures now found with sufficient probability in
what proportion are formed the isomerides of other compounds, for
instance in the nitration of o- and of p-bromotoluene.
The above mentioned order of the substituents towards the decreasing
substitution velocity caused by the same now becomes :
OH > NH, >I >Cl> Br > CH.
Hence, the ratio OH: NH, and NH,:1 still remains to be deter-
mined. As, however, in the nitration of the iodoanilines great diffi-
culties may be expected, A. F. H. Losry pe Bruyn has taken
in hand a quantitative research of the nitration of o- and p-chloro-
aniline in the above direction. As a preliminary result of his experi-
849
ments it may be mentioned that the ratio Cl: NH, is very large.
The above velocity series must, therefore, be resolved into two parts:
OH and NH, which cause a great substitution velocity and which
are presumably of the same order of magnitude; on the other side
the halogens and CH, with a lesser velocity, also of the same order
of magnitude. A more detailed description of the above experiments
will be published in the Recueil.
Oct. ’14. Org. Chem. Lab. University Amsterdam.
Physics. — “The reduction of aromatic ketones. UI. Contribution
to the knowledge of the photochemical phenomena.” By Prof. J.
BörseKEN and Mr. W. D. Conrn. (Communicated by: Prof An
F. HoLLEMAN).
(Communicated in the meeting of October 31, 1914).
I. The reduction of the aromatic ketones in a perfectly neutral medium.
per.
In our former communications ') we have shown that the reduction
of the aromatic ketones does not proceed any further than to pinacone,
which is presumably formed from the primary generated half pina-
cone molecule by rapid polymerisation. The fact that in an alkaline
medium hydrol is always obtained, must be attributed to the rapid
transformation of the pinacone, under the influence of the hydroxyl-
ions, into an equimolecular mixture of hydrol and ketone, the latter
of which can be again reduced to pinacone.
This explanation was confirmed by the study of the reduction of
ketones by means of aluminium amalgam.
Here is formed a mixture of pinacone and hydrol; the proportion
in which these two substances are formed differs from ketone to
ketone and now it appeared that the quantities of hydrol ran strictly
parallel to the velocities with which the diverse pinacones are con-
verted into a mixture of ketone and hydro! under the influence of
sodium ethoxide.
Hence, aluminium amalgam in 80°/, alcohol may by no means
be considered as a neutral reducing agent.
The only modus operandi that gives the necessary guarantee that
complete neutrality would prevail during and after the reduction is
the action of the aromatic ketone on an aleohol under the cooperation
of sunlight. The original intention of this part of the research,
1) Proc. XVI p. 91 and 962 (1913).
56*
namely the tracing of the progressive change of the reduction, was
soon attained by applying this method.
A series of ketones dissolved in a great variety of alcohols and
a few other substances, was exposed to sunlight (or to the light of
the quartz-lamp) ; in all cases where reduction set in, not a trace of
hydvol was obtamed.
The ketone was usually quantitatively converted into pinacone ;
occasionally, namely with benzylaleohol and a prolonged exposure
to sunlight a combination of the half pinacone molecule with a
group of the benzylaleohol, namely triphenylglycol was obtained as
a by-product. ’).
When to the alcohol some ethoxide was added hydrol was formed,
as was to be fully expected.
Hence, we arrive at the result that in the reduction of aromatic ketones
the hydrogen unites exclusively with the oxygen.
The experiments were carried out as follows :
Quantities of 5 grams of the ketone were dissolved in 30 cesar
aleohol rendered carefully anhydrous *) and exposed in sealed tubes
of common glass to direct sunlight.
The drying of the lower terms was performed by successively
boiling with CaO, allowing to remain over metallic calcium at 0%
and distilling; the higher ones were purified by distillation and both
were then immediately sealed into the tube together with the ketone.
After exposure to the light for some time, during which the course
of the reduction could be traced by noticing the deposition of the
sparingly soluble pinacone, the tube was opened, the pinacone was
filtered off, the filfrate distilled, the residue united with the pinacone
and in the distillate the aldehyde or ketone was tested and in
some cases determined quantitatively.
The exact details will be published elsewhere by one of us, a
few remarks may suffice here.
First of all was investigated the behaviour of benzophenone in
regard to methyl, ethyl, n-propyl, sec.-propyl, tso-butyl, n-heptyl,
sec.-octyl and cetylaleohol. The latter only was not attacked, not
1) This had already been noticed by CramrcraN and Sier (B. 36, 1577 (1903)) ;
the formation thereof is moreover a confirmation of our conception that as the
first reaction product the half pinacone molecule is formed.
2) Water acts in this reaction in a remarkable manner as a powerful negative
catalyst; in 80°/) alcohol no reduction takes place after exposure for months,
whereas in absolute alcohol in the same conditions, about two grams of pinacone
are formed during ten hours’ action of sun-light.
851
even at higher temperatures ; the mixture however, was of a fairly
strong yellow colour.
The other alcohols reduced the benzophenone in some sunny
spring days, with the exception of methylalcohol which required a
much longer time.
The research was then continued with allylalcohol, geraniol, cyclo-
hexanol, benzylealeohol, phenylmethylaleohol, benzhydrol and cinna-
mylaleohol. Of these, the saturated alcohols reduced rapidly and
quantitatively ; the allylaleohol was attacked more slowly with form-
ation of acraldehyde (even after two months’ exposure to light, the
acraldehyde was unchanged, thus showing that the light alone does
not exert a polymerising influence on this mobile substance).
The geraniol was also oxidized very slowly, the cinnamylalcohol
remained unaffected (we will refer to this behaviour later). |
A few tertiary alcohols were also investigated; it was expected
that these would remain unaffected and indeed this was the case
with the dimethyiethylearbinol after two months’ exposure; during
that period, diaethylmethylearbinol had generated 0,3 gram of pina-
cone; with methyl-di-n-propylearbinol the separation of pinacone
started after a few days and after two months 0.7 gram had formed.
From this we notice that when the chain becomes longer, the
activity of the hydrogen of tertiary alcohols gets enhanced, which
enables it, with the cooperation of sun-light, to attack an aromatic
ketone; what gets formed from the alcohol has not been investigated
by us.
From observations of CiawiciAn and SitBer') it is known that the
hydrogen of some hydrocarbons, such as toluene, is already active
enough to cause this reduction. We have been able to show that
also the hydrogen of the eycloherane is transferred to the ketone,
on the other hand, hydrogen itself was not capable of acting.
Besides benzophenone some other ketones — particularly those
that were previously subjected by us to the action of aluminium
amaleam — were subsequently exposed in alcoholic solution to
the light.
