Henry's law

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In chemistry, Henry's law is a gas law that states that the amount of dissolved gas is proportional to its partial pressure in the gas phase. The proportionality factor is called the Henry's law constant. It was formulated by the English chemist William Henry, who studied the topic in the early 19th century. In his publication about the quantity of gases absorbed by water,^[1] he described the results of his experiments:

..."water takes up, of gas condensed by one, two, or more additional atmospheres, a quantity which, ordinarily compressed, would be equal to twice, thrice, &c. the volume absorbed under the common pressure of the atmosphere."

An example where Henry's law is at play is in the depth-dependent dissolution of oxygen and nitrogen in the blood of underwater divers that changes during decompression, leading to decompression sickness. An everyday example is given by one's experience with carbonated soft drinks, which contain dissolved carbon dioxide. Before opening, the gas above the drink in its container is almost pure carbon dioxide, at a pressure higher than atmospheric pressure. After the bottle is opened, this gas escapes, moving the partial pressure of carbon dioxide above the liquid to be much lower, resulting in degassing as the dissolved carbon dioxide comes out of solution.

Fundamental types and variants of Henry's law constants

There are many ways to define the proportionality constant of Henry's law, which can be subdivided into two fundamental types: One possibility is to put the aqueous phase into the numerator and the gaseous phase into the denominator ("aq/gas").^[2] This results in the Henry's law solubility constant ${\displaystyle H}$. Its value increases with increased solubility. Alternatively, numerator and denominator can be switched ("gas/aq"), which results in the Henry's law volatility constant ${\displaystyle K_{\rm {H}}}$. The value of ${\displaystyle K_{\rm {H}}}$ decreases with increased solubility. There are several variants of both fundamental types. This results from the multiplicity of quantities that can be chosen to describe the composition of the two phases. Typical choices for the aqueous phase are molar concentration (${\displaystyle c_{\rm {a}}}$), molality (${\displaystyle b}$), and molar mixing ratio (${\displaystyle x}$). For the gas phase, molar concentration (${\displaystyle c_{\rm {g}}}$) and partial pressure (${\displaystyle p}$) are often used. It is not possible to use the gas-phase mixing ratio (${\displaystyle y}$) because at a given gas-phase mixing ratio, the aqueous-phase concentration ${\displaystyle c_{\rm {a}}}$ depends on the total pressure and thus the ratio ${\displaystyle y/c_{\rm {a}}}$ is not a constant.^[3] To specify the exact variant of the Henry's law constant, two superscripts are used. They refer to the numerator and the denominator of the definition. For example, ${\displaystyle H^{cp}}$ refers to the Henry solubility defined as ${\displaystyle c/p}$.

Henry's law solubility constants ${\displaystyle H}$

Henry solubility defined via concentration (${\displaystyle H^{cp}}$)

Atmospheric chemists often define the Henry solubility as

${\displaystyle H^{cp}=c_{\text{a}}/p}$.^[2]

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.^{[citation needed]}

The SI unit for ${\displaystyle H^{cp}}$ 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).^{[citation needed]}

The dimensionless Henry solubility ${\displaystyle H^{cc}}$

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 ${\displaystyle c_{\text{g}}}$:

${\displaystyle H^{cc}=c_{\text{a}}/c_{\text{g}}}$.^[2]

For an ideal gas, the conversion is:

${\displaystyle H^{cc}=H^{cp}\times RT}$ ,^[2]

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

Sometimes, this dimensionless constant is called the "water-air partitioning coefficient" ${\displaystyle K_{\text{WA}}}$.^[4] It is closely related to the various, slightly different definitions of the "Ostwald coefficient" ${\displaystyle L}$, as discussed by Battino (1984).^[5]

Henry solubility defined via aqueous-phase mixing ratio (${\displaystyle H^{xp}}$)

Another Henry's law solubility constant is

${\displaystyle H^{xp}=x/p}$ .^[2]

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:

${\displaystyle c_{\text{a}}\approx x{\frac {\varrho _{\mathrm {H_{2}O} }}{M_{\mathrm {H_{2}O} }}}}$ ,^[2]

