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The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)

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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):

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)               ..(6.41)

For simplicity considering a single reaction occurring in a closed system one can rewrite the last equation using eqn. 8.3:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)       ..(8.40)

 

It follows that: The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)            ..(8.41)

 

On further applying the general condition of thermodynamic equilibrium given by eqn. 6.36b it follows that:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                        ...(8.42)

Hence by eqn. 8.41 and 8.42:   

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                        ...(8.43)

Since the reactive system is usually a mixture one may use the eqn. 6.123: 

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                        ...(6.123)

Integration of this equation at constant T from the standard state of species i to the reaction pressure:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                        ...(8.44)

The ratioThe Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) is called the activity aˆi of species i in the reaction mixture, i.e.:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                        ...(8.45)

Thus, the preceding equation becomes: The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)               ..(8.46)

 

Using eqns. 8.46 and 8.44 in eqn. 8.43 to eliminate �i gives 

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                        ...(8.47)

On further re-organization we have:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)      

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                                ...(8.48)

Where, ∏ signifies the product over all species i. Alternately:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.49)
The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.50)


On comparing eqns. 8.49 and 8.50 it follows:
The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.51)

The parameter KT is defined as the equilibrium constant for the reaction at a given temperature. Since the standard Gibbs free energy of pure species The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) depends only on temperature, the equilibrium constant KT is also a function of temperature alone. On the other hand, by eqn. 8.50 Kis a function of  The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) 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 Kremaining constant. Equation (8.51)  may also be written as:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.52)

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.53)

Taking a differential of eqn. 8.53:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.53)

Now using eqn. 8.18:

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.54)

On further use of eqn. 8.13:

 

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) 

Lastly, upon integration one obtains the following expression:

 

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                         ...(8.55)

Where, KT0 is the reaction equilibrium constant at a temperature T0 

If The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) is assumed independent of T (i.e. The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) , over a given range of temperature (T2 −T1) , a simpler relationship follows from eqn. 8.54:

 

The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE)                          ...(8.55)

The above equation suggests that a plot of ln KT 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 of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) , the equilibrium constant decreases with increasing temperature. On the other hand, if the reaction is endothermic, i.e., The Equilibrium Constant of Reactions Notes | Study Additional Documents & Tests for Civil Engineering (CE) - Civil Engineering (CE) , 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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