USEFUL WORK AND THE GIBBS AND HELMHOLTZ FUNCTIONS

7.5 USEFUL WORK AND THE GIBBS AND HELMHOLTZ FUNCTIONS

Thus far we have observed that the Gibbs and Planck functions provide the criteria of spontaneity and equilibrium in isothermal changes of state at constant pressure. If we extend our analysis to systems in which other constraints are placed on the system, and therefore work other than mechanical work can be performed, we find that the Gibbs and Helmholtz functions also supply a means for calculating the maximum magnitude of work obtainable from an isothermal change.

Isothermal Changes We can begin with Equation (7.1) as a statement of the combined first and second

laws, which were rearranged to

(7 :72) As we are concerned with isothermal changes, 2SdT can be added to the left side of

dU

Equation (7.74) without changing its value:

(7 :73) As constraints other than the constant pressure of the environment are now

dA dW

considered, the one-to-one relationships between reversibility and equilibrium on the one hand and irreversibility and spontaneity on the other hand are no longer valid. A spontaneous change of state or the opposite change, a nonspontaneous change, can be carried out reversibly by the appropriate adjustment of a constraint, such as an electrical voltage. As before, the inequality applies to an irreversible process and the equality to a reversible process. If the change of state is spontaneous,

dA is negative, work can be performed on the surroundings, and DW is negative. The

EQUILIBRIUM AND SPONTANEITY FOR SYSTEMS AT CONSTANT TEMPERATURE

value for dA is the same for a change of state whether it is carried out reversibly or irreversibly. The reversible work, DW rev , then is equal to dA, whereas the irreversible work, DW irrev , is algebraically greater than dA but smaller in magnitude. For a macro- scopic change, we can write

DA ¼ W rev

DA , W irrev

The change in the Helmholtz function thus provides a limiting value for the magni- tude of the total work (including work against the pressure of the atmosphere) obtain- able in any spontaneous, isothermal process. That is,

(7 :75) and the magnitude of the reversible work is a maximum. If the change of state is not

jW rev j . jW irrev j

spontaneous, dA is positive, work must be performed on the system to produce the change, and DW is positive. Then W rev is the minimum work required to carry out

a nonspontaneous change of state. An interesting alternative demonstration of Equation (7.75) can be carried out on the basis of isothermal cycles and of the Kelvin – Planck statement of the second law. Consider two possible methods of going from State a to State b, a spontaneous change of state, in an isothermal fashion (Fig. 7.1): (1) a reversible process and (2) an irreversible process.

For each path, the first law of thermodynamics is valid:

DU rev ¼Q rev þW rev DU irrev ¼Q irrev þW irrev DU rev ¼ DU irrev

Figure 7.1. An isothermal process.

177 TABLE 7.6. Isothermal Cycle

7.5 USEFUL WORK AND THE GIBBS AND HELMHOLTZ FUNCTIONS

Irreversible Process Reversible Process Net for Both

Processes Heat absorbed

(Forward)

(Backward)

Q irrev 2Q rev .0 Work performed

Q irrev

2Q rev

W irrev 2W rev ,0 Both processes begin and end at the same points, and U is a state function; thus,

W irrev

2W rev

Q rev þW rev ¼Q irrev þW irrev

or

(7 :76) Let us assume that the spontaneous irreversible process gives work of a greater mag-

nitude than the spontaneous reversible one. In that case

jW irrev j . jW rev j

and, from Equation (7.78), as W is negative for both alternatives,

Q irrev .Q rev

Let us use the irreversible process (which goes in only one direction) to carry the system from State a to State b and the reversible process to return the system to its initial state. We can construct a table for the various steps (Table 7.6). As we can see from Table 7.6, the net result is that a positive amount of heat has been absorbed and work has been performed on the surroundings in an isothermal cycle. However, such a consequence is in contradiction to the Kelvin – Planck statement of the second law of thermodynamics, which denies the possibility of converting heat from a reser- voir at constant temperature into work without some accompanying changes in the reservoir or its surroundings. In the postulated cyclical process, no such accompany- ing changes have occurred. Hence, the original assumption is incorrect and the irre- versible work cannot be greater in magnitude than the reversible work:

(7 :77) Thus, the reversible work is a limiting maximum value for the magnitude of work

jW rev

irrev j

obtainable in an isothermal change, with the equality applying to the limit when the process becomes reversible.

Changes at Constant Temperature and Pressure Equation (7.74) can be rewritten to include explicit reference to DW net , the net useful

(non-PdV ) work, by substituting 2P 0 dV þDW net for DW. That is,

dU þP 0 dV net

EQUILIBRIUM AND SPONTANEITY FOR SYSTEMS AT CONSTANT TEMPERATURE

For a constant-pressure process, PdV can be substituted for P 0 dV , and VdP can

be added without changing the value of the expression. As the temperature is con- stant, 2SdT also can be added. With these additions and substitutions, Equation

(7.78) becomes dU

net

or [see Equation (7.15)]

(7 :79) For a spontaneous change at constant T and P, dG is negative, work can be obtained,

dG DW net

and DW net is negative. The value of dG is the same for a given change of state whether it proceeds irreversibly in the absence of additional constraints, or whether it follows a reversible path or proceeds irreversibly when subjected to additional con- straints (for example, electrical). If the process is reversible, the equality in Equation (7.79) applies. If the process is irreversible, the inequality applies. Thus DW net,rev is equal to dG, whereas DW net,irrev is greater algebraically than dG but smaller in mag- nitude. For a macroscopic change, we can write

DG ¼ W net,rev

(7 DG , W :80)

net,irrev

If the change of state is spontaneous, jDGj is equal to the maximum magnitude of non-PdV work that can be obtained. If the change of state is nonspontaneous, DG is equal to the minimum non-PdV work that must be performed to carry out the change.

Relationship between DH P and Q P When Useful Work is Performed We repeatedly have used the relationship [Equation (4.6)]