T H E THIRD LAW OF THERMODYNAMICS AND CAICULATION O F ENTROPIES BY T. J . WEBB
The evaluation, from thermal data, of the free energy change accompanying a reaction, has been of fundamental importance since 1888, when LeChatelier integrated the Gibbs-Helmholtz equation :l AF=AH+T
AFdT - TdAF T2
,):;(
~
AHaT -
-
T2 AHdT T2
-AF
T
s
+
AHdT I T (2) T2 I is a constant of integration, and upon its evaluation, as well as upon a knomledge of AH, depended the usefulness of the above equation. Nernst2 in 1906 in a paper, “Uber die Bereclznung chemischer Gleichgewichte aus thermischen Messungen”, brought forth evidence to show that the integration constant in the above equation is zero for all reactions in which only pure solids and liquids are involved. The evidence for the assumption that the integration constant is zero arose from the fact that AF and AH rapidly approach a constant value, as the temperature is lowered. It was assumed that tiAF/dT and dAH/dT, which are always of opposite sign, approach zero as the absolute zero of temperature is approached. A H in the above equation may be expressed by a equation such as the following: AH = AH, AI’,T ArlT2 Q Ar2T3 . . . . (3) where the Ar’s stand for the increments of heat capacity terms, and AH, is therefore the heat of reaction at the absolute zero. Substituting this value of AH in equation ( z ) , and integrating,A F = AH, - Ar,T In T - $ ArlT2 - ArzT3 . . . . IT Then
+
=
++
+
+
*
.__ dAF
dT
- - AI’,(l+lnT)
-
ArlT -
+
4 AI’zT2+I
+
(gal
and ’AH - - - AI’, dT
+ AF1T + AI’zT2 +
....
The symbols used are those considered standard. Cf. Lewis and Randall: “Thermodynamics”, McGraw-Hill Book Co. :Nernst: Nachr. kgl. Wiss. Gottingen. Math.-physik. Kiasse, 1906, I ,
T H E THIRD LAW OF THERMODYSAMICS
817
Introducing the assumption that:
it follows that
+ AriT + ArzT2 = o - AI',lnT - ArlT - Ar2T2+ I = AI',
and
Ar,
0.
In order that the tu70 expressions may be simultaneously equal to zero a t T = 0, it is necessary that I = o and Ar, = o This is the original way in which the third law was formulated by Nernst in 1g06. Accurate specific heat data over the entire range of temperature were almost non-existent, and therefore accurate tests of the law were not readily forthcoming. -4much more wieldy method of testing is furnished by entropy relations, directly deducible from the fundamental assumptions outlined above. From the definitions of free energy and entropy,
aAF dT where A F and AS denote respectively free energy and entropy increases. From Kirchhoff's law, the following relation is obtained : _ _ - -AS,
dAH
=
AC,
Consideration of the assumption that
__
~
dT
dAF
dT
and
dAH __ approach zero a t the abdT
solute zero brings to light the relation that -AS approaches zero as well as AC,. The fact that AS approaches zero means that in any reaction a t the absolute zero the entropies are additive: at T = 0,the entropy of a compound is the sum of the entropies of its elements. The general principle of the additivity of entropies, however, necessitates that all elements and compounds have zero entropy a t the absolute zero. The methods of testing the third law have been varied. In all attempts the ultimate object of all investigators has been to show that the integration constant in the above equation is zero. In this paper, evidence will be brought forward to shorn that the inaccuracies attending measurement of temperature coefficients of reversible galvanic elements in many cases are of sufficient magnitude as to render them useless in many such calculations. A typical example is furnished by the silver-iodine combination, which will be taken up in considerable detail. Practically all experiments were in agreement to within 30 calories as regards the free energy change, while the value for the total energy change varied from 14,565 calories to 15,zjo calories. The discordance of thermochemical measurements rendered the thermodynamic properties of this combination all the more uncertain.