Nothing but pinacone was ever obtained, but the phenomena
occurring in these photo-reactions induced us to systematically repeat
a large part of these purely qualitative observations in such a manner
that on using a very simple modus operand: a relatively-quantitative
result was still obtained.
1) B. 48, 1537 (1910),
852
Il. The photo-reaction : ketone + hydrogen'= pinacone.
In order to obtain a relatively-quantitative result we could make
use of a constant source of light and allow this to act on the differ-
ent solutions under the same conditions; for this purpose a small
7 em. quartz-mercury lamp was at our disposal. Yet we have em-
ployed this method but rarely, for instance in continuous dark
weather, because on account of the unequal distribution of the light,
at most two little tubes could be placed in front of the lamp in
such a manner that it might be assumed that they existed under
equal conditions.
When it had been ascertained by us that the reduction took place
quite as well in ordinary white glass as in quartz, from which it
appeared that a very large part of the actinic rays was situated in
the visible spectrum, the experiments intended for comparison were
carried out as follows:
A number of equally wide tubes of the same kind of glass and
having walls of approximately the same thickness were filled with
the same quantity of solution, and all placed at the same distance
in front of a white sereen, which was placed close to a large labora-
tory window.
In this manner it was attained that the quantity of light that
fell in the same time on each solution was practically the same,
and perfectly comparable results were thus obtained.
It speaks for itself that even then only the figures of a same
experimental series were mutually comparable. *).
A photo-reaction is distinguished from a reaction in the dark by
two points.
It is of a lower and frequently of the O order in regard to the
substance which is being activated and the temperature coefficient
is small. 7)
As we found that the active light was situated in the visible
spectrum and that the alcohols do not absorb visible rays, the
ketones are in this reaction the sensitive substances, and so we could
expect that the quantities of pinacone would be independent of the
1) Also comp. O. Gross Z. phys. Ch. 37, 168 (1901) and E. GoLpBerG Z phys.
Ch. 44, 1 (1902).
*) The first property is due to the activation occurring in the outer layer; from
the sensitive substance only a limited number of molecules can be raised by the
same quantity of light to the same degree of activily; even at a moderate dilution,
lie reaction becomes, on this ‘account, indevendent of the concentration of the
sensilive substance and therefore of the 0 order. This applies to slowly progress-
ing reactions where the sensitive substance can be rapidly supplied by diffusion
from the dark interior to the light zone.
853
ketone concentration (in regard to the sensitive substance a reaction
of the 0 order).
By selecting the alcohol itself as a solvent the change in con-
centration thereof could be eliminated. (Table I).
In order to determine the order of the reaction in regard to the
alcohol, benzene was chosen as being a general, non-absorbing and
non-reducing solvent. (Table Iw) (Chronologically these last experiments
were made after the position of the active light in the spectrum
had been ascertained; we, however, state them here because they
enabled us to give a complete image of the course of the reaction.)
We notice that this reaction is indeed independent of the con-
centration of the ketone, but not independent, however, of the con-
centration of the alcohol. As the quantities thereof had been chosen in
such a manner that they were amply sufficient even at the slightest
concentration, it follows from the figures obtained that the velocity
of the pinacone formation is proportional to the concentration of
the alcohol. *)
Thus we may represent the reaction by the kinetic equation :
en
dt
With a constant light-quantity, the velocity of the pinacone forma-
tion thus becomes proportional to the aleohol concentration; how
many molecules of the ketone act simultaneously cannot be ascer-
tained in this manner. As, however, pinacone and aldehyde are
formed and as according to the above equation one molecule of
aleohol is attacked simultaneously, the reaction scheme becomes:
C,H,OH -++ 2(C,H,),CO = C,H,0 + (C,H,),(COH), ”).
In order to learn the temperature coefficient the ordinary tubes
(16 mm. diameter) were enclosed and sealed into a second tube
(24 mm. internal diameter); the intervening space was filled with
conductivity water and now two of these tubes were exposed to
light as deseribed, one of them being kept at 25°—28° and the
other at 792 —-78° 2):
1) Here we have assumed that the change in concentration of the alcohol during
each of the four experiments was so slight that it could be regarded as being
constant; this, of course, is not correct and we really ought to have taken each
time a portion from larger apparatus. In that case, however, the experiments
would become much more complicated, because the light-quantily did not then
remain constant during the experiment. Hence, we have rested content with the
above modus operandi which is sufficiently accurate for our purpose.
*) For a mixture of ketone and benzhydrole we have proved this reaction scheme
yet in another manner (see next communication).
3) Compare R. LurHER and F. Wereert, Z. phys. Ch. 53, 400 (1905).
854
TAB LE aw
a Concentration of the ketone Quantity of |
N°. in 25 cc. alcohol pinacone Remarks
ist Series
l 0.1 gr. benzophenone 0.09 gr.
entirely converted
2 025 : 0:25" 5,
3 0.50 ” ” 0.34 ”
4 OS, 5 0.36 „
5 l= fy ” 0.36,
2nd Series
1 1 gr. benzophenone 0.47 gr
2 age x 0.49 „
3 3 ” ” 0.49 ”
3rd Series
1 0.1 gr.ochlorobenzophenone 0.09 gr
entirely converted
2 O25; ie 0.24 „
3 0:50"; ne 0:38 „
4 ir er Pf 0:39 .,,
5 l= ” ” 0.39 ”
6 2— ,, De 0.88 „
Ht 3.— 5, e 0.39 =,
8 A ,, > 0.41 „
TA BIE de:
Concentration of C‚Hs5OH in .
No, | the benzene solution of 2 gr. or T'atios
(CgH;)2CO per 25 cc. P
1 0:2527 gror 1 eq. 0.08 gr. 1
2 0.5054 Fe fa a 018.5, 2.25
3 LOTS teer 0:36, 4.50
4 2OZ1 OV rt, NGN 0.66 ,, 8.25
5 oo (pure alcohol) 0.69 „
855
More accurate experiments were not considered necessary as we
did not care for the absolute value, but only for the order of mag-
nitude of the temperature coefficient.
Adjacent to the jacketed tube was also suspended an ordinary
tube to ascertain whether the presence of the jacket had any influence
on the pinacone formation.
The subjoined table II gives a survey of some series of experiments.
From these results it follows that the method is sufficiently accurate
for our purpose, the ketone reduction is indeed a photo-reaction
with a small temperature coefficient; this still falls below the mean
stated by Piotnixow') of 1,17 per 10°.
