Henry's law

Atmospheric chemists often define the Henry solubility as

${\displaystyle H_{\rm {s}}^{cp}={\frac {c_{\text{a}}}{p}}}$.

Here ${\displaystyle c_{\text{a}}}$ is the concentration of a species in the aqueous phase, and ${\displaystyle p}$ is the partial pressure of that species in the gas phase under equilibrium conditions.

The SI unit for ${\displaystyle H_{\rm {s}}^{cp}}$ is mol/(m³·Pa); however, often the unit M/atm is used, since ${\displaystyle c_{\text{a}}}$ is usually expressed in M (1 M = 1 mol/dm³) and ${\displaystyle p}$ in atm (1 atm = 101325 Pa).

The Henry solubility can also be expressed as the dimensionless ratio between the aqueous-phase concentration ${\displaystyle c_{\text{a}}}$ of a species and its gas-phase concentration ${\displaystyle c_{\text{g}}}$:

${\displaystyle H_{\rm {s}}^{cc}={\frac {c_{\text{a}}}{c_{\text{g}}}}}$.

For an ideal gas, the conversion is:

${\displaystyle H_{\rm {s}}^{cc}=RTH_{\rm {s}}^{cp},}$

where ${\displaystyle R}$ is the gas constant, and ${\displaystyle T}$ is the temperature.

Sometimes, this dimensionless constant is called the water–air partitioning coefficient ${\displaystyle K_{\text{WA}}}$.[5] It is closely related to the various, slightly different definitions of the Ostwald coefficient ${\displaystyle L}$, as discussed by Battino (1984).[6]

Another Henry's law solubility constant is:

${\displaystyle H_{\rm {s}}^{xp}={\frac {x}{p}}}$ .

Here ${\displaystyle x}$ is the molar mixing ratio in the aqueous phase. For a dilute aqueous solution the conversion between ${\displaystyle x}$ and ${\displaystyle c_{\text{a}}}$ is:

${\displaystyle c_{\text{a}}\approx x{\frac {\varrho _{\mathrm {H_{2}O} }}{M_{\mathrm {H_{2}O} }}}}$.

where ${\displaystyle \varrho _{\mathrm {H_{2}O} }}$ is the density of water and ${\displaystyle M_{\mathrm {H_{2}O} }}$ is the molar mass of water. Thus

${\displaystyle H_{\rm {s}}^{xp}\approx {\frac {M_{\mathrm {H_{2}O} }}{\varrho _{\mathrm {H_{2}O} }}}H_{\rm {s}}^{cp}}$.

The SI unit for ${\displaystyle H_{\rm {s}}^{xp}}$ is Pa⁻¹, although atm⁻¹ is still frequently used.

It can be advantageous to describe the aqueous phase in terms of molality instead of concentration. The molality of a solution does not change with ${\displaystyle T}$, since it refers to the mass of the solvent. In contrast, the concentration ${\displaystyle c}$ does change with ${\displaystyle T}$, since the density of a solution and thus its volume are temperature-dependent. Defining the aqueous-phase composition via molality has the advantage that any temperature dependence of the Henry's law constant is a true solubility phenomenon and not introduced indirectly via a density change of the solution. Using molality, the Henry solubility can be defined as

${\displaystyle H_{\rm {s}}^{bp}={\frac {b}{p}}.}$

Here ${\displaystyle b}$ is used as the symbol for molality (instead of ${\displaystyle m}$) to avoid confusion with the symbol ${\displaystyle m}$ for mass. The SI unit for ${\displaystyle H_{\rm {s}}^{bp}}$ is mol/(kg·Pa). There is no simple way to calculate ${\displaystyle H_{\rm {s}}^{cp}}$ from ${\displaystyle H_{\rm {s}}^{bp}}$, since the conversion between concentration ${\displaystyle c_{\text{a}}}$ and molality ${\displaystyle b}$ involves all solutes of a solution. For a solution with a total of ${\displaystyle n}$ solutes with indices ${\displaystyle i=1,\ldots ,n}$, the conversion is:

