4079
INTERPRETATION OF HEATCAPACITY MEASUREMENTS
The Interpretation of Heat Capacity Measurements through Two-Component, Liquid-Liquid Critical Regions by Robert E. Barieau Helium Research Center, Bureau of Mines, U.S. Department of the Interior, Amarillo, Texaa 70108 (Received April 10, 1968)
The heat capacity, per mole, at constant volume and constant number of moles of a two-component, threephase (two liquid phases and a vapor phase) system is given by Cv(3 phases) = -u1T(d201/dT2) - (1 ul)T(d2&/dT2) (VT/n)T(d2P/dT2),where u1 is the filling mole fraction of component 1, VT is the total volume, and n is the total number of moles. 81 and 82 are partial molal Gibbs free energies or chemical potentials of components 1 and 2, respectively. The above equation is derived and it is shown how heat capacity measurements can be determined so that the three quantities d281/dT2,d2dz/dT2,and dzP/dT2can be calculated from the measurements.
+
Because of the interest in the properties of critical points, thermal measurements have been made through critical regions of ferromagnetic materials, liquid-vapor equilibria of pure materials, and liquid-liquid equilibria of two-component systems. It is the interpretation of measurements on the latter system that the author wishes to discuss. RowlinsonJ2discussed the heat capacity of a liquidliquid, two-component system, which is accurately applicable to a system at constant pressure or to a system of partially miscible liquids at a low saturation pressure that may be treated as a constant zero pressure. The author wishes to discuss the interpretation of heat capacities determined at constant volume and a constant number of moles for a two-component, three-phase (two liquid phases and a vapor phase) system under conditions when it is not assumed that the saturation pressure has the constant value zero. From the phase rule, such a system has one degree of freedom. As long as three phases are present, the system therefore has a unique vapor pressure as a function of temperature. The system carbon tetrachloride-perfluoromethylcyclohexane is an example of such a system. Schmidt, Jura, and Hildebrand3 made thermal measurements on this system through the critical region using a Bunsen ice calorimeter to measure the heat removed from the sample on cooling to 0”. They interpreted their measurements on assumptions that are only valid a t negligible vapor pressure of the system. When a small amount of heat is added to such a system, it may be set equal to TdST, where STis the total entropy of the system, as the addition of the heat may be considered reversible. The total heat capacity of the system it3 then defined as
To obtain the total heat capacity of the contents of the calorimeter, the heat capacity of the bomb or container must, of course, be subtracted. We will assume that this has been done and that eq 1 represents the total heat capacity of the contents of the calorimeter. We will also assume that the internal volume of the container does not change with temperature. The measured heat capacity is then made at total constant volume. Dividing eq 1 by the total number of moles, n, in the calorimeter, we obtain for the heat capacity, per mole, at constant volume
(F)
T dST Cv(3 phases) = ; VTm
where XT in eq 2 represents the total entropy of the contents of the calorimeter. The following equation is applicable to our system dGT = -&dT
+ VTdP +
2
G,dnc
(3)
i= 1
where GT is the total Gibbs free energy of the contents of the calorimeter, VT is the total inside volume of the calorimeter, 8, is the partial molal Gibbs free energy or chemical potential of the ith component, and ni is the total number of moles of the ith component in the calorimeter. During a heat capacity measurement the total number of moles of each component remains constant in the calorimeter. We then have dGT = -STdT
+ VTdP
(4)
(1) “Critical Phenomena,” National Bureau of Standards Miscellaneous Publication 273, U. S. Government Printing Office, Washington, D. C., 1966. (2) J. S. Rowlinson, “Liquids and Liquid Mixtures,” Butterworth and Co. Ltd., London, 1959, Chapter 6. (3) H. Schmidt, G. Jura, and J. H. Hildebrand, J. Phys. Chem., 6 3 , 297 (1959).