TABLE Il. Time of exposure 2—3 days.
K
NO, Contents of inner tube | = Pf? ep Ben Ros |
Ist Series
1 2 gr. (CgH;), CO in 25 ec. alcohol | 25°—28° 0.45 gr.
without jacket |
2 Blind nde ict ib gacket st 0.76,
|+ 50° 1.06
3 ad Meee NSC Ie. Gr tie Ibe 4020,
2nd Series
1 ee, ets ve Withoût jacket | 25°—28° 0.44 gr.
2 NR Der os 3s WHILE CRE | oe | 0:70,
[+ 50° | 1.065
3 ke Lt Td 0.96 ,,
3rd Series .
1 2er.(CICsHy)2CO in 25cc. alcohol | 25°—28° 0.27 gr.
without jacket |
2 RiP) Alert sy veitheraekel iyi ail | 0.35 ,, |
+) 5074) | 1.095
3 23 MOEN PME ROU ee oC ne ORE Ber sel 0.55 ,,
4th Series
1 ome, he” Without: jacket 2525 0.24 gr.
2 LL aA ages eles Wikka abt rr 0.30 ,,
| 50° | 1.10
3 Ee Ane PE ee A 0:50
1) Jon, Prornikow. Photochemische Versuchstechnik. p. 273 (1912),
856
That the temperature coefficient for orthochlorobenzophenone is
really somewhat higher than for benzophenone seems to us rather
probable, but this can only be ascertained by more delicate measu-
rements *).
The independence of the concentration and the very small tempe-
rature coefficient now enables us to continue following this very
simple method in the quantitative investigation as to the influence
of the ketone to be reduced as well as of the reducing alcohol.
Influence of the alcohol.
The alcohols, as described above, were carefully dried over calcium
and, after distillation, poured at once into the tubes containing two
grams of ketone. These were then sealed and exposed to the light.
These tubes were suspended at such a distance that they could
not interfere with each other.
The subjoined table gives two series of experiments, the first series
was exposed for three and the second one for six days: particularly
during the first days it was sunny spring weather.
What strikes us here in the first place is the agreement in the
action of the alcohols 2—6; the secondary propyl alcohol gets
oxidised somewhat more rapidly. the amyl alcohol a little more
slowly. In the latter case a strong yellow coloration sets in.
Very much smaller is the velocity of the pinacone formation in
the case of methyl and allyl alcohol; as no interfering yellow
coloration occurred here and as the conditions were moreover quite
equal, this different behaviour must be attributed to the particular
position these alcohols occupy.
Although we cannot yet enter here into an explanation of the
process, it is obvious that the reduction of benzophenone will proceed
all the more readily when in the conversion of alcohol into aldehyde
(or ketone) more energy is set free.
The absolute extent of this energy is unknown to us, but still
some thermic data point to the existence of a parallelism in the
1) The remarkably greater reduction velocity in the jacketed as compared with
that in an ordinary tube, must be attributed to the larger quantity of light which,
owing to refraction in the jacket filled with water, falls on the inner tube. In fact
nothing could be noticed of this inner tube when the tube was entirely filled; it
looked as if the alcoholic solution has the width of the outer tube. In harmony
with this observation, it appears that the ratios of the velocities in the four series
namely 76:45, 70:44, 35:27 and 30:24 do not greatly diverge and are about
equal to the proportions of the sections of the outer and inner tube 24; 16.
(Compare Lurner and Welcert |. c. p. 391).
oe
857
PAB LE HI
list evel 2 gr. benzophenone in 25 cc. | Parente | Remarks
1 methyl alcohol | 0.29 ar.
2 ethyl alcohol 0.84 ,,
3 n-propyl alcohol 0:85, faint yellow
coloration
+ sec. propyl alcohol 0 |
5 n. butyl alcohol Ee en
| ‚| _ strong yellow
6 amyl alcohol (Bp. 130 -133 ) OL 1S 5; | Soro
1 allyl alcohol fe: Oel
2nd Series | | siete
Lf methyl alcoho! 0.49 2E 4) 1.69
Ze ethyl alcohol 1.46*- |; 1.74
| yellow coloration
Si n. propyl! alcohol Ae so tlie? not much Vie
| | increased
4’ sec. propyl alcohol aes Se) des | 1.58
; | Il | i
6 amyl alcohol (a. ab.) | denn 7e ye ae 1.40
fie allyl alcohol MAK ARE a 1.68
velocity of the reduetion and the extent of the difference of the
molecular heat of combustion of aleohol and the correlated aldehyde
(or ketone).
The greater this difference the more energy will be represented
by the hydrogen atoms playing a rôle in that transformation.
As the heats of evaporation of the aleohols on one side and of
the aldehydes on the other side do not sensibly differ and as all we
require here are a few figures for comparison, a correction for this
may be omitted here. We then find for these differences (according
to data from the tables of LaNnporr-BÖRNstTEIN-Rorn). (See table IV.)
The heat of combustion of acraldehyde is not known, neither
that of formaldehyde. There is, however, a statement as to meta-_
formaldehyde: if from this one calculates the molecular combustion
as if it were a monomeride, the difference amounts to 47 cal. As,
however, this also includes the heat of polymerisation the difference
is presumably considerably less than 47 calories,
858
TABLE IV,
E Z | methyl alcohol—(meta)formaldehyde < 41.0 cal.
2 5 | ethyl alcohol —aldehyde Ala os, |
e E n. propyl alcohol—propionaldehyde | 20.0 „ |
5 5 sec. propy! alcohol—acetone | 51 Os; |
2 © | amyl alcohol (?)—valeraldehyde I |
The two series of experiments of table III were started at the
same moment, the first was investigated after three and the latter
after six days; when the converted quantity of substance is proportional
to the quantity of light and no secondary hindrances occur, the
proportion of the quantities of pinacone at each of the numbers
1:1’, 2:2’ ete. must be the same; these ratios have been inserted
in the last column of the second series.
We notice that this ratio is indeed almost constant except in the
case of amyl alcohol, where a hindrance in the form of an increasing
yellow coloration is distinctly observed.
Influence of the ketone.
The tubes were filled with solutions of one gram of ketone in
50 ce. of absolute ethyl alcohol. Two series were exposed simulta-
neously to the action of the light; the first was investigated after
three, the second after six days. Some pinacones remain very long
in supersaturated solution, hence the alcohol was always distilled
off and the residue shaken with 80°/, alcohol so as to remove all
unconverted ketone
The subjoined table V gives the results obtained and the ratios of
the velocities with those of the benzopinacone formation as unit.
