15.6

Specific Heat Capacities

In this section the first law of thermodynamics is used to gain an understanding of the factors that determine the specific heat capacity of a material. Remember, when the temperature of a substance changes as a result of heat flow, the change in temperature and the amount of heat Q are related according to (Equation 12.4). In this expression c denotes the specific heat capacity in units of J/(kg · C°), and m is the mass in kilograms. Now, however, it is more convenient to express the amount of material as the number of moles n, rather than the number of kilograms. Therefore, we replace the expression with the following analogous expression:

| (15.6)

where the capital letter C (as opposed to the lowercase c) refers to the molar specific heat capacity in units of J/(mol · K). In addition, the unit for measuring the temperature change is the kelvin (K) rather than the Celsius degree (C°), and , where Tf and Ti are the final and initial temperatures. For gases it is necessary to distinguish between the molar specific heat capacities CP and CV, which apply, respectively, to conditions of constant pressure and constant volume. With the help of the first law of thermodynamics and an ideal gas as an example, it is possible to see why CP and CV differ.
To determine the molar specific heat capacities, we must first calculate the heat Q needed to raise the temperature of an ideal gas from Ti to Tf. According to the first law, . We also know that the internal energy of a monatomic ideal gas is (Equation 14.7). As a result, . When the heating process occurs at constant pressure, the work done is given by Equation 15.2: . For an ideal gas, , so the work becomes . On the other hand, when the volume is constant, , and the work done is zero. The calculation of the heat is summarized below:

The molar specific heat capacities can now be determined, since Equation 15.6 indicates that :

| (15.7)

| (15.8)

The ratio of the specific heats is

| (15.9)

For real monatomic gases near room temperature, experimental values of CP and CV give ratios very close to the theoretical value of .
Many gases are not monatomic. Instead they consist of molecules formed from more than one atom. The oxygen in our atmosphere, for example, is a diatomic gas, for it consists of molecules formed from two oxygen atoms. Similarly, atmospheric nitrogen is a diatomic gas consisting of molecules formed from two nitrogen atoms. Whereas the individual atoms in a monatomic ideal gas can exhibit only translational motion, the molecules in a diatomic ideal gas can exhibit translational and rotational motion, as well as vibrational motion at sufficiently high temperatures. The result of such additional motions is that Equations 15.7–15.9 do not apply to a diatomic ideal gas. Instead, if the temperature is sufficiently moderate that the diatomic molecules do not vibrate, the molar specific heat capacities of a diatomic ideal gas are and , with the result that .
The difference between CP and CV arises because work is done when the gas expands in response to the addition of heat under conditions of constant pressure, whereas no work is done under conditions of constant volume. For a monatomic ideal gas, CP exceeds CV by an amount equal to R, the ideal gas constant:

| (15.10)

In fact, it can be shown that Equation 15.10 applies to any kind of ideal gas-monatomic, diatomic, etc.

Test your understanding of the central ideas presented in Sections 15.1–15.6:

· | The Zeroth and First Laws of Thermodynamics
· | Specific Heat Capacities
· | Thermal Processes

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.