The Temperature Dependence of The Equilibrium Constant

Although The exact calculation of the effect of temperature changes upon an equilibrium is somewhat more involved than we can treat here, we can make several approximate relationships that are very useful. Remember that the equilibrium constant is related to the standard free energy change of the reaction

And recall that the free energy change upon reaction is related to both the enthalpy and entropy changes

Although the free energy change is strongly dependent on temperature, usually the enthalpy and entropy changes upon reaction are NOT temperature dependent (or at least not very much so). This implies that the free energy is a linear function of temperature. Combining the above two equations:

substituting in the equilibrium constant

and finally

Remember that the enthalpy and entropy changes are regarded as constants with respect to temperature, so the equilibrium constant is exponentially dependent on temperature. The relative change in equilibrium constant as a function of temperature is usually not expressed as above but as the van't Hoff equation

This equation looks very reminiscent of the Clausius-Clapeyron equation for vapor pressures and the Arrhenius equation for the temperature dependence of reaction rate constants, doesn't it? This form of temperature dependence (exponential w.r.t. 1/T) comes up over and over in chemistry.

Example:

Consider the Reaction of NO with Cl2:

The Thermodynamic quantities associated with this reaction are as follows:

At what temperature is the equilibrium constant, Kp, for this reaction equal to 1.0 x 103 ?

Sol'n:

Assuming the enthalpy and entropy values are independent of temperature, we need to calculate the temperature at which the free energy change will imply the desired equilibrium constant.

Using the given information


One for the Bag of Tricks... Successive Approximation

Often in the course of solving for equilibrium concentrations, the equilibrium equation appears to be an impossibly complicated equation in x, the equilibrium advancement. Sometimes the method of successive approximation is useful

Example

The reaction

This reaction has a KC of 7.4 x 10-3 at 1000K. If 1 mole each of methane and water are introduced into a 10 L vessel at 1000K, what is the equilibrium number of moles of H2 in the vessel?

Sol'n

The setup for this problem is quite straightforward. We must first parameterize the number of moles and concentrations of each species in terms of the initial conditions and the advancement of the reaction.

Inserting the information we are given and calling x the number of moles of reaction in the forward direction

OK the equilibrium number of moles of H2 is just 4x, but how do we solve this equation for x?

First, lets simplify this a little by multiplying out all those powers of the volume

Now, lets assume that x and 2x are much less than 1.00, the initial number of moles of the reactants. Then

This equation can easily be solved for an approximate value of x0 = (0.74/256)1/5 = 0.31. But 0.31 is not very much smaller than 1.00 so our initial approximation is not a very good one. So we put this first guess back into the equation where we had assumed x was zero initially

Our next best guess for the value of x1 is ((0.74)(0.69)(0.38)2/(256))1/5 = 0.20. This is better approximation of the value of x, but we can iterate this procedure until the value of x remains constant to within the significant figures of the problem. Our next best guess will be the solution to the equation.

Which yields a value of x= 0.24. Continuation of this method yields x= 0.22, x4 = 0.23, x5 = 0.23 etc. so we may stop after the fifth iteration if we want the answer to two significant figures. Cute, huh?

The final solution? The number of moles of H2 at equilibrium = 4 (0.23) = 0.92 moles.


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