Table VI gives a similar double series; most of the ketones
investigated here were not attacked.
Table VIL gives a survey of the results obtained in amyl alcohol
as solvent and as reducing agent.
1st. The velocity of the pinacone formation, according to this
survey, is greatly dependent on the nature and on the position of the
substituent. As regards the nature, there is only one group (the
methyl group on the two para-positions (N°. 7)) that appears to
accelerate the reduction velocity somewhat, for the rest the substitution —
causes a decrease in velocity.
This decrease is strongest when the substituting group isa phenyl
859
1) These ketones had not entirely passed in solution in the alcohol.
TABDE Ve
Bl ; 5 Quantitiés ge pinacone He | Ratio | Ratio of the Re-
No. Name of the ketone en | Senes le) Series | [Series Il, I) with that of
| In grams In millimols. alte Ee ae unt
l benzophenone 0.41 | 0.85 1.12 23 ||) 2005 1.—
2 | 2 chlorobenzophenone 0.12 0.25 0.28 0.58 2.01 0.25
3 | 3 chlorobenzophenone ? +010 — 0.23 -~- +0.1 (from II)
4 4 chlorobenzophenone 0.32 | 0.75 0.74 Wei) 2.34 0.66
5 | 4 methoxybenzophenone | 0.39 0.80 0:92 1.88 2.04 0.82
6 | 4 methylbenzophenone | 0.41 0.86 1.04 2.18 2.09 0.93
1 | 44’ dimethylbenzophenone | 0.48 0.95(off), 1.19 -- — 1.06
8 | 4 bromobenzophenone 0.51 0.98(off), 1.— — —— 0.90
91) | 44’ dichlorobenzophenone ? 0.73 — 1.45 — 0.63 (from II)
10 | 22/44 tetrachlorobenzoph. 0.22 0.47 0.34 0.72 212 0.30
11 | 2 chloro 4’ methyl P 0.27 0.55 0.58 1.18 2.03 0.52
121) | 4 chloro 4’ methyl 5 0.19 0.70 0.41(?)| 1.50 3.66(?) 0.64 (from II)
TABLE VE
13 | benzophenone 0.85 0.98(off) 2.35 — — !
1141) | 4 phenylbenzophenone — — -- — — 0
15 | phenyl-z-naphtylketone — — — — _ 0
16 | phenyl-2- RE — — — _ — 0
17 | 2 methylbenzophenone _ — a — — 0
18 | 3 methylbenzophenone 0.80 0.96 (off) 2.03 oo — 0.89
19 | 2424 tetramethyl „ — — — — — 0
20 | fluorenone -- — — — — 0
TABLE Vil Amyl alcohol as solvent.
21 | benzophenone 0.75 0.97(off)) 2.05
22 | 2 chlorobenzophenone 0.22 0.33 0.51
23 | 4 chlorobenzophenone 0.65 0.96 1.49
24 | 4 methylbenzophenone 0.74 0.96 (off), 1.88
25 | phenyl «-naphthylketone — — —
560
group, because 4-phenylbenzophenone (14) and the two phenyl-
naphthylketones (15 and 16) are not reduced.
Halogen atoms and methyl groups do diminish the single substitution,
but (with one exception) do not prevent the same. Para substitution
has the least influence, ortho the greatest; this, however does not
apply to the chlorobenzophenones (3), so that we-can hardly speak
of a universal rule.
The symmetry of the molecule seems to accelerate the velocity.
Whereas the 4-methylbenzophenone has a smaller velocity than
the benzophenone (5), the 4.4’-dimethylbenzophenone has a somewhat
greater one. The fairly considerable decrease in velocity in the
d-chlorobenzophenone (4) is not continued in the 44’-dichlorobenzo-
phenone (9). In connexion with the considerable decrease in the
2-chlorobenzophenone (2) that in the 2.2’.4.4’-tetrachlorobenzophenone
(10) is unexpectedly high.
A remarkable fact is the slight influence of the methyl group on
the meta position (18) in regard to the great one of the chlorine atom (3).
2d. Of more importance is the fact that the alcohol, the reducing -
agent, is of very secondary significance as regards the ratio of the
reduction velocities; this is shown from the comparison of tables
V and VII. The ratio of the velocities in ethyl and amyi aicohol
is practically the same. We have completed these observations with
a few on methyl and propyl alcohol, selecting methy! aleohol because
the velocities therein are generally much less, whereas o-chloro-
benzophenone with benzophenone were compared as ketones, because
the velocities in ethyl (and amyl) alcohol differ strongly.
The subjoined table VIII gives a survey of the results.
PAB LE Vill
Sol. C‚HsOH, | Sol.CsH,,OH, | Sol. CH, OH, | Sol. nC3H7OH,
Ist Series | _3rd Series || 4th Series 5th Series
Quantity Quantity Quantity | Quantity
in Ratio | in Ratio |; in Ratio || in Ratio
m.mols | m.mols ‚m.mols m.mols
Benzophenone | 2.32 | 1.0 2405 40 MY 4-01 1 2:03 14
2 Chloros 0.58 | 0.25 051 KO. 25 0.46 | 0.24 1.08 | 0.27
Lao po dy LE heal TS OBE Saar sao aa KOE
4 methyl _,, 2:18 | 0:03 || ABS 0201
phenyl «-naph-
thyl ketone ° 0 0 0
nn - a
861
First of all it follows from this constant ratio that the ketone is
prominent in the photo-reaction, that this passes into a photo-active
condition. Further, that the diverse ketones are activated in a
perfectly analogous manner in such a way that either a number of
molecules (the same for all ketones) become photo-active, which molecules
then react with the alcohol with a velocity specifie for the ketone;
or, a number of molecules specifie for each ketone becomes
activated which, with a definite velocity which is independent of
the ketone, dehydrogenises the alcohol.
A choice from these alternatives can only be made by a further
study of the photo-reaction.
The active light of the ketone reduction.
The first attempt to ascertain the position of the active light in
the spectrum has been made by Ciamician and Sier *), They in-
vestigated, for instance, the reduction of benzophenone and alcohol,
employing two photo-filters.
As a red photo-filler was used a cold saturated solution of fluo-
rescein in alcohol (thickness of layer 15 mm.) which extinguishes
all light to 0.510; by adding gentian-violet the absorption could
be raised to 0.620 u.