${\displaystyle c_{\text{a}}={\frac {b\varrho }{1+\sum _{i=1}^{n}b_{i}M_{i}}},}$

where ${\displaystyle \varrho }$ is the density of the solution, and ${\displaystyle M_{i}}$ are the molar masses. Here ${\displaystyle b}$ is identical to one of the ${\displaystyle b_{i}}$ in the denominator. If there is only one solute, the equation simplifies to

${\displaystyle c_{\text{a}}={\frac {b\varrho }{1+bM}}.}$

Henry's law is only valid for dilute solutions where ${\displaystyle bM\ll 1}$ and ${\displaystyle \varrho \approx \varrho _{\mathrm {H_{2}O} }}$. In this case the conversion reduces further to

${\displaystyle c_{\text{a}}\approx b\varrho _{\mathrm {H_{2}O} },}$

and thus

${\displaystyle H_{\rm {s}}^{bp}\approx {\frac {H_{\rm {s}}^{cp}}{\varrho _{\mathrm {H_{2}O} }}}.}$

According to Sazonov and Shaw,[7] the dimensionless Bunsen coefficient ${\displaystyle \alpha }$ is defined as "the volume of saturating gas, V1, reduced to T° = 273.15 K, p° = 1 bar, which is absorbed by unit volume V₂* of pure solvent at the temperature of measurement and partial pressure of 1 bar." If the gas is ideal, the pressure cancels out, and the conversion to ${\displaystyle H_{\rm {s}}^{cp}}$ is simply

${\displaystyle H_{\rm {s}}^{cp}=\alpha {\frac {1}{RT^{\text{STP}}}}}$,

with ${\displaystyle T^{\text{STP}}}$ = 273.15 K. Note, that according to this definition, the conversion factor is not temperature-dependent. Independent of the temperature that the Bunsen coefficient refers to, 273.15 K is always used for the conversion. The Bunsen coefficient, which is named after Robert Bunsen, has been used mainly in the older literature, and IUPAC considers it to be obsolete.[3]

According to Sazonov and Shaw,[7] the Kuenen coefficient ${\displaystyle S}$ is defined as "the volume of saturating gas V(g), reduced to T° = 273.15 K, p° = bar, which is dissolved by unit mass of pure solvent at the temperature of measurement and partial pressure 1 bar." If the gas is ideal, the relation to ${\displaystyle H_{\rm {s}}^{cp}}$ is

${\displaystyle H_{\rm {s}}^{cp}=S{\frac {\varrho }{RT^{\text{STP}}}}}$,

where ${\displaystyle \varrho }$ is the density of the solvent, and ${\displaystyle T^{\text{STP}}}$ = 273.15 K. The SI unit for ${\displaystyle S}$ is m³/kg.[7] The Kuenen coefficient, which is named after Johannes Kuenen, has been used mainly in the older literature, and IUPAC considers it to be obsolete.[3]

A common way to define a Henry volatility is dividing the partial pressure by the aqueous-phase concentration:

${\displaystyle H_{\rm {v}}^{pc}={\frac {p}{c_{\text{a}}}}={\frac {1}{H_{\rm {s}}^{cp}}}.}$

The SI unit for ${\displaystyle H_{\rm {v}}^{pc}}$ is Pa·m³/mol.

Another Henry volatility is

${\displaystyle H_{\rm {v}}^{px}={\frac {p}{x}}={\frac {1}{H_{\rm {s}}^{xp}}}.}$

The SI unit for ${\displaystyle H_{\rm {v}}^{px}}$ is Pa. However, atm is still frequently used.

The Henry volatility can also be expressed as the dimensionless ratio between the gas-phase concentration ${\displaystyle c_{\text{g}}}$ of a species and its aqueous-phase concentration ${\displaystyle c_{\text{a}}}$:

${\displaystyle H_{\rm {v}}^{cc}={\frac {c_{\text{g}}}{c_{\text{a}}}}={\frac {1}{H_{\rm {s}}^{cc}}}.}$

In chemical engineering and environmental chemistry, this dimensionless constant is often called the air–water partitioning coefficient ${\displaystyle K_{\text{AW}}}$.