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

${\displaystyle H^{xp}\approx {\frac {M_{\mathrm {H_{2}O} }}{\varrho _{\mathrm {H_{2}O} }}}\times H^{cp}}$ .^[2]

The SI unit for ${\displaystyle H^{xp}}$ is Pa⁻¹, although atm⁻¹ is still frequently used.^[2]

Henry solubility defined via molality (${\displaystyle H^{bp}}$)

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

${\displaystyle H^{bp}=b/p.}$

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 ${\displaystyle H^{bp}}$ is mol/(kg Pa). There is no simple way to calculate ${\displaystyle H^{cp}}$ from ${\displaystyle H^{bp}}$, 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

${\displaystyle H^{bp}\approx H^{cp}/\varrho _{\mathrm {H_{2}O} }.}$

The Bunsen coefficient ${\displaystyle \alpha }$

According to Sazonov and Shaw, 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."^[6] If the gas is ideal, the pressure cancels out, and the conversion to ${\displaystyle H^{cp}}$ is simply

${\displaystyle H^{cp}=\alpha \times {\frac {1}{RT^{\text{STP}}}}}$ ,

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

The Kuenen coefficient ${\displaystyle S}$

According to Sazonov and Shaw, 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."^[6] If the gas is ideal, the relation to ${\displaystyle H^{cp}}$ is

${\displaystyle H^{cp}=S\times {\frac {\varrho }{RT^{\text{STP}}}}}$ ,^{[citation needed][original research?]}

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.^[6] The Kuenen coefficient, which is named after Johannes Kuenen, has been used mainly in the older literature, and IUPAC considers it to be obsolete.^[7]

Henry's law volatility constants ${\displaystyle K_{\text{H}}}$

The Henry volatility defined via concentration (${\displaystyle K_{\text{H}}^{pc}}$)

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

${\displaystyle K_{\text{H}}^{pc}=p/c_{\text{a}}=1/H^{cp}.}$

The SI unit for ${\displaystyle K_{\text{H}}^{pc}}$ is Pa m³/mol.

The Henry volatility defined via aqueous-phase mixing ratio (${\displaystyle K_{\text{H}}^{px}}$)

Another Henry volatility is

${\displaystyle K_{\text{H}}^{px}=p/x=1/H^{xp}.}$

The SI unit for ${\displaystyle K_{\text{H}}^{px}}$ is Pa. However, atm is still frequently used.

The dimensionless Henry volatility ${\displaystyle K_{\text{H}}^{cc}}$

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 ${\displaystyle c_{\text{a}}}$:

${\displaystyle K_{\text{H}}^{cc}=c_{\text{g}}/c_{\text{a}}=1/H^{cc}.}$

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

Values of Henry's law constants

A large compilation of Henry's law constants has been published by Sander (2015).^[2] A few selected values are shown in the table below:

Henry's law constants (gases in water at 298.15 K)

equation:	${\displaystyle K_{\text{H}}^{pc}={\frac {p}{c_{\text{aq}}}}}$	${\displaystyle H^{cp}={\frac {c_{\text{aq}}}{p}}}$	${\displaystyle K_{\text{H}}^{px}={\frac {p}{x}}}$	${\displaystyle H^{cc}={\frac {c_{\text{aq}}}{c_{\text{gas}}}}}$
unit:	${\displaystyle {\frac {{\text{L}}\cdot {\text{atm}}}{\text{mol}}}}$	${\displaystyle {\frac {\text{mol}}{{\text{L}}\cdot {\text{atm}}}}}$	${\displaystyle {\text{atm}}}$	(dimensionless)
O2	770	1.3×10−3	4.3×104	3.2×10−2
H2	1300	7.8×10−4	7.1×104	1.9×10−2
CO2	29	3.4×10−2	1.6×103	8.3×10−1
N2	1600	6.1×10−4	9.1×104	1.5×10−2
He	2700	3.7×10−4	1.5×105	9.1×10−3
Ne	2200	4.5×10−4	1.2×105	1.1×10−2
Ar	710	1.4×10−3	4.0×104	3.4×10−2
CO	1100	9.5×10−4	5.8×104	2.3×10−2

