# CRE: Equilibrium Constants and the Gibbs Free Energy

### Equilibrium Constants and $\Delta g_{rxn}^o$

For reactive equilibrium, we then require that:

$\displaystyle{\frac{dG}{d\xi}=0}$

which we can write as

$\displaystyle{\frac{dG}{d\xi}=0=\frac{d}{d\xi}\left(\sum_in_i\mu_i\right)=\sum_i\mu_i \frac{dn_i}{d\xi}=\sum_i\mu_i \frac{d(n_{i_o}+\nu_i\xi)}{d\xi}=\sum_i\mu_i \nu_i}$

so our criterion for reactive equilibrium is

$0=\sum_i \nu_i \mu_i$

recalling that $\mu_i$ is given by (at standard state):

$\mu_i = g_i^o + RT \ln \left [\frac{\hat f_i}{f_i^o} \right ]$

we can combine these expressions to get:

$0 = \sum_i \nu_i\left [g_i^o + RT \ln \left [\frac{\hat f_i}{f_i^o} \right ]\right ]$

Rearranging we get:

$-\frac{\sum_i \nu_i g_i^o}{RT} = \sum_i \nu_i \ln \left [\frac{\hat f_i}{f_i^o} \right ]$

##### Definition:

We define the Gibbs Free Energy change of reaction ($\Delta g_{rxn}^o$) in a manner similar to $\Delta h_{rxn}^o$ (from Hess's Law) so that it is:

$\displaystyle{\Delta g_{rxn}^o = \sum_i \nu_i g_i^o}$.

Using this definition and two ln rules (the first is that $a\ln[x] = \ln\left [x^a\right]$, while the second is the inverse of the product rule), we get:

$-\frac{\Delta g_{rxn}^o}{RT} = \ln \left ( \Pi_i \left [\frac{\hat f_i}{f_i^o} \right ]^{\nu_i} \right )$

##### Note:

The $\Pi_i$ operator denotes the product of all $i$ components (much like $\sum_i$ denotes the sum of components $i$).

Because $\Delta g_{rxn}^o$ is at standard state, it is a function only of $T$. For simplicity, we then define:

##### Definition:

The Reaction Equilibrium Constant ($K$) is also a function only of temperature and is defined as:

$\displaystyle{\ln K = -\frac{\Delta g_{rxn}^o}{RT}}$

Combining this definition with our equation thus far we get:

$K = \left { \Pi_i \left [\frac{\hat f_i}{f_i^o} \right ]^{\nu_i} \right }$

##### Note:

This is essentially what we are used to as a typical equilibrium expression (from Freshman Chemistry, for example), except that now we are explicitly accounting for species and mixing non-idealities (by using fugacities).

##### Outcome:

Use thermochemical data to calculate the equilibrium constant and its dependence on temperature

##### Test Yourself:

Use tabulated values of $\Delta g_{rxn}^o$ to determine the equilibrium constant at 25C for the following reaction:

$C_2H_4(g)+H_2O(g) \Longleftrightarrow C_2H_5OH(g)$