William M. White - Geochemistry

where S 0is the entropy at 0 K (the configurational, or third law entropy) and ΔS φ is the entropy change associated with any phase change. Compilations for S 298are available for many minerals. Table 2.2lists some heat capacity constants for the power series formula as well as other thermodynamic data for a few geologically important minerals. Example 2.6illustrates how entropy and enthalpy changes are calculated.

If the heat capacity of steam can be represented by a three-term power series:

with constants , , and , and the enthalpy of vaporization at 100°C is 40.6 kJ/mol, calculate the S and H changes when 1 mol of liquid water at 100°C and 1 atm is converted to steam and brought to 200°C and 3 atm. Assume that with respect to volume, steam behaves as an ideal gas (which, in reality, it is certainly not).

Answer : We need to calculate entropy and enthalpy associated with three changes: the conversion of water to steam, raising the steam from 100°C to 200°C, and increasing the pressure from 1 atm to 3 atm. Since both S and H are state variables, we can treat these three processes separately; our answer will be the sum of the result for each of these processes and will be independent of the order in which we do these calculations.

1 Conversion of water to steam. This process will result in ΔH of 40.6 kJ. For entropy, . We converted centigrade to Kelvin, or absolute, temperature.

2 Raising the steam from 100°C to 200°C (from 373 K to 473 K) isobarically. Since heat capacity is a function of temperature, we will have to integrate eqn. 2.112over the temperature interval:Evaluating this, we find that . The entropy change is given by:Evaluating this, we find that .

3 Increasing pressure from 1 atm to 3 atm (0.1 MPa to 0.3 MPa) isothermally. We can use eqn. 2.114to determine the enthalpy change associated with the pressure change. On the assumption of ideal gas behavior, we can substitute 1/T for α. Doing so, we find the equation goes to 0; thus, there is no enthalpy change associated with a pressure change for an ideal gas. This is in accord with assumptions about an ideal gas: namely, that there are no forces between molecules, hence no energy is stored as potential energy of attraction between molecules.

The isothermal pressure dependence of entropy is given by eqn. 2.106. We substitute 1/ T for α and R T/P for V and integrate from P 1to P 2:

The whole enthalpy and entropy changes are the sum of the changes in these three steps:

We can now introduce two free energy functions, the Helmholtz free energy and Gibbs free energy. Gibbs free energy is one of the most useful functions in thermodynamics.

We can rearrange eqn. 2.58to read d U − T d S = − P d V . The − P d V term is the work term and the T d S term is the heat function. T d S is the energy unavailable for work. Therefore, d U − T d S is the amount of internal energy available for work , or the free energy . We define it as A , the Helmholtz free energy:

(2.119)

As usual, we are interested in the differential form (since we are more interested in changes than in absolutes):

(2.120)

or substituting eqn. 2.58into 2.120:

(2.121)

The Gibbs free energy is perhaps misnamed. By analogy to the Helmholtz free energy, it should be called the free enthalpy (but enthalpy is an energy), because it is derived as follows:

(2.122)

and

(2.123)

or

which reduces to:

(2.124)

Notice the similarity to the Helmholtz free energy; in that case we subtracted the TS term from the internal energy; in this case we subtracted the TS term from the enthalpy. The Gibbs free energy is the energy available for nonPV work (such as chemical work) . It has two other important properties: its independent variables are T and P , generally the ones in which we are most interested in geochemistry, and it contains the entropy term (as does the Helmholtz free energy), and hence can be used as an indication of the direction in which spontaneous reactions will occur.

For a finite change at constant temperature, the Gibbs free energy change is:

(2.125)

The free energy change of formation, , is related to the enthalpy and entropy change of reaction:

(2.126)