As a blue filter served a 10°/, solution of cobalt chloride in alcohol
which transmits rays of a wavelength less than 0,480 u; a green
band at + 0,560 w and a red one at + 770 u remain, however,
unextinguished. They arrive at the result that all the reactions with
which they were engaged, took place under the influence of blue
light. We have used a larger number of photo-filters and carried
out the research in jacketed tubes; the inner tubes were those which
were used by us in the other experiments; the intervening space
was 15 mm. Above the liquid in the jacket the outer tubes were
covered with black lacquer, so that none but filtered light could
penetrate into the inner tube.
As photo-filters were selected :
I. Red: aqueous solution of chrysoidin *),
He rGreenis 5, n », potassium dichromate + acid green
B. extra. ®)
Blue and violet.
HI. 10°/, aleoholie solution of CoCl,.
IV. Cold saturated aqueous solution of erystal violet 5 B. O. ay
ne i 9 5 „ acid violet 4 B. N. ?).
VI. Solution of iodine in CCL.
tj B. 35, 3593 (1902)
2) Colouring matters from the “Gesellschaft f, chem. Ind. Basel”.
862
In agreement with that found by Criamician and SiBer for the red
fluorescein filter we found that the filters I and II which only
transmit red (690—598 wu) or red and green (> 500 uy) absorbed
all actinic rays.
Also V, which besides red rays of about 700 uu still transmitted
blue and violet >> +483 nu, completely prevented the reduction in the
inner tube. On the other band an important reduction took place with
the filters I], IV, and VI which transmitted rays to the extreme, visible
violet +
400 uu.
The series of experiments were conducted in this way that a set
of four jacketed tubes with photo-filters were exposed to sun-light
for some days in front of the white sereen: the results are contained
in the subjoined table.
TABLE IX,
7 | | Quantity |
: 6 Photo- of
NOR Ketone in the inner tube filter | pinacone Remarks
x | EREN: i id in gr. : nd
Ist Series
1 2 gr.benzophenone in 25 cc.C;H;OH I 0
2 II 0
entirely
: It 2 converted
4 IV 0.67
2nd Series
5 Ill 0.85 | From the com-
parison of the
6 V 0 figures for Ili
and IV with the
1 VI 0.48 | controlling tube
with conduct-
conduct- ivity water it
8 ivity 1.28 | appears that
water there always
q takes place a
TET RE ° ; partial absorpt-
3rd Series ion of the actinic
| rays; this, how-
9 _\2gr. o-chlorobenzophenone Ill 0.33 | ever,isrelatively
| | small and is pro-
10 Vv 0 bably based on
a general ab-
11 VI 0.28 sorption, which
| in a spectros-
| (conduct- copic investigat-
| 12 ivity 0.42 | ion was readily
|
water
observed.
Cr
86
Now with this method we can only get a very rough determination
of the position of the active region, still it appears that the active
rays are presumably situated in the violet and have a wavelength
smaller than + 480 gu. In order to see whether in the beginning
of the ultraviolet active rays were still present, a small jacketed
tube was constructed from quartz, the alcoholic benzophenone solution
was put into the inner tube and in the jacket a cold saturated
solution of nitrosodimethylaniline') which absorbs all visible violet
and blue rays and transmits ultra-violet ones of 400—280 uu.
Neither in sun-light, nor in front of the quartz lamp did any
reduction set in; from this we could conclude that the active rays
were not situated in the ultra violet, but in the visible spectrum
< 430 and > 400 uu.
A fortunate incident now came to our aid when we were engaged
in determining the correct position of the active light.
We had noticed that the ketones were converted with compara-
tively great rapidity into pinacones by means of the Hrerävs quartz-
mercury lamp.
The mercury spectrum must thus contain a great quantity of the
chemically active rays. This spectrum exhibits a very intensive blue
line at 436—434 uu and two violet ones at 407,8 wu and 404.7 up’).
Photo-filter V completely removes the violet lines and leaves the
blue ones unchanged; as this filter in sunlight as well as in front
of the quartz lamp prevents all conversion of benzophenone as well
as of o-chlorobenzophenone, and as we have noticed that the ultra-
violet light of the lamp is inactive we may conclude that the active
light for the photochemical reduction of the aromatic ketones is
situated in the extreme end of the visible violet.
The fact that the nature of the source of light has no principal
influence on the reduction process is shown from the subjoined table,
in which are given the ratios of the quantities of pinacone that are
formed from diverse ketones when exposed either to sun-light or
mercury-light. |
The exposure to mercury-light was carried out by placing a solution
of 0.5 gram of ketone in 15 ce of ethyl alcohol at a distance of
5 em from and parallel to the quartz lamp and exposing these for
10 hours; hence, the quantity of light was approximately the same
for all ketones. |
The close agreement of these ratios also renders it probable that
1) Compare Prornikow etc. p. 19.
2) LEHMANN, Phys. Zeitschr. 11, 1039 (1910).
or
~]
Proceedings Roya! Acad. Amsterdam. Vol. XVII.
864
TABLE.
| l iba | Quantity of pinacone on exposure | Quantity of
| | to Hg light ‚__pinacone on
ees ; — exposure to sun-
No. Name of the ketone inlet light ee
in grams benzopinacone | pinacone as
as unit | unit
1 benzophenone OFo0 | 1
2 2 chlorobenzophenone 0.10 0.23 | 0.25
3 4 chlorobenzophenone 0.30 | 0.70 | 0.66
4 2 methylbenzophenone | 0 | 0 | 0
5 4 methylbenzophenone 0.34 | 0.89 | 0.93 |
6 phenyl z-naphthylketone 0 0 0
| 7 fluorenone 0 0 : 0 |
the active rays are situated for the greater part at 407.8 and 404.7 uu *).
The action of light on murtures of ketones.
The phenomena observed by us during the exposure to light of
ketone mixtures in absolute alcohol divulged a very strong mutual
influence.
In order to better understand these observations, the following
should precede :
We have noticed that the chemically active light comprises a very
limited part of the „spectrum, yet, therein are rays of different
frequency and intensity.
A. We can now suppose that each of the ketones present wants
its own active rays without absorbing rays intended for the other
ketone; then — as the pinacone formation is independent of the
concentration — there will have formed in the tube with the mixture
the sum of the quantities of pinacone that are formed in the separate
tubes under the same conditions.
Those quantities within certain limits must also be independent
of the proportion of the concentrations of the ketones in the tube
containing the mixture. |
1) Presumably, the action is in a high degree selective, as a layer of 3 dm 40/,
benzophenone in absolute alcohol certainly caused a very distinct fading of these {wo
mercury lines, whereas nothing could be noticed of a curtailing or fading at the
violet side of the arc lamp spectrum through that same liquid layer. We attach,
moreover, not much value to this subjective observation, for only an accurate
speetrophotometrie investigation of the absorption spectra of the ketones can
properly determine the connexion between absorption and chemical action.