818
T . J. WEBB
The method of testing the third law, initiated herein by the silver-iodine combination, and illustrated subsequently by the calculation of the entropy of iodine, is as follows. An accurate knowledge of the free energy change accompanying the process must be obtained. This may be obtained in various ways : the most convenient one (where possible) is by setting up a reversible galvanic element employing the process under investigation. The free energy change of any element which is reversible and reproducible can be measured by potentiometric means with an accuracy of 50 calories (corresponding to an accuracy of 0.15 entropy unit at 298’). Instead of calculating the total energy change from the free energy change and its variation with temperature, it is proposed that an accurate calorimetric determination be substituted. An accuracy of j o calories can be obtained without an excessively elaborate apparatus. The chief errors arise from heat capacity determinations and from stirring corrections. The former can be minimized by accurate measurement of the electrical energy (i. e., with potentiometer) and the latter by the choice of an efficient stirrer. A Beckmann thermometer is amply accurate for the temperature measurements. The difference between the total energy change and the free energy change gives TAS, (or T a(-AF) dT
). The entropy change thus ob-
tained may then be compared with that obtained from specific heat measurements. The tables compiled by Miething (referred to later) facilitate the calculation of AS from specific heat data. The tables give the numerical value of the following functions:
E =h,,dT
F = -TJ’+;IT T
E is the total energy increase from T = o to T = T and F is the free energy increase from T = Q to T = T. It is obvious from t>heequations that E and F correspond respectively to AH and AF, previously used. Substitution in the fundamental equation, AF = AH - TAS, gives immediately the value of AS between T = o and T = T, which is usually spoken of as simply the entropy of the substance at temperature T. The third law is involved in that an integration constant appears in the equation defining F. It is clear that if the third law is assumed, and if the entropies of all of the substances except one are known, the entropy of that substance may be calculated. This is the method used in the calculation of the entropy of iodine (vide infra). The method of testing the third lam by comparing observed temperature coefficient with that calculated from equation (3a) may prove most hazardous, as exemplified by the work of Jones and Hartman who found that the integration constant was 2.33 calories per degree in the silver-iodine combination. It seems that these unusual results may be attributed entirely to inaccurate calculations of the temperature coefficient of the liquid potential and osmotic work connections. In most cases, however, where such corrections are unnecessary, the entropy relations may be obtained from the temperature coefficient of the cell a
THE THIRD L A W O F THERMODYNAMICS
819
with considerable accuracy. The third law may be tested by calculating in several ways the entropy of a compound from temperature coefficients of A F of several of its reaction processes. Obtaining the same value in various reactions is evidence of the validity of the third law. 11. Entropy Relations applied to Cadmium Chloride and Iodide
The absence of accurate specific heat data for these two salts prevented their employment in a purposed test of the third law (as originally put forward by Kernst). The principle of the additivity of entropies simplifies the calculations. In the case of cadmium chloride, the following reactions have been investigated :
* Cd + zhgCl + 2.5H20 = CdClz2.5HzO(Solid)+ zhg ** Cd + HgsClz + 2.;H20 = CdC122,5HzO(Sslid) + 2Hg * Cd + PbC1, + z.jHtO = CdCle2.;HzO(Solid) + P b
(4)
(5)
(6)
If the third law is assumed true, that is, if it is assumed that every cyrstalline substance has zero entropy at the absolute zero, the entropy of cadmium chloride hydrate may be calculated by three independent methods from the following equations :
+ SzhgC! -I- S2.jHZO= SCdClz z.gH20 + S2-4gSCd + SHg2C12 f Sz.;H20= $ICdC!2, z.$TgO+ SzHg-
SCd
SCd
+ SPbC12 + Sz.gH20= SCdC1g.z.5H20+ S P b -
(AS,)
(7)
(As,)
(8)
(As31
(9)
AS,, AS,, and 4 S a are the entropy increases accompanying each reaction, and are measured by the cell temperature coefficients, i. e., a(-AF) __- - AS, where a'ir
A F is the increase in free energy accompanying the reaction process. In the cells mentioned above, cadmium amalgam was used instead of pure cadmium; electrolytic silver chloride instead of the precipitated form; and water from a saturated solution of the hydrate instead of pure water. The entropy differences between these various forms must be taken into account. For the difference in entropy between cadmium and cadmium amalgam, the data of Hulett' may be employed. If however his data are used h combination with the measured values, to calculate the electromotive force of the cell, Cd I CdCle . z.$H2(g)- E = Ht, is arbitrarily zero, both facts are represented by the following equation:
+
+
+
%H,
+ x B r z (I)
=
H+
+ Br-
AF,,, = -24595
Before combining this equation with the one mentioned previously, AgBr
+ :hH,
=
I
HBr
(.OI
M)
+ Ag, AF298 =
-3622
the value, - 24595, for the free energy decrease must be corrected to the value that corresponds to the formation of HBr at .OI molality. The activity COefficient of .OI HBr solut>ionis given as .93; the corresponding activity is .0093. The correction is made by means of the following equations: Lewis and Storch: J. Am. Chem. Soc., 39, 2544 (1917). Lewis and Randall: “Thermodynamlrs,” XXX, Table 7.
