Читать книгу Geochemistry - William M. White - Страница 54

Constant pressure and constant temperature heat capacities are different because there is energy associated (work done) with expansion and contraction. Thus how much energy we must transfer to a substance to raise its temperature will depend on whether some of this energy will be consumed in this process of expansion. These energy changes are due to potential energy changes associated with changing the position of an atom or molecule in the electrostatic fields of its neighbors. The difference between C_V and C_p reflects this energy associated with volume. Let us now determine what this difference is.

We can combine relations 2.67 and 2.68 as:

(2.70)

From this, we may derive the following relationship:

(2.71)

and further:

(2.72)

It can also be shown that, for a reversible process:

(2.73)

(∂U/∂V)_T is the energy associated with the volume occupied by a substance and is known as the internal pressure (P_int, which we introduced earlier in our discussion of the van der Waals law, e.g., eqn. 2.17). It is a measure of the energy associated with the forces holding molecules or atoms together. For real substances, energy changes associated with volume changes reflect potential energy increases associated with increased separation between charged molecules and/or atoms; there are no such forces in an ideal gas, so this term is 0 for an ideal gas. Substituting eqn. 2.73 into 2.72, we obtain:

(2.74)

Thus, the difference between C_p and C_v will depend on temperature and pressure for real substances. The terms on the right will always be positive, so that C_p will always be greater than C_v. This accords with our expectation, since energy will be consumed in expansion when a substance is heated at constant pressure, whereas this will not be the case for heating at constant volume. For an ideal gas, .

As it is impractical to measure C_v for solids and liquids, only experimentally determined values of C_p are available for them, and values of C_V must be obtained from eqn. 2.74 when required.

We found earlier that C_p is the variation of heat with temperature at constant pressure. How does this differ from the variation of energy with temperature at constant volume? To answer this question, we rearrange eqn. 2.71 and substitute C_V for (∂U/∂T)_V and Vα for (∂V/∂T)_P. After simplifying the result, we obtain (on a molar basis):

(2.75)

For an ideal gas, the term PVα reduces to R, so that (∂U/∂T)_P = C_p − R. C_p − R may be shown to be equal to C_V, so the energy change with temperature for an ideal gas is the same for both constant pressure and constant volume conditions. This is consistent with the notion that the difference between C_p and C_v reflects the energy associated with, and changing distances between, atoms and molecules in the presence of attractive forces between them. In an ideal gas, there are no such forces, hence .