Entropy Change for Temperature Equilibration Processes
C. W. Pyun
Carnegie-Mellon University Pittsburgh, Pennsylvania 15213
I n most elementary textbooks of physical chemistry and thermodynamics, we find examples and problems of calculating entropy change for irreversible temperature equilibration processes, in which two systems a t different temperatures are brought into thermal contact under either constanbvolume or constant-pressure conditions. Typically, one is asked to consider an iron ball a t 90°C with a heat capacity of 60.0 cal deg-' added to 360 g of water a t 20°C, the combined system enclosed in an adiabatic container. Then one is asked to calculate the entropy change accompanying the irreversible heat transfer between the metal and water. Sometimes, one of the systems combined may be a constanttemperature heat bath. This type of numerical computation illustrates in a concrete manner that although the entropy change is negative for the component subsystem which had higher initial temperature and positive for the other, the total entropy change of the combined system as a whole is positive for irreversible adiabatic temperature equilibration processes. Some students may, however, want a more general proof of this for any physically possible set of initial temperatures and heat capacities. Since it does not seem usually available in textbooks, an elementary derivation is given here. One interesting feature of the proof is the appearance of the classical arithmetic-mean-geometric-mean inequality in connection with the entropy inequality. For the systems with one component as a heat bath, expansion of ln(1 - x) and (1 - x)-I in power series of x is employed. Two Systems with Finite Heat Capacity
A system of heat capacity1 Cl a t temperature TI is brought into thermal contact with another system with Cz and a t Ta. Take Tl < T2 and assume that Cx and Cz are independent of temperature. The equilibrium temperature TI of the combined adiabatic system is obtained from heat balance equation CI(TJ - TI) = CdTs
- TJ)
+
+
T, = (CLT, C%Td/(C, Cd It is seen that the final temperature TI is the arithmetic mean of Tl and T2 with weighting factor of Cl and Cz, respectively. Now the entropy change for this temperature equilibration process is
This is then further transformed as follows
We note that T,is nothing but the geometric mean of Tl and Tz weighted by C1 and Cz, respectively. Now, one of the most famous inequalities in classical analysis asserts that the geometric mean cannot be greater than the arithmetic mean.? That is T, Z T, and therefore AS
>0
where the equality sign holds for TI = T2mly. Therefore, AS for the processes of this type is always positive except when TI = T2, in which case the two component systems were already a t the same temperature from the beginning and therefore AS = 0. Two Systems, One with Infinite Heat Capacity
If one of the component systems is a constant temperature heat bath, we may visualize it as a system with an infinite heat capacity. Alternatively, a finite but sufficiently large amount of a two phase system such as a water-ice mixture can serve a similar p u r p o ~ e . ~ First consider the case when the heat bath temperature T, is higher than that of the other component system TI (TI < T,). Then A S for the latter is again Cl ln(Tb/Tl) and AS for the bath is -Cl(Tb - T1)/Tb. The total entropy change for the combined system is therefore4
1 Take CC, far a constant-pressure process and C, for a constantvolume process. 9 For the special case of C, = CSthis means
cl/*,cn
+ Td 2 dm
which can be proved immediately far any positive values of TI and T2. For more general cases, see, for example, BECICENBACH, E., AND BELLMAN, R., "An Introduction to Inequalities," Random House, N. Y., 1961. This is a volume in the New Mathematical Library intended for high school students and laymen. m y combining the result of this section with that of the preceding section, we can extend the proof of AS > 0 to slightly mare complex cases, such as that of ice-water mixture as one subsystem but with insufficient amount ,of ice or water to serve as a constant temperature bath. * T E E HAAR,D., AND WERGELAND, H., "Elements of Thermodynamics," Addison-Wesley Publishing Company, Reading, Massschusetts, 1966, p. 28. Volume 46, Number 10, Odober 1969
/
677
If we put z = (Tb- T l ) / T bor T l / T b = 1 can be rewritten as AS/C,
=
-ln(l
- x, this Putting y
O