865
B. As soon, however, as rays for the one ketone are also con-
sumed by the other one, the quantity of pinacone will be less than
the sum in question and, moreover, the proportions of concentrations
will no longer be a matter of indifference.
For in the layer where the photo-reaction takes place each molecule
of the one ketone requires a part of the light-energy also wanted by
the second ketone, so that the lindrance experienced by the latter
will become greater when its relative concentration gets less.
C. The extreme case would be that both ketones require just
the same rays; we should then obtain a quantity of each of the
ketones which in equimolecular concentration is equal to half the
quantity that forms in the tube with the separate ketone (always
supposing that no other obstacles occur).
The phenomena recorded by us are now best understood from
the supposition B: a ketone does require specifie rays from its
neighbour. Some of the observations approach to A, others to C,
some even exceed this extreme case, showing that the action is
more complicated than was at first supposed, as will appear from
the subjoined tabulated survey.
FABLE. XI
Ist Solution of 2 gr. of o-chlorobenzo- ne Quantity
Series phenone and varying quantities of aicobel pinacone Remarks
No. | phenyl z-naphthylketone in gr.
1 | 2 gr. o-Cl benzophenone pure 0.84 Se ESR
u Va — .
2 § -+0.1 gr. phenyl z-naphthylketone 0.30 52% g 25
vv is} =
3 5 0:25, 5 0.12 (ASSR
S893 6
4 \ ” 0.50 ” ” ” 0 = „SE 2S
—
5 ” in ” ” ” 0 ve S gv Me
O's SO =
ES gaa
ig Oe
| Os 0
| LES mn AO
2nd : | :
a As above o-Cl benzophenone and varying Quantity
patie quantities of o-methylbenzophenone pinacone Remarks
1 2 gr. o-Cl benzophenone pure 0.37
2 Ed + 0.1 gr.o-CH3benzophenone 0.36
2 s + 0.25 4 | 0.30 , As above
4 | a + 0.50 é | 0.26 |
| |
5 : ig : 0.15 —
| ú |
866
We notice that when one of the ketones does not get reduced it
exerts a very powerful retarding action on the reduction of the
other ketone.
This action, particularly with phenyl-a-naphtylketone is much
more important than we should expect even in the extreme case C;
besides the elimination of the chemically active rays, the molecules
of the naphtylketone must cause an impediment, which may, perhaps
be put on a par with the obstruction caused by oxygen in the
photo-halogenations.
TABLE XI.
ea Reals El 388 e ie |
| s ¥ 5 FG z = Ee TE
| Solutions of various ketones, which are |= o © | bo 8 oh | SES |
N f SEE S25 oe) o2s.| Remark
reduced separately in 50 cc. solution. S65 | §3* SOS\eGa
(One ZOE ITO IE Ore
MARE ERE ee
= AT = zn — a = SSS SS ==
©
Sl ERE:
is)
1 2gr. benzophenone VSG, Ros a | — — | Boe.
| | | =— U) nn?
22> > + 2gr.o-Cl benzophenone | 2.31 4.16 Sexe
| | 1} 2.37 38 vsa
re > del» > legen | 3.46 Tes
| | | ow o>
ay i bee
55 SE
U) | ~ | +r
1 | 2 gr. benzophenone 1:66 | 1.66.0 =S == eas §
| | | Se ane
2/2» > +2gr. p Clbenzophenone 2.18 ‚9,19 | s=sE
| 2.76 1.2 | OSES
302 > En > 1.69 | 3.19 Ee So
| 2 mS,
en DN ——- -- — - - SES
ae | sian
65 | ‚AES
u) | | nog
1 2gr. benzophenone 10485: | OQ R85: en ae
| | | | DAS ow
22 > + 2gr.pBrbenzophenone | 1.65 | 23.9 area E
| 1.95 | 19.0 |vE5
3 2 > » + 1 » » 1.05 | ile! | fa a
From Table XII it appears that, in the case when both ketones
are reduced, we have demonsirated a considerably less impediment
than in the ease that one of them is not reduced. Still, there is
always a negative influence, we obtain in all cases a quantity less
than the sum of the quantities which we should have obtained in
separate tubes; we are always dealing with case B.
In this we notice the smallest mutual hindrance in the mixture
of benzophenone and o-chlorobenzophenone, yet we notice plainly
that the impediment increases when one of the ketones is present
in large excess (18 series N°. 3) and that in such a case that present
in the smallest amount is the most strongly impeded. Much more
pronounced is the hindrance observed with mixtures of benzophenone
and p. Cl- or p. Br-benzophenone; the sum of the pinacones remains
here far below the calculated quantity. The halogen ketone has as
a rule a stronger impeding action than benzophenone, for even in
smaller quantities than the molecular ones (N°. 2 of the three series)
the halogen pinacone in the mixture is predominant. Only with a
considerable excess of benzophenone the halogen pinacone is repelled
and mostly so in the cases where the greatest hindrance is present
(compare N°. 3 of the three series).
We thus find in rough traits what we could expect; there occur,
however, particularly when one of the ketones is not reduced
separately, such great hindrances that they cannot be satisfactorily
TABLE ZH.
Quantity | Quantity of
Contents inner tube Contents outer tube | :
ind me pinacone pinacone in the
N?.|2 gr. benzophenone in| 50 cc. abs. alcoholic k ‘inner tube with
at | 38 Y the blank tube
20 ec. abs. aleohol | solution of 4 grams: (EE g ae it
ZE | /
B | |
WY | |
1 alcohol (blank-exper.) | 0.64 size | 1.—
2 p CH; benzophenone | 0.28 | 0.66 | 0.44
3 oCl benzophenone ‚0.18 | 0.19, 0.28
4 phenyl naphthylketone trace | 0 | trace
| | |
ar a ETA Ai Pier eal ORE
kek | |
Sh | |
ND |
1 alcohol (blank-exper.) | 1.05 — | 1. |
2 | o CH3 benzophenone | 0.50 dl 0.48
1 # | phenyls-naphthylketone 0.28 0 0.27
4 | fluorenone 0 0 0
| | |
AI EET At owe) ? ? 4 |
Mole |
dn |
03 S|
1 alcohol (blank-exper.) | 0.79 — 1.—
es p Br benzophenone | 0.42 1.18 | 0.54
| 3 | | p Cl benzophenone 0.38 | 0.83 | 0.47 |
| 4 | benzophenone 0.29 | 0.93 | 0.37 |
| U
868
explained in the above cited manner. There seems to be a connexion
here between the extent of the impediment and the non-appearance
of the photo-reduction.
