# CRE: Liquid and Solid Phase Reaction Equilibrium

### Reaction Equilibrium in the Solid and Liquid Phases

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

Here, the $f_i^o$ value is the fugacity of species $i$ at the reference pressure of 1 bar.

As with liquids and solids in the phase equilibrium sections of the course, we will consider this reference value to be the pure species fugacity ($f_i$), but now it will be specifically at the reference pressure of 1 bar ($f_i^o$).

If we expand our expression for the mixture fugacity (numerator) in terms of activity coefficients, we get:

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

##### Note:

Since liquids and solids are essentially incompressible, the pure species fugacity at any pressur is typically very close to the same as the pure species fugacity at 1 bar. This is why we would typically choose the saturation fugacity ($f_i = P_i^{sat}$) to be the "correct" value of pure species fugacity.

##### Definition:

If we have very high pressure reactions, we must use the Poynting correction that states:

$\frac{f_i}{f_i^o} = exp \left [\frac{1}{RT}\int_{P^o}^P v_i dP\right ]$

For most problems, we can still assume the liquid/solid to be incompressible (so $v_i$ is constant), so that this reduces to

$\frac{f_i}{f_i^o} = exp \left [\frac{v_i}{RT}(P-P^o)\right ]$

Combining these we get:

$K = \Pi_i \left [\gamma_i x_i exp \left [\frac{v_i}{RT}(P-P^o)\right ] \right ]^{\nu_i}$

which at low pressures gives us:

$K = \Pi_i \left [\gamma_i x_i \right ]^{\nu_i}$

which at low pressures and for an ideal solution gives us:

$K = \Pi_i \left [ x_i \right ]^{\nu_i}$

##### Note:

This last form is what you may be familiar with from Chemistry classes and is typically called the law of mass action. Remember that it assumes that the fugacity is independent of pressure and that the solution is ideal!

##### Outcome:

Determine the equilibrium composition for a single-phase, single-reaction system in liquid phase reactions