Temperature dependence

When the temperature of a system changes, the Henry constant also changes. The temperature dependence of equilibrium constants can generally be described with the van 't Hoff equation, which also applies to Henry's law constants:

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

where ${\displaystyle \Delta _{\text{sol}}H}$ is the enthalpy of dissolution. Note that the letter ${\displaystyle H}$ in the symbol ${\displaystyle \Delta _{\text{sol}}H}$ refers to enthalpy and is not related to the letter ${\displaystyle H}$ for Henry's law constants. Integrating the above equation and creating an expression based on ${\displaystyle H^{\circ }}$ at the reference temperature ${\displaystyle T^{\circ }}$ = 298.15 K yields:

${\displaystyle H(T)=H^{\circ }\times \exp \displaystyle \left[{\frac {-\Delta _{\text{sol}}H}{R}}\left({\frac {1}{T}}-{\frac {1}{T^{\circ }}}\right)\right].}$

The van 't Hoff equation in this form is only valid for a limited temperature range in which ${\displaystyle \Delta _{\text{sol}}H}$ does not change much with temperature.

The following table lists some temperature dependencies:

Values of ${\displaystyle -\Delta _{\text{sol}}H/R}$ (in K)

O2	H2	CO2	N2	He	Ne	Ar	CO
1700	500	2400	1300	230	490	1300	1300

Solubility of permanent gases usually decreases with increasing temperature at around room temperature. However, for aqueous solutions, the Henry's law solubility constant for many species goes through a minimum. For most permanent gases, the minimum is below 120 °C. Often, the smaller the gas molecule (and the lower the gas solubility in water), the lower the temperature of the maximum of the Henry's law constant. Thus, the maximum is at about 30 °C for helium, 92 to 93 °C for argon, nitrogen and oxygen, and 114 °C for xenon.^[8]

Effective Henry's law constants H_eff

The Henry's law constants mentioned so far do not consider any chemical equilibria in the aqueous phase. This type is called the "intrinsic" (or "physical") Henry's law constant. For example, the intrinsic Henry's law solubility constant of formaldehyde can be defined as

${\displaystyle H^{{\ce {cp}}}={\frac {c({\ce {H2CO}})}{p({\ce {H2CO}})}}.}$

In aqueous solution, methanal is almost completely hydrated:

${\displaystyle {\ce {{H2CO}+ {H2O}<=> {H2C(OH)2}}}}$

The total concentration of dissolved methanal is

${\displaystyle c_{{\ce {tot}}}=c({\ce {H2CO}})+c({\ce {H2C(OH)2}}).}$

Taking this equilibrium into account, an effective Henry's law constant ${\displaystyle H_{{\ce {eff}}}}$ can be defined as

${\displaystyle H_{{\ce {eff}}}={\frac {c_{{\ce {tot}}}}{p({\ce {H2CO}})}}={\frac {c({\ce {H2CO}})+c({\ce {H2C(OH)2}})}{p({\ce {H2CO}})}}.}$

For acids and bases, the effective Henry's law constant is not a useful quantity because it depends on the pH of the solution.^{[verification needed]} In order to obtain a pH-independent constant, the product of the intrinsic Henry's law constant ${\displaystyle H^{{\ce {cp}}}}$ and the acidity constant ${\displaystyle K_{{\ce {A}}}}$ is often used for strong acids like hydrochloric acid (HCl):

${\displaystyle H'=H^{{\ce {cp}}}\times K_{{\ce {A}}}={\frac {c({\ce {H+}})\times c({\ce {Cl^-}})}{p({\ce {HCl}})}}.}$

Although ${\displaystyle H'}$ is usually also called a Henry's law constant, it should be noted that it is a different quantity and it has different units than ${\displaystyle H^{{\ce {cp}}}}$.

