### 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

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:

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