State functions reversible/irreversible processes

This section demonstrates calculations of changes in macroscopic properties caused during several specific reversible processes in ideal gases. These will serve as auxiliary calculation pathways for evaluating changes in state functions during irreversible processes. We use this procedure extensively in Chapter 13 on spontaneous processes and the second law of thermodynamics. 

The so-called Grand Potential function J = A — G has occasionally been used to characterize thermodynamic states. Show under what conditions this quantity can be related to reversible performance of work and characterize the properties of this function when irreversible processes occur. 

Equation (2.66) indicates that the entropy for a multipart system is the sum of the entropies of its constituent parts, a result that is almost intuitively obvious. While it has been derived from a calculation involving only reversible processes, entropy is a state function, so that the property of additivity must be completely general, and it must apply to irreversible processes as well. 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

To calculate a change in entropy for a process we find a reversible path between the initial and final states. It is immaterial whether the actual process is irreversible or reversible. Because entropy is a state function, the change for that path will be the same as that for the irreversible path. 

For an irreversible adiabatic expansion in which some work is performed, the work performed is less in magnimde than that in the reversible process because the external pressure is less than the pressure of the gas by a finite amount. Thus, if the final volume is the same as in the reversible process, the final temperature will not be as low in the irreversible process, because, according to Equation (5.47), the temperature drop depends directly on the work performed by the expanding gas. Similarly, from Equations (5.42) and (5.44), AC7 and A//, respectively, also must be numerically smaller in the intermediate expansion than in the reversible expansion. In the adiabatic expansion, from a common set of initial conditions to the same final volume, the values of Af7 and A//, as well as the values of the work performed, seem to depend on the path (see summary in Table 5.2). At first glance, such behavior seems to contradict the assumption that U and H are state functions. Careful consideration shows that the difference occurs because the endpoints of the three paths are different. Even though the final volume can be made the same, the final temperature depends on whether the expansion is free, reversible, or intermediate (Table 5.2). 

The definition of entropy requires that information about a reversible path be available to calculate an entropy change. To obtain the change of entropy in an irreversible process, it is necessary to discover a reversible path between the same initial and final states. As S is a state function, AS is the same for the irreversible as for the reversible process. 

The second law of thermodynamics for an isolated system can be stated as follows There exists a function of the state of a uniform phase called the entropy which is conserved for any reversible process and which increases for any irreversible process over all of the phases participating in the process... 

The state function which measures disorder is the entropy, S, and the second law of thermodynamics may be stated as follows The entropy of the universe or of an isolated system always increases when a spontaneous irreversible process occurs entropy remains constant in a reversible process, i.e., a process which remains at equilibrium for every step along the way,... 

Since entropy is a state function, we can use a reversible process with the same initial and final states to calculate the entropy change that occurs in an irreversible process. 

The entropy of the phase transition, AtiansiS , taking place at a fixed temperature, can be calculated using equation (13.14), Frame 13 and since pressure is constant state function, the enthalpy of transition, Alrans// will be identical both for reversible and irreversible changes, so that ... 

Since the heat supplied, Aq at constant pressure, P is equal to the change in enthalpy, AH, which is itself a state function - and is therefore identical for both reversible and irreversible processes - hence we can write ... 

The equations developed in this section have been derived for mechanically reversible nonflow processes involving ideal gases. However, those equations which relate state functions only are valid for ideal gases regardless of the process and apply equally to reversible and irreversible flow and nonflow processes, because changes in state functions depend only on the initial and Anal states of the system. On the other hand, an equation for Q or W is specific to the case considered in its derivation. 

When a system undergoes an irreversible process from one equilibrium state to another, the entropy change of the system AS is stilt evaluated by Eq. (A). In this case Eq. (A) is applied to an arbitrarily chosen reversible process that accomplishes the same change of state. Integration is not carried out for the original irreversible path. Since entropy is a state function, the entropy changes of the irreversible and reversible processes are identical. 

In the above we have invented a new function of state, A = E — TS, involving state functions that had been previously introduced. A is called the Helmholtz (free) energy function. The right-hand side follows from Eq. (1.12.7d). As is seen, changes in A are tracked by the reversible performance of work at constant T. If no work is involved, but irreversible (and therefore, uncontrollable) processes are allowed to occur at constant temperature, we find from the above that... 

The successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

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