The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev

Thermodynamics

Civil Engineering (CE) : The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev

The document The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Thermodynamics.
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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 Civil Engineering (CE) Notes | EduRev               ..(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 Civil Engineering (CE) Notes | EduRev       ..(8.40)

 

It follows that: The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev            ..(8.41)

 

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

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                        ...(8.42)

Hence by eqn. 8.41 and 8.42:   

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                        ...(8.43)

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

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                        ...(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 Civil Engineering (CE) Notes | EduRev                        ...(8.44)

The ratioThe Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev is called the activity aˆi of species i in the reaction mixture, i.e.:

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                        ...(8.45)

Thus, the preceding equation becomes: The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev               ..(8.46)

 

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

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                        ...(8.47)

On further re-organization we have:

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev      

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                                ...(8.48)

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

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(8.49)
The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(8.50)


On comparing eqns. 8.49 and 8.50 it follows:
The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(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 Civil Engineering (CE) Notes | EduRev 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 Civil Engineering (CE) Notes | EduRev 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 Civil Engineering (CE) Notes | EduRev                         ...(8.52)

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(8.53)

Taking a differential of eqn. 8.53:

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(8.53)

Now using eqn. 8.18:

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(8.54)

On further use of eqn. 8.13:

 

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev 

Lastly, upon integration one obtains the following expression:

 

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                         ...(8.55)

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

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

 

The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev                          ...(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 Civil Engineering (CE) Notes | EduRev , the equilibrium constant decreases with increasing temperature. On the other hand, if the reaction is endothermic, i.e., The Equilibrium Constant of Reactions Civil Engineering (CE) Notes | EduRev , 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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