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The Equilibrium Constant of Reactions
Since chemical composition of a reactive system undergoes change during a reaction, one may use the eqn. 6.41 for total differential of the Gibbs free energy change (for a single phase system):
..(6.41)
For simplicity considering a single reaction occurring in a closed system one can rewrite the last equation using eqn. 8.3:
..(8.40)
It follows that: ..(8.41)
On further applying the general condition of thermodynamic equilibrium given by eqn. 6.36b it follows that:
...(8.42)
Hence by eqn. 8.41 and 8.42:
...(8.43)
Since the reactive system is usually a mixture one may use the eqn. 6.123:
...(6.123)
Integration of this equation at constant T from the standard state of species i to the reaction pressure:
...(8.44)
The ratio is called the activity aˆi of species i in the reaction mixture, i.e.:
...(8.45)
Thus, the preceding equation becomes: ..(8.46)
Using eqns. 8.46 and 8.44 in eqn. 8.43 to eliminate �_{i} gives
...(8.47)
On further reorganization we have:
...(8.48)
Where, ∏ signifies the product over all species i. Alternately:
...(8.49)
...(8.50)
On comparing eqns. 8.49 and 8.50 it follows:
...(8.51)
The parameter K_{T} is defined as the equilibrium constant for the reaction at a given temperature. Since the standard Gibbs free energy of pure species depends only on temperature, the equilibrium constant K_{T} is also a function of temperature alone. On the other hand, by eqn. 8.50 K_{T }is a function of which is in turn a function of composition, temperature and pressure. Thus, it follows that since temperature fixes the equilibrium constant, any variation in the pressure of the reaction must lead to a change of equilibrium composition subject to the constraint of K_{T }remaining constant. Equation (8.51) may also be written as:
...(8.52)
...(8.53)
Taking a differential of eqn. 8.53:
...(8.53)
Now using eqn. 8.18:
...(8.54)
On further use of eqn. 8.13:
Lastly, upon integration one obtains the following expression:
...(8.55)
Where, K_{T0} is the reaction equilibrium constant at a temperature T_{0}
If is assumed independent of T (i.e. , over a given range of temperature (T_{2} −T_{1}) , a simpler relationship follows from eqn. 8.54:
...(8.55)
The above equation suggests that a plot of ln K_{T} vs. 1/ T is expected to approximate a straight line. It also makes possible the estimation of the equilibrium constant at a temperature given its values at nother temperature. However,
eqn. 8.55 provides a more rigorous expression of the equilibrium constant as a function of temperature. Equation 8.54 gives an important clue to the variation of the equilibrium constant depending on the heat effect of the reaction. Thus, if the reaction is exothermic, i.e., , the equilibrium constant decreases with increasing temperature. On the other hand, if the reaction is endothermic, i.e., , equilibrium constant increases with increasing temperature. As we shall see in the following section, the equilibrium conversion also follows the same pattern.
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