4.3 Partial derivatives of internal energy and enthalpy

 

Consider some of the most important relationships for the partial derivatives of internal energy and enthalpy. From Eq. (4.14) we obtain:

 

                                                                                                         (4.24)

 

Substituting into Eq. (4.24) the value of  from Maxwell's relation (4.23), we find that

 

                                                                                                         (4.25)

 

This relationship characterizes the dependence of internal energy on volume in an isothermal process.

Proceeding as above, we obtain the relationship characterizing the dependence of internal energy on pressure in an isothermal process:

 

                                                                                               (4.26)

 

whence, with account taken of Eq. (4.22), we find:

 

                                                                                                (4.27)

 

Since

 

                                                               

 

it is obvious that

 

                                                                                                                 (4.28)

 

and, consequently, Eq. (4.14) takes the following form:

 

                                                                                                                         (4.29)

 

whence

 

                                                                                                               (4.30)

 

Taking into account Eq. (4.22), we obtain:

 

                                                                                                           (4.31)

 

The above relation characterizes the dependence of enthalpy on pressure in an isothermal process.

Proceeding in a similar manner, we can derive the formula characterizing the dependence of enthalpy on volume in an isothermal process:

 

                                                                                               (4.32)

 

and, allowing for Eq. (4.23), we obtain:

 

                                                                                               (4.33)

 

As was already shown in Chapter 2, one of the properties of an ideal gas consists in that its caloric quantities are independent of thermal parameters:

 

                                                 and   

 

The following "proof” of the independence of the internal energy of an ideal gas on volume (and, respectively, of enthalpy on pressure) can sometimes be encountered in the literature devoted to the subject: since from Clapeyron's equation

 

                                                                 

 

it is clear that

 

                                                                 

 

substituting this value of  in equation (4.25) we get:

 

                                                                  

 

This proof, however, is deceptive. In fact, the dependence of internal energy u on volume v constitutes, as was already mentioned in Chapter 2, an independent, special property of an ideal gas and it is not related in any way to its other property, the property of an ideal gas obeying Clapeyron's equation. In Chapter 3 the independence of the internal energy of an ideal gas on volume was used to show that the ideal-gas temperature scale is identical with the Kelvin absolute thermodynamic scale. It is precisely this identity that permits us to use Clapeyron's equation in any thermodynamic calculations. Thus, the fact that  was already taken into account in Clapeyron's equation when the ideal-gas temperature was replaced in the equation with the absolute thermodynamic temperature (Sec. 3.5). Consequently, the above "proof” only states once more the previously known fact.

The equations derived in this section, especially Eqs. (4.25) and (4.31), are of paramount importance for thermodynamic studies of the properties of substances. Making use of the thermal properties of a substance (specific volume as a function of temperature and pressure), Eqs. (4.25) and (4.31) permit us to find the caloric quantities, i.e. the internal energy and enthalpy, and also to solve the reverse problem, i.e. using the known caloric quantities we can calculate the thermal properties of substances. At a given pressure p and temperature T the enthalpy of a substance is determined by integrating Eq. (4.31):

 

                                                                               (4.34)

 

where h (p0, T) is the enthalpy of the substance in some initial state, characterized by the same temperature but by another pressure p0. By analogy,

 

                                                                                     (4.35)

 

where u (v0, T) is the internal energy of the substance in a state characterized by the same temperature T but by another specific volume v0.

If data on the thermal properties of a substance are available, one can calculate the integrals present in the right-hand sides of Eqs. (4.34) and (4.35), with it being necessary then to first calculate the magnitudes of the derivatives  or . In both cases the integration is performed along an isotherm.

It should be emphasized that Eqs. (4.34) and (4.35) permit us to calculate not the values of h and u themselves (this problem cannot be solved by thermodynamic methods alone) but only the difference between the value of h (or u) in a given state and its value in any other state (initial) on the same isotherm.

To solve the reverse problem, i.e. to calculate the thermal quantities of a substance on the basis of the known caloric properties, we must transform Eqs. (4.25) and (4.31) in the following way:

 

                                                                                                           (4.36)

 

and

 

                                                                                                                  (4.37)

 

Solving Eqs. (4.36) and (4.37), we obtain, respectively:

 

                                                                                    (4.38)

 

and

 

                                                                                   (4.39)

 

where p0 (v, T0) and v0 (p, T0) are the values of p and v in some initial state having the same specific volume [as applied to Eq. (4.38)] or the same pressure [as applied to Eq. (4.39)] as in the unknown state. It will be noted that the integral in Eq. (4.38) is taken along an isochor, and the integral in Eq. (4.39) along an isobar. The partial derivatives of caloric quantities present in the integrands are calculated from the known caloric properties of the substance.

Equations (4.38) and (4.39) are rarely used in practice, whereas Eqs. (4.34) and (4.35) find wide application in calculating the thermodynamic properties of substances.

In conclusion let as calculate some important derivatives of entropy:

 

   and

 

From Eq. (4.14) it follows that

 

                                                                                                                                                                    (4.40)

 

and

 

                                                                                                                                                                  (4.41)

 

and from equation (4.29) we get:

 

                                                                                                                                                                          (4.42)

 

and

 

                                                                                                                                                                     (4.43)