THE THIRD LAW O F THERMODYXAMICS
A F - AF, = RT In aHBr AF 24595 = ItT In .oog3 A F = - 24593 - 2 7 5 5 = - 2 7 3 5 0 A F A ~ B ~A F M H ~= A F A ~ AFHB~ AFn A F A ~ B ~0 = O - 27330 3620 A F A ~ B=~- 23 730 calories
+
+
+
+
+
+
The entropy of silver bromide may now be found by employment of the heat of formation of silver bromide; the relation, A F = AH - TAS; and the entropies of the elements of silver bromide. The heat of formation has been determined in two ways. The first is a direct calorimetric method. -1bulb containing a weighed amount of bromine was broken in a concentrated solution of ammonium bromide, containing finely divided silver in suspension. Silver bromide was thus formed in solution. The heat of solution of silver bromide in the same solution n-as found and correction made (the heat of solution was found to be zero, however). Amt. Bri added
Expt.
Expt.
Rise due to Reaction
4.908 grms.
I .29
I .
830 2.720 4.301
'487 ,669 1.045
Heat Capacity of Calorimeter
Heat of Reaction
1126 1124
I 2
3 4
23,710 23,910 23,810 ~3~j z o
I210
1210
The average was taken between the first three values and found to be 23,810 calories. The heat of solution was found by measuring the heats of the following reactions : Agx03 AgIYOa
+ iVH4Br(conc.) +AgBr(Disso1ved) + K&NOs + + 1IH4Br(onc.) AgBrppt.) + n"aN03 + QZ
&I
-+=
(sat. with AgBr) Subtraction gives the heat of solution - AgBr(ppt.) +AgBr(Disso1ved)
+
(&I
Expt.
-
Qd &SO3 Added
Heat Capacity of Calorimeter
Rise due to reaction
,428 '413 .288
I
4.263
1141
2
4.022 2.778
1111
3
II02
Heat of Reaction
(QJ (&I) (Q2)
19~46~ 19,530 19,430
The heat of solution was therefore taken to be zero. This was confirmed by dropping finely powdered silver bromide into a concentrated solution of am.
832
T. J. WEBB
monium bromide. N o change of temperature, other than that due t o stirring, was detected. The heat of forniation of silver bromide as determined by this method was then compared with that determined electrometrically by Krahmer', employing the cell reaction, P b zAgBr = PbBrz zAg, and the heat of formation of lead bromide. The value obtained by him is 24,190 calories. The mean of the two values was taken, viz., 23,990 calories. Substituting this value for AH, along with the value of -23730 for AF, in the equation, A F = AH - TAS, AS is found to be - A 7 entropy unit. When this value is used in the following equation, the entropy of silver bromide is obtained: Ag x B r 2 = AgBr AS - 3 7 10.25 16.3 -k SAgBr -k .87 ASA~B = ~25.68 entropy units.
+
+
+
+
It may be pointed out that the above calculation suggests a method for an accurate determination of the potential of the bromine electrode. A specific heat curve over the entire range of temperature would give an accurate and independent value for the entropy of silver bromide. The above calculations could then be reversed, and the value of the bromine potential calculated. This is important, owing to the difficulties attending any cell reversible with respect to Br-ion involving bromine, due to the formation of Br3 and Br5 ions. Summary The third law of thermodynamics has been subjected t o a test by the application of the entropy relations of cadmium chloride hydrate and cadmium iodide. (The data were obtained froin temperature coefficients of the free energy changes of their cell reactions.) A slightly different method of testing the third law has been proposed, via., obtaining AS of the reaction from an accurate calorimetric determination of the heat of reaction and the free energy change, and comparing this with AS obtained from specific heat measurements. The heat of formation of silver iodide has been redetermined. The solvents used were concentrated sodium and potassium iodide solutions. A repetition of the experiments of the investigators, giving the most accurate values recorded, has also been made. The value, concluded correct, from these experiments is 14,975 calories. The maximum error is thought to be 50 calories. The previously accepted values are over I O O calories higher. The entropy of iodine has been calculated ( I ) from the energetics of the silver-iodine combination, ( 2 ) from the energetics of the zinc-iodine combination, and (3) from the specific heat of iodine. The values obtained are 13.95, 14.05,and 14.24 entropy units. The heat of formation of silver bromide has been determined. From the energetics of the reaction, Hz aAgBr = zHBr zAg, and from the value
+
Krahmer: Z . Elektrochem. 26, 97 (1920).
+
T H E THIRD LAW O F THERMODYKAMICS
83 3
of the potential of the bromine electrode, the entropy of silver bromide has been calculated. An accurate method for the determination of the potential of the bromine electrode has been indicated. NOTE: Since the w~itingof the above paper, an article has appeared by Langel, giving a new and apparently satisfactory set of specific heat data for iodine. E
=
To
CpdT =
1527
calories; F
=
-T
= 13.77
-- - 2465
calories;
0
0
whence S293
J’B
and
S298 = 13.95.
I wish to acknowledge my gratitude to Profs. G. A. Hulett and H. S. Taylor of this laboratory for their helpfulness and interest in connection with this problem. Princeton, N . J . Frit5 Lange: Z. physik. Chem. 110, 343 (1924).
