How to derive van 't Hoff equation for Henry's law constant?

Question

While studying about Henry's law, I encountered this equation:

$$\frac{\mathrm d\ln H}{\mathrm d(1/T)}= \frac{-\Delta_\text{sol}H }{R},$$

where $\displaystyle\mathrm d(1/T) = -\frac{\mathrm dT}{T^2}$ and the $\displaystyle\frac{\Delta H}{RT^2}$ reminds me of some kind of $\Delta G$ equation.

I didn't know how to derive it, so I searched about it and got nothing. I tried to substitute $H$ (Henry's law constant) by $\gamma\cdot f_i$ where $\gamma$ is the activity coefficient at infinite dilution and $f_i$ is the fugacity of pure species.

But still I have no idea what to do to obtain the formula that I'm familiar with, so I welcome any idea about how to derive or how to reverse engineer the formula.

Answer 1

Assuming Henry's coefficient $\mathcal{H}_i$ is defined by the following relation:

$$ \mathcal{H}_i = \gamma_i^\inf \cdot p_i^\mathrm{sat} $$

Where $\gamma_i^\inf$ is the activity coefficient for an infinite diluted solution and $p_i^\mathrm{sat}$ is the vapor pressure of pure component $i$.

One can try to extrapolate Henry's coefficient at another temperature by assessing the following ratio (this is a common approach in Chemistry, not always valid):

$$ \frac{\mathcal{H}_i(T_1)}{\mathcal{H}_i(T_0)} \approx \frac{\gamma_i^\inf(T_1) \cdot p_i^\mathrm{sat}(T_1)}{\gamma_i^\inf(T_0) \cdot p_i^\mathrm{sat}(T_0)} $$

The RHS maybe rearranged in order to have the well know form of ratio of mixed equilibrium constants $K_i$ considering that bulk concentration are close and ratio of activities also simplifies (over a small range of temperature).

$$ \frac{\mathcal{H}_i(T_1)}{\mathcal{H}_i(T_0)} \approx \frac{\frac{\gamma_i(T_1)\cdot p_i(T_1)}{\eta_i(T_1)\cdot x_i(T_1)}}{\frac{\gamma_i(T_0)\cdot p_i(T_0)}{\eta_i(T_0)\cdot x_i(T_0)}} =\frac{K_i(T_0)}{K_i(T_0)} $$

That is, we are stating that ratio of Henry's constants approximates to the ratio of equilibrium constant because we assume that only ratio of pressure is significant and other ratios tend to unity. This is, off course, not always acceptable, but in some restrained conditions it might hold.

Where the concerned reaction is the converse of gas solubilization:

$$ \ce{X_{i,(aq)} <=> X_{i,(\mathrm{g})}}{\quad \Delta_\mathrm{R}H = -\Delta_\mathrm{sol}H} $$

And its mixed equilibrium constant:

$$ K_i = \frac{\gamma_i\cdot p_i}{\eta_i \cdot x_i} $$

Then one may apply van 't Hoff relation using enthalpy $\Delta_\mathrm{sol}H$ of the concerned reaction, from its integrated form, it comes:

$$ \frac{\mathcal{H}_i(T_1)}{\mathcal{H}_i(T_0)} \approx \exp\left[\frac{\Delta_\mathrm{sol}H}{R}\left(\frac{1}{T_1}-\frac{1}{T_0}\right)\right] $$

This formulae is generally a good approximation around $\pu{20°C}$ over a small range of temperature. Do not forget that $\Delta_\mathrm{sol}H$ is not a constant but rather a function of temperature.

If you wish to get your definition, you just have to differentiate the last relation with respect to temperature:

$$ \frac{\mathrm{d}\ln(\mathcal{H}_i)}{\mathrm{d}T} \approx \frac{\Delta_\mathrm{R}H}{RT^2} = -\frac{\Delta_\mathrm{sol}H}{RT^2} $$

Or equivalently: $$ \frac{\mathrm{d}\ln(\mathcal{H}_i)}{\mathrm{d}\left(\frac{1}{T}\right)} \approx -\frac{\Delta_\mathrm{R}H}{R} = \frac{\Delta_\mathrm{sol}H}{R} $$