Does the equilibrium constant of an ideal liquid mixture reaction depend on pressure?

I want to calculate the equilibrium constant K[T] for a the following reaction:


$$\ce{CH3OH(l) + C4H8(l) -> C5H12O(l)}$$


It is assumed that the mixture is ideal and the reaction takes place at 14 bar pressure.


K[T] is defined as the product of the thermodynamic activities to the power of the stoichiometric coefficient:


$$\prod\_j {a\_j^{\nu\_j}} = \prod\_j {\left( \frac{\hat f\_j[T, p, \mathbf{x}] }{ f\_j^{\mathrm{ref}}[T, p^{\mathrm{ref}}, \mathbf{x}^{\mathrm{ref}}]} \right)}^{\nu\_j}$$


The fugacity is given by Raoult's law,


$$f\_i = x\_i \cdot p\_i^{\*}$$


where x is the molar fraction and p\* the vapor pressure, which are given.


How can I determine the reference fugacity, and where in the calculations do I use the 14 bar pressure at which the reaction takes place?




The equilibrium constant for a chemical reaction is only a function of temperature. However, if you want to calculate the equilibrium constant $K$ via an equilibrium calculation, e.g. using the equation you posted, then we need to find the pressure dependence on the fugacity of the species $i$ as a liquid.




---


We write the relationship between the fugacity of a pure species $i$ with the Gibbs energy, for two different situations distinguished by a zero superscript
\begin{align}
 g\_i &= \Gamma(T) + RT\ln(f\_i) \tag{1} \\
 g\_i^0 &= \Gamma(T) + RT\ln(f\_i^0) \tag{2}
\end{align}
Subtracting Eq. (2) from Eq. (1)
$$ g\_i - g\_i^0 = RT\ln\bigg(\frac{f\_i}{f\_i^0}\bigg) \tag{3} $$
Equation (3) links two different states:


1. The fugacity of a pure species at pressure $p$ and temperature $T$.
2. The fugacity of a pure species at pressure $p^0$ and temperature $T$. The reference state for a pure liquid, in the context of this equation, is generally chosen as that where the substance at temperature $T$ is a **saturated liquid**, and thus $p^0 = p^\mathrm{sat}$.


Now we use the fundamental thermodynamic relation for pure substances
$$ \mathrm{d}g = v\mathrm{d}p - s\mathrm{d}T \tag{4} $$


The evolution $(p^\mathrm{sat},T)\to(p,T)$ at constant temperature, for a species labeled with $i$, is carried out by integrating Eq. (4)
$$ g\_i - g\_i^\mathrm{sat} = \int\_{p\_i^\mathrm{sat}}^p
 v\_i\mathrm{d}p \tag{5} $$


The liquid molar volume will change between these two pressures. For a liquid, we make the assumption that if the pressure range is not that large, it will remain constant and equal to the *liquid saturated volume* at the saturation pressure $ v\_i = v\_i^\mathrm{sat} $. Hence, Eq. (5) turns into
$$ g\_i - g\_i^\mathrm{sat} = v\_i^\mathrm{sat}(p - p^\mathrm{sat}) \tag{6} $$
Combining Eqs. (3) and (6)
\begin{equation}
 RT\ln\bigg(\frac{f\_i}{f\_i^\mathrm{sat}}\bigg) = v\_i^\mathrm{sat}(p - p^\mathrm{sat})
 \to
 \frac{f\_i}{f\_i^\mathrm{sat}} = \exp\bigg[\frac{v\_i^\mathrm{sat}
 (p - p\_i^\mathrm{sat})}{RT}\bigg] \tag{7}
\end{equation}
The right-hand side of Eq. (7) is the famous **Poynting factor**.




---


We return to the equilibrium constant. As stated, the mixture is ideal. We deal with liquids, so the fugacity coefficient of species $ i $ in a mixture is $\hat{f}\_i = x\_i\gamma\_if\_i= x\_i f\_i $, because $ \gamma\_i = 1 $. Summarizing the reaction in the fashion $\ce{A + B -> C}$
\begin{equation}
 K = \dfrac{\dfrac{\hat{f}\_C}{f\_C^\mathrm{sat}}}
 {\left(\dfrac{\hat{f}\_B}{f\_B^\mathrm{sat}}\right)
 \left(\dfrac{\hat{f}\_A}{f\_A^\mathrm{sat}}\right)} =
 \dfrac{\dfrac{x\_Cf\_C}{f\_C^\mathrm{sat}}}
 {\left(\dfrac{x\_Bf\_B}{f\_B^\mathrm{sat}}\right)
 \left(\dfrac{x\_Af\_A}{f\_A^\mathrm{sat}}\right)} \tag{8}
\end{equation}
Now we use Eq. (7) for the three ratios in Eq. (8)
\begin{equation}
 \boxed{K =
 \dfrac{x\_C \exp\bigg[\dfrac{v\_C^\mathrm{sat}
 (\color{blue}{p} - p\_C^\mathrm{sat})}{RT}\bigg]}
 {x\_A \exp\bigg[\dfrac{v\_A^{sat}(\color{blue}{p} - p\_A^{sat})}
 {RT}\bigg] \cdot
 x\_B \exp\bigg[\dfrac{v\_B^{sat}(\color{blue}{p} - p\_B^{sat})}
 {RT}\bigg]}} \tag{9}
\end{equation}


In blue is emphasized the pressure dependence in the fugacity.


Eq. (9) shows that additional information is needed when we deal with liquid-phase reactions:


* The saturation pressure $p\_i^\mathrm{sat}$ for every species.
* The saturated liquid molar volumes $v\_i^\mathrm{sat}$ for every species, evaluated at $p\_i^\mathrm{sat}$.


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