Volume 78, Number 1 8 November 1968
4080
ROBERT E, BARIEAU
or
We now let u1 be the filling composition. That is, we let nl/n = u1
(14)
Then
We may write GT =
nlGl
+ nzGz
n2/n = 1
(6)
- u1
(15)
It is not necessary to identify the partial molal Gibbs
Substituting eq 14 and 15 into eq 13, we have
free energies with particular phases because, for thermodynamic equilibrium
Cv(3 phases) = -ulT
G,I
=
Gt1 = @I1
(7) (1
and GzI =
= (JZIII
(8)
where the superscripts identify the three different phases. Differentiating eq 6 with respect to the temperature at constant total volume and constant nl and nz, we have
(g) -
- ul)T (%) + dT2
An
VTT = u1
ACv(3 phases)In = -T Au1
n = nl
+ n2
(12)
we have
where ET is the total internal energy of the contents of the calorimeter. The Journal of Physical Chemistry
(16)
(17)
(g) (18)
n'n"
We may thus determine dzP/dT2 for a three-phase, two-component system by means of eq 18. We may also make heat capacity measurements a t the same constant volume and the same total number of moles but vary the feed composition. Under these conditions we have, from eq 16
[ As long as three phases are present, &I, 8 2 , and P are independent of the total volume and of the number of moles present. The restrictions on the right-hand side of eq 11 can thus be dropped. Multiplying eq 11 by T and dividing by the total number of moles
(g)
d2P (dCv(3 phases)) - - cVTT ( - @ ) dn lll or with finite differences we have
[ACv(3 phases)]
Differentiating eq 10 with respect to the temperature at constant total volume and constant n1 and n2, we have
T
Equation 16 shows that if we make two heat capacity measurements at constant volume and constant filling composition but vary the total number of moles in the calorimeter, we may evaluate
Substituting eq 9 into eq 5, we have
(10)
-
(d2G1 - '3) -
dT2
dT2
From eq 18, 18, and 19, we can then solve for the three quantities d2Gl/dT2, d2G2/dT2, and d2P/dT2. Equations 16, 18, and 19 apply only when three phases are present. If the saturation pressure is negligible, the third term in eq 16 may be neglected and the resulting equation is equivalent to Rowlinson's treatment. Also, from eq 16 one can see that the smaller VT/n the smaller the contribution of the third term to the measured heat capacity will be. One must, however, have some idea of the magnitude of the third term to justify ignoring it. To ensure that it is possible for the liquid-liquid critical point to exist within the calorimeter, the feed composition and VT/n values must be chosen with care. Let x1 be the mole fraction of component 1 in the liquid phase I, y1 be the mole fraction of component 1 in the liquid phase 11, z1 be the mole fraction of component 1 in the vapor phase 111, 'VI be the molal
INTERPRETATION OF HEATCAPACITY MEASUREMENTS volume of the liquid phase I, VI1 be the molal volume of liquid phase 11, and V"' be the molal volume of vapor phase 111. Let the corresponding critical values for the system when only two phases are present VI11c. be given by XI.^, yl,c; VI,, VI',; Z I , ~ , Under conditions where the liquid-liquid critical point is present in the calorimeter, we have from the conservation of the total number of moles
+
n = n1IC
(20)
nI1Ic
where nlIo is the number of moles present in the critical point liquid and nnI1ICis the number of moles present in the vapor. From the conservation of component one, we may write
uln
=
yl,cnllc
+
(21)
xl,,nI1Ic
VT
= n
,V
I1
+~
z
~
(22) ~
Solving eq '20 and 21 for n", and nI1Ic and substituting in eq 22, we find for the relationship between u1 and VT/n u1 =
z l , c [ ( ~ T / ~-~ )VIIc] VI11
0
+ yl,c[V1I1o - (VT/n)l - V1IC
between the critical liquid composition and the composition of the vapor in equilibrium with the critical liquid. I n addition VT/n and u1 must be related by eq 23. If this equation is not satisfied, it is impossible for the critical liquid composition to appear in the calorimeter and a single liquid phase will appear a t a temperature above or below the true critical solution temperature. Again from the conservation equations, it is possible to show that liquid phase I will disappear when the following equation is satisfied
VT n
- (21
- vi)
(23)
=
(21
- UJV'I
+ (UI - y1)V1I1
(29)
Similarly, liquid phase I1 will disappear when the following equation is satisfied
VT - ($1
Conservation of volume gives I1
4081
n
~
- 21)
~
=
(UI - XJVI
v
~
+ (XI - Ui)V1I1 ~
~
(30)
~
When a single liquid phase and a vapor phase of a two-component system are in equilibrium, the following equation applies to the measured heat capacity at constant total volume
Cv(2 phases) = -ulT
The expressioin for the volume fraction filled with liquid is given by Equation 31 may be written Equation 24 shows that it is possible to have the critical liquid present from a volume fraction of 0-1 by choosing appropriate vrtlues of the filling composition, u1. If u1 = then vT/n = VI11C
(25)
and
nllcV1lo -- - 0 VT If u1 = y1,c
VT/n = VI1,
Equations 16 and 32 are similar, the difference being that the derivatives appearing on the right-hand side of eq 16 are for a system with one degree of freedom, while eq 32 is for a system with two degrees of freedom. Superficially eq 32 appears to represent a system with three degrees of freedom but from the conservation equations we may show that VT/n and u1 are related by
(27) u1
and
n1ICV1Ic -- - 1 VT We may thus state that in order for the critical liquid to be present, the filling composition must be
=
+-Yl[V1rl - (vT/n)l VI1
Zl[(VT/n) - VI1] VI11
(33)
which reduces the degrees of freedom to two. It is hoped that this paper will aid experimenters in interpreting and understanding heat capacity measurements a t total constant volume, through the liquidliquid critical point.
Volume 7% Number 19 November 1968