Dependence on ionic strength (Sechenov equation)

Values of Henry's law constants for aqueous solutions depend on the composition of the solution, i.e., on its ionic strength and on dissolved organics. In general, the solubility of a gas decreases with increasing salinity ("salting out"). However, a "salting in" effect has also been observed, for example for the effective Henry's law constant of glyoxal. The effect can be described with the Sechenov equation, named after the Russian physiologist Ivan Sechenov (sometimes the German transliteration "Setschenow" of the Cyrillic name Се́ченов is used). There are many alternative ways to define the Sechenov equation, depending on how the aqueous-phase composition is described (based on concentration, molality, or molar fraction) and which variant of the Henry's law constant is used. Describing the solution in terms of molality is preferred because molality is invariant to temperature and to the addition of dry salt to the solution. Thus, the Sechenov equation can be written as

${\displaystyle \log \left({\frac {H_{0}^{bp}}{H^{bp}}}\right)=k_{\text{s}}\times b({\text{salt}}),}$

where ${\displaystyle H_{0}^{bp}}$ is the Henry's law constant in pure water, ${\displaystyle H^{bp}}$ is the Henry's law constant in the salt solution, ${\displaystyle k_{\text{s}}}$ is the molality-based Sechenov constant, and ${\displaystyle b({\text{salt}})}$ is the molality of the salt.

Non-ideal solutions

Henry's law has been shown to apply to a wide range of solutes in the limit of "infinite dilution" (x → 0), including non-volatile substances such as sucrose. In these cases, it is necessary to state the law in terms of chemical potentials. For a solute in an ideal dilute solution, the chemical potential depends only on the concentration. For non-ideal solutions, the activity coefficients of the components must be taken into account:

${\displaystyle \mu =\mu _{c}^{\circ }+RT\ln {\frac {\gamma _{c}c}{c^{\circ }}}}$,

where ${\displaystyle \gamma _{c}={\frac {K_{{\text{H}},c}}{p^{*}}}}$ for a volatile solute; c° = 1 mol/L.

For non-ideal solutions, the activity coefficient γ_c depends on the concentration and must be determined at the concentration of interest. The activity coefficient can also be obtained for non-volatile solutes, where the vapor pressure of the pure substance is negligible, by using the Gibbs-Duhem relation:

${\displaystyle \sum _{i}n_{i}d\mu _{i}=0.}$

By measuring the change in vapor pressure (and hence chemical potential) of the solvent, the chemical potential of the solute can be deduced.

The standard state for a dilute solution is also defined in terms of infinite-dilution behavior. Although the standard concentration c° is taken to be 1 mol/l by convention, the standard state is a hypothetical solution of 1 mol/l in which the solute has its limiting infinite-dilution properties. This has the effect that all non-ideal behavior is described by the activity coefficient: the activity coefficient at 1 mol/l is not necessarily unity (and is frequently quite different from unity).

All the relations above can also be expressed in terms of molalities b rather than concentrations, e.g.:

${\displaystyle \mu =\mu _{b}^{\circ }+RT\ln {\frac {\gamma _{b}b}{b^{\circ }}},}$

where ${\displaystyle \gamma _{b}={\frac {K_{{\text{H}},b}}{p^{*}}}}$ for a volatile solute; b° = 1 mol/kg.

The standard chemical potential μ_m°, the activity coefficient γ_m and the Henry's law constant K_H,b all have different numerical values when molalities are used in place of concentrations.

Solvent mixtures

Henry law constant H_{2, M} for a gas 2 in a mixture of solvents 1 and 3 is related to the constants for individual solvents H₂₁ and H₂₃:

${\displaystyle \ln H_{2M}=x_{1}\ln H_{21}+x_{2}\ln H_{23}-a_{13}x_{1}x_{3}}$

where a₁₃ is the interaction parameter of the solvents from Wohl expansion of the excess chemical potential of the ternary mixtures.

Miscellaneous

In geochemistry

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

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

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.

Comparison to Raoult's law

Henry's law is a limiting law that only applies for "sufficiently dilute" solutions. The range of concentrations in which it 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

${\displaystyle p=K_{\text{H}}x.}$

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 K_H = 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: ${\displaystyle \lim _{x\to 0}\left({\frac {p}{x}}\right)=K_{\text{H}}.}$

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