Henry's law solubility constant ${\displaystyle H_{\rm {s,2,M}}^{xp}}$ for a gas 2 in a mixture M of two solvents 1 and 3 depends on the individual constants for each solvent, ${\displaystyle H_{\rm {s,2,1}}^{xp}}$ and ${\displaystyle H_{\rm {s,2,3}}^{xp}}$ according [11] to:

${\displaystyle \ln H_{\rm {s,2,M}}^{xp}=x_{1}\ln H_{\rm {s,2,1}}^{xp}+x_{3}\ln H_{\rm {s,2,3}}^{xp}+a_{13}x_{1}x_{3}}$

Where ${\displaystyle x_{1}}$, ${\displaystyle x_{3}}$ are the molar ratios of each solvent in the mixture and a₁₃ is the interaction parameter of the solvents from Wohl expansion of the excess chemical potential of the ternary mixtures.

A similar relationship can be found for the volatility constant ${\displaystyle H_{\rm {v,2,M}}^{px}}$, by remembering that ${\displaystyle H_{\rm {v}}^{px}=1/H_{\rm {s}}^{xp}}$ and that, both being positive real numbers, ${\displaystyle \ln H_{\rm {s}}^{xp}=-\ln(1/H_{\rm {s}}^{xp})=-\ln H_{\rm {v}}^{px}}$, thus:

${\displaystyle \ln H_{\rm {v,2,M}}^{px}=x_{1}\ln H_{\rm {v,2,1}}^{px}+x_{3}\ln H_{\rm {v,2,3}}^{px}-a_{13}x_{1}x_{3}}$

For a water-ethanol mixture, the interaction parameter a₁₃ has values around ${\displaystyle 0.1\pm 0.05}$ for ethanol concentrations (volume/volume) between 5% and 25%.[12]

In geochemistry, a version of Henry's law applies to the solubility of a noble gas in contact with silicate melt. One equation used is

${\displaystyle {\frac {C_{\text{melt}}}{C_{\text{gas}}}}=\exp \left[-\beta \left(\mu _{\text{melt}}^{\text{E}}-\mu _{\text{gas}}^{\text{E}}\right)\right],}$

where

C is the number concentrations of the solute gas in the melt and gas phases,

β = 1/k_BT, an inverse temperature parameter (k_B is the Boltzmann constant),

µ^E is the excess chemical potentials of the solute gas in the two phases.

Henry's law is a limiting law that only applies for "sufficiently dilute" solutions, while Raoult's law is generally valid when the liquid phase is almost pure or for mixtures of similar substances.[13] The range of concentrations in which Henry's law applies becomes narrower the more the system diverges from ideal behavior. Roughly speaking, that is the more chemically "different" the solute is from the solvent.

For a dilute solution, the concentration of the solute is approximately proportional to its mole fraction x, and Henry's law can be written as

${\displaystyle p=H_{\rm {v}}^{px}x.}$

This can be compared with Raoult's law:

${\displaystyle p=p^{*}x,}$

where p* is the vapor pressure of the pure component.

At first sight, Raoult's law appears to be a special case of Henry's law, where H_v^px = p*. This is true for pairs of closely related substances, such as benzene and toluene, which obey Raoult's law over the entire composition range: such mixtures are called ideal mixtures.

The general case is that both laws are limit laws, and they apply at opposite ends of the composition range. The vapor pressure of the component in large excess, such as the solvent for a dilute solution, is proportional to its mole fraction, and the constant of proportionality is the vapor pressure of the pure substance (Raoult's law). The vapor pressure of the solute is also proportional to the solute's mole fraction, but the constant of proportionality is different and must be determined experimentally (Henry's law). In mathematical terms:

Raoult's law: ${\displaystyle \lim _{x\to 1}\left({\frac {p}{x}}\right)=p^{*}.}$

Henry's law: ${\displaystyle \lim _{x\to 0}\left({\frac {p}{x}}\right)=H_{\rm {v}}^{px}.}$

Raoult's law can also be related to non-gas solutes.