Now, in order to eliminate the hindrance which might eventually
take place owing to the mixing, the oft-quoted jacketed tubes were
filled in such a manner that in the inner tube was always inserted
a definite ketone and in the jacket diverse other ketones.
The light then first traversed a = 2 mm. thick layer of a ketone
then to exert its action on the benzophenone; in this way we could
form a better opinion as to the absorption of rays of light by the
one ketone (in the jacket) which were needed for the other ketone
(in the inner tube).
There exists no doubt that a// ketones absorb actinic rays intended
for the benzophenone; the degree of this absorption is certainly
very different and specific.
The ketones which were attacked in the jacket were, during the
experiment, reduced in concentration, so that the conditions for the
reducing of the benzophenone in the inner tube gradually became
more favourable; this causes, however, that we can only consider
as fairly comparable the experiments where no reduction takes place
in the jacket. Hence, a few ketones have been placed in the jacket
in benzene solution whilst in the inner tube was again present a
solution of 2 grams of benzophenone in 20 ee. of absolute alcohol;
the following result was thus obtained:
TABLE XV,
| In the inner tube In the jacket a | Quantity of |
N°.| 2 gr. (CgH;)2CO | N/4 benzene solu- pinacone in the | Remarks
in 20 cc. abs. alc. tion of | inner tube. |
f |
| |
1 ieee (blank) A Be i oes |
| | | | The benzene
2 | benzophenone | 0.45 | 0.34 |" Pentone
jacket was al-
3 | o-chloro » 0.53 | 0.40 | | ways coloured
En 5 | pale yellow
| 2 | E Ee LEE | which colour
rey c | | ‚| again faded in
|. | | |
It appears that several ketenes absorb rays of light which effect
the benzophenone activation, even when they are dissclved in
benzene.
The reversible yellow coloration of this benzene solution, however,
tells us to be careful, for the activated ketone can form with benzene
869
a light-screen, thus causing the absorption effect to be greater than
when the ketone had been present in alcoholic solution.
In each case a circumstance occurs owing to which the phenomenon
becomes more complicated, so that from these last experiments we
may at most draw the conclusiun that rays of light are indeed always
absorbed by the one -ketone, which the other required for the oxidation
of alcohols.
The most powerful absorbing ketones appear mostly — but not
always — to oxidise the alcohol slowly or not at all, so that we
gain the impression that a liberal absorption does take place, but
that the possibility of the setting in of a reaction and its velocity
does not only depend on the alcohol, but in a great measure on
the ketone.
It speaks for itself that the experiments on this almost quite
unexplored region can only bear a very provisional character ; still
we believe we have attained, with very simple means and methods,
some results which will prove of importance for the insight into
the photochemical reactions.
For the moment, however, we wish to refrain from an attempt to
explain the phenomena observed until more accurate spectrophoto-
metric data are at our disposal.
SUMMARY.
I From aromatic ketones and alcohol are formed, in the light,
exclusively pinacones ; these latter are, therefore, the products
to be first isolated in the reduction. Hydrols are, in the reduct-
ion of the aromatic ketones, always formed secondarily (see
Proc. XVI 91 and 962) either from the pinacones or from
the primarily formed half pinacone molecules.
Il 1. The photo-reduction of the ketone by alcohols was studied
by exposing simultaneously to the light a set of tubes of
equal dimensions and filled with equal quantities of liquid,
thus causing the light-quantity (i.t) for each object of a
serial experiment to be equal.
2. The velocity of the pinacone formation appeared to be inde-
pendent of the concentration of the benzophenone and propor-
tional to the concentration of the alcohol. Hence, it satisfied
the equation :
d pinacone
dt
= KL. [alcohol] and, therefore, the reaction scheme:
2 ketone + alcohol = pinacone + aldehyde.
870
3. The temperature coefficient was small: 1.06—1.11 for 10°.
4. The velocity of the pinacone formation is greatly dependent on
the alcohol; for instance, the methyl aleohol and the allyl
alcohol were oxidised much more slowly than other primary
and secondary alcohols.
5. The velocity of the pinacone formation is eet dependent
on the ketone, the benzophenone is attacked rapidly, most of
the ketones as yet examined less rapidly, many not at all.
6. The ratio of these velocities in different aleohols is constant.
7. The active light of the ketone reduction is sure to be situated
in the spectrum between 400 and 430 uu and very probably
in, or adjacent to, the rays 404.7 and 407.8 of the mercury
quartz lamp.
8. The ratio of the velocities of the pinacone formation in sun-
light and in mercury light is the same.
9. When two ketones are present simultaneously one of them
absorbs a part of the rays required by the other ketone; this
also appears when the light passes through a solution of the
one ketone and falls on that of the other.
Particularly in the case of the powerfully absorbing ketones
the hindrances are stronger than was to be expected.
Delft, October 1914.
Physics. — “Simplified deduction of the formula from the theory
of combinations which PLANCK uses as the basis of his radiation-
theory.” By Prof. P. Exrenrest and Prof. H. KAMERTLINGH ONNes.
(Communicated in the meeting of Oct. 31, 1914).
We refer to the expression
ww (Nel Py
CASS (4)
Pp PKN—1)!
which gives the number of ways in which .V monochromatic reso-
nators A&,, Lt... Ry may be distributed over the various degrees
of energy, determined by the series of multiples 0, ¢, 2e... of the
unit energy ¢, when the resonators together must each time contain
the given multiple Pe. Two methods of distribution will be called
identical, and only then, when the first resonator in the one distri-
bution is at the same grade of energy as the same resonator in the
second and similarly the second, third,.... and the Nt resonator
are each at the same energy-grades in the two distributions.
Taking a special example, we shall introduce a symbol for the
distribution. Let VW = 4, and P=7. One of the possible distributions
871
is the following: resonator ft, has reached the energy-grade 4e (R,
contains the energy 4e), R, the grade 2e, R, the grade Oe (contains
no energy), A, the grade ¢. Our symbol will, read from left to right,
indicate the energy of R,, A, R,. R, in the distribution chosen, and
particularly express, that the total energy is 7e. For this case the
symbol will be:
[2220224004
or also more simply:
eer Oee OO ||
With general values of V and P the symbol will contain P times
the sign & and (V—1) times the sign O'). The question now is,
how many diferent symbols for the distribution may be formed in
the manner indicated above from the given number of ¢ and O?
