# Explain how Henry's law works?

Nov 18, 2016

Henry's Law only applies to the solute in ideally-dilute solutions, where a gas has been dissolved into a liquid to a small extent, and it is obvious which is the solvent and which is the solute (Raoult's law applies to the solvent in either an ideal or ideally-dilute solution).

Henry's Law for ideally-dilute solutions states:

$\boldsymbol{{P}_{i} = {\chi}_{i}^{l} {k}_{H , i}}$

or for real, dilute solutions:

$\boldsymbol{{P}_{i} = {\gamma}_{I I , i} {\chi}_{i}^{l} {k}_{H , i} = {a}_{I I , i} {k}_{H , i}}$

where:

• ${P}_{i}$ is the partial pressure of the vapor above the solution.
• ${\chi}_{i}^{l}$ is the $\boldsymbol{\text{mol}}$ fraction of the gas that is dissolved in the liquid.
• ${k}_{H , i}$ is the Henry's Law constant, and it is equal to ${P}_{i}^{\text{*}}$, the partial pressure of the pure gas, when one extrapolates back to ${\chi}_{i}^{l} = 0$.
• ${\gamma}_{I I , i}$ is the activity coefficient of the gas in solution in reference to the ideally-dilute solution (rather than an ideal solution).
• ${a}_{I I , i} = {\gamma}_{I I , i} {\chi}_{i}^{l}$ is the activity of the gas in solution in reference to the ideally-dilute solution.

The partial pressure of the vapor above the solution is also related to the $\text{mol}$ fraction of the vapor that is above the solution and the total pressure:

$\boldsymbol{{P}_{i} = {\chi}_{i}^{v} P}$

where ${\chi}_{i}^{v}$ is the $\text{mol}$ fraction of the vapor that is above the solution and $P$ is the total pressure.

This assumes that the gas is ideal (which is a pretty good assumption compared to assuming a liquid is "ideal"), and that the solution is highly dilute.

Using this information, you can, for example, solve the following problem for an ideally-dilute solution:

A solution of ethanol (eth) and chloroform (chl) at ${45}^{\circ} \text{C}$ with ${\chi}_{e t h} = 0.9900$ has a vapor pressure of $\text{177.95 torr}$. At this high dilution of chloroform, the solution can be assumed to be essentially ideally dilute. The vapor pressure of pure ethanol at ${45}^{\circ} \text{C}$ is $\text{172.76 torr}$.

a) Find the partial pressures of the gases in equilibrium with the solution.

b) Find the mole fractions in the vapor phase.

c) Find the Henry's Law constant for chloroform in ethanol at ${45}^{\circ} \text{C}$.

d) Predict the vapor pressure and vapor-phase mole fractions at ${45}^{\circ} \text{C}$ for a chloroform-ethanol solution with ${\chi}_{e t h} = 0.9800$ (using the Henry's Law constant from part c). Compare with the experimental values of $P = \text{183.38 torr}$ and ${\chi}_{e t h}^{v} = 0.9242$.

a) ${P}_{e t h} = \text{171.03 torr}$, ${P}_{c h l} = \text{6.92 torr}$
b) ${\chi}_{e t h}^{v} = {0.9611}_{1}$, ${\chi}_{c h l}^{v} = {0.0388}_{9}$
c) ${k}_{H , c h l} = \text{692 torr}$
d) $P = \text{183.14 torr}$, ${\chi}_{e t h}^{v} = 0.9244$