The answer is
MET EP)
P(N—1)! *
Proof: first considering the (V—1- P) elements e...e, 0...0
as so many distinguishable entities, they may be arranged in
REA Roe See eS
different manners between the ends ae Next note, that each time
NEARS PIN EEE celia Sg RE
(1)
of the combinations thus obtained give the same symbol for the
distribution (and give the same energy-grade to each resonator), viz.
all those combinations which are formed from each other by the
permutation of the P elements ¢ 7’) or the (W—41) elements O. The
number of the different symbols for the distribution and that of the
1) We were led to the introduction of the (N—1) partitions between the MN
resonators, in trying to find an explanation of the form (V — 1)! in the denomi-
nator of (A) (compare note 1 on page 872). PrancK proves, that the number of
distributions must be equal to the number of al! ‘‘combinations with repetitions
of N elements of class P” and fur the proof, that this number is given by the
expression (A), he refers to the train of reasoniug followed in treatises on com-
binations for this particular case. In these treatises the expression (A) is ariived
at by the aid of the device of “transition from ton +1”, and this method taken
as a whole does not give an insight into the origin of the final expression.
2) See appendix,
872
distributions themselves required is thus obtained by dividing (2) by
Cyqre id).
ABP EN Dex
The contrast between PraNok’s hypothesis of the energy-grades and
EINSTEIN’s hypothesis of energy-quanta.
The permutation of the elements e is a purely formal device, just as the per-
mutation of the elements O is. More than once the analogous, equally formal device
used by PLANCK, viz. distribution of P energy-elements over N resonators, has by
a misunderstanding been given a physical interpretation, which is absolutely in
conflict with PLANCK’s radiation-formula and would lead to Wren’s radiation formula.
As a matter of fact PLANCK’s energy-elements were in that case almost entirely
identified with Erysrery’s light-quanta and accordingly it was said, that the difference
between PLANCK and EINSTEIN consists herein that the latter assumes the existence
of mutually independent energy-quanta also in empty space, the former only in
the interior of matter, in the resonators. The confusion which underlies this view
has been more than once pointed out ®). EINSTEIN really considers P similar quanta,
existing independently of each other. He discusses for instance the case, that they
distribute themselves irreversibly from a space of NV, cm over a larger space of
Ny em? and he finds using BoLTzMAn’s entropy-formula: S= klog W, that this
produces a gain of entropy ?):
UNS Ee
S— 8, = blog (57 ] a gk Oe he OR
1
1) It may be added, that the problem of the distribution of N resonators over
the energy grades corresponds to the following: On a rod, whose length is a mul-
tiple Pe of a given length -, notches have been cut at distances ¢, 2e, etc. from
one of the ends. At each of the notches, and only there, the rod may be broken,
the separate pieces may subsequently be joined together in arbitrary numbers and
in arbitrary order, the rods thus obtained not being distinguishable from each other
otherwise than by a possible difference in length. The question is, in how many
different manners (comp. Appendix) the rod may be divided and the pieces distri-
buted over a given number of boxes, to be distinguished from each other as the
Ist 2nd,.... Nth, when no box may contain more than one rod. If the boxes,
which may be thought of as rectangular, are placed side by side in one line, they
form together as it were an oblong drawer with (N—1) partitions, formed of two
walls each, (comp. the above symbol in its first form, from which the second
form was derived by abstracting from the fact, that each multiple of « forms one
whole each lime), and these double partitions may be imagined to be mutually
exchanged, the boxes themselves remaining where they are. The possibility of this
exchange is indicated by the form of the symbol chosen.
As a further example corresponding to the symbol we may take a thread on
which between P beads of the same kind, (N—1) beads of a different kind are
strung, which divide the beads of the first kind in a Ist, 2nd... Nth group.
*) P. ErRENFesT, Ann. d. Phys. 36, 91, 1911, G. KrurKow, Physik. Zschr. 15,
133, 363, 1914.
5) A. EINSTEIN, Ann. d. Phys. 17, 132, 1905,
873
Le. the same increase as in the analogous irreversible distribution of P similar,
independent gas-molecules, for the number of ways in which P quanta may be
distributed first over Ny, then over Ny cells in space, are to each other in the ratio
NEE NE ; (3)
If with PrLANcK the object were to distribute P mutually independent elements
s over NV resonators, in passing from AN, to Ns resonators the number of possible
distributions would in this case also increase in the ratio (.-) and correspondingly
the entropy accord.ng to equ.tion (4). We know, however, that PLANCK obtains
the totally different formula
ie Py Wte
(N,—1)! PL (N-I! PI ()
(which -only coincides approximately with (@) for very large values of P) and a
corresponding law of dependence of the entropy on N. This can be simply
explained as follows: PLANCK does not deal with really mutuaily free quanta «,
the resolution of the multiples of ¢ into separate elements s, which is essential in
his method, and the introduction of these separate elements have to be taken
“cum grano salis”; it is simply a formal device entirely analogous to our permut-
ation of the elements « or O. The real object which is counted remains the number
of all the different distributions of N resonators over the energy-grades O,¢, 2, …
with a given total energy P:. If for instance P= 3, and N=2, Einstein has to
distinguish 2% —8 ways in which the three (similar) light-quanta A, B, C can be
distributed over the space-cells 1, 2.
Hr | 1 ae Î
2
TE iet 2
VAIS avant |
Vine 2 1 2
A 1
Vile 22 oe
PrarcK on the other hand must count the three éases Ii, IL and V as a single
one, for all three express that resonator Ay is at the grade 2, Ry ate; similarly
he has to reckon the cases IV, VI and VII as one; A, has here ¢ and R 2e.
Adding the two remaining cases [ (, contains 3e, R, Oe) and II (R, has Oe, Ry 3e)
one actually obtains
(N--1--P)!/ (2—1+83)!
(N= By Peo STE MER
different distributions of the resonators #,, R, over the energy-grades.
We may summarize the above as follows ; EINSTELN’s hypothesis leads necessarily
to formula (z) for the entropy and thus necessarily to WrieN’s radiation-formula,
not PLANCK’s PLANCK’s formal device (distribution of P energy-elements < over
N resonators) cannot be interpreted in the sense of EtnstErn’s lignt-quanta.
(December 24, 1914).
KONINKLIJKE AKADEMIE
VAN WETENSCHAPPEN
-- TE AMSTERDAM -:-
PROCEEDINGS OF THE
SECTION OF SCIENCES
VOLUME XVII
( — IST PART — )
JOHANNES MULLER :—: AMSTERDAM
: DECEMBER 1914:
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PAE Ke
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_ (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Na t
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