# Partial pressure

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Dalton's law states that each gas in a mixture of ideal gases has a partial pressure which is the pressure that the gas would have if it alone occupied the same volume at the same temperature. The total pressure of a gas mixture is the sum of the partial pressures of each individual gas in the mixture.

Henry's law states that at a constant temperature, the partial pressure of a gas in equilibrium with a liquid solution containing some of the gas is directly proportional to the concentration of that gas in the liquid solution.

## Dalton's law of partial pressures

John Dalton, an English chemist, meteorologist, and physicist, first propounded his law of partial pressures in 1803 and published it in 1805. His statement that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas in the mixture can be expressed mathematically. For example, for a mixture of three ideal gases (denoted as gases a, b and c):

(1)     ${\displaystyle p_{t}=p_{a}+p_{b}+p_{c}}$
 where: ${\displaystyle p_{t}}$ = total pressure of the gas mixture ${\displaystyle p_{a}}$ = partial pressure of gas a ${\displaystyle p_{b}}$ = partial pressure of gas b ${\displaystyle p_{c}}$ = partial pressure of gas c

Dalton's law applies only to gases that behave in accordance with the ideal gas law which is applicable for hypothetical gases with no intermolecular forces. The ideal gas law is a useful approximation for predicting the behavior of many gases over a wide range of temperatures and pressures.

However, real gases can deviate considerably from ideal gas behavior because of the intermolecular attractive and repulsive forces. The deviation is especially significant at low temperatures or high pressures. In other words, Dalton's law is not applicable for real gases at conditions where they deviate significantly from ideal gas behavior.

Dalton's law is also applicable only to gases at conditions under which they are mutually inert (i.e., they do not react with each other).

## Ideal gas mixtures

The mole fraction of an individual gas component in an ideal gas mixture can be expressed in terms of the component's partial pressure or the moles of the component:

(2)     ${\displaystyle x_{i}={\frac {p_{i}}{p_{t}}}={\frac {n_{i}}{n}}}$

and the partial pressure of an individual gas component in an ideal gas can be obtained using this expression:

(3)     ${\displaystyle p_{i}=x_{i}\cdot p_{t}}$
${\displaystyle x_{i}}$ where: = mole fraction of any individual gas component in a gas mixture = partial pressure of any individual gas component in a gas mixture = total pressure of the gas mixture = moles of any individual gas component in a gas mixture = total moles of the gas mixture

Using equation (2), Dalton's law as expressed in equation (1) may also be expressed as:

(3)     ${\displaystyle p_{t}=(x_{a}\cdot p_{t})+(x_{b}\cdot p_{t})+(x_{c}\cdot p_{t})}$

The mole fraction of a gas component in a gas mixture is equal to the volumetric fraction of that component in a gas mixture.

## Henry's law and the solubility of gases

William Henry, an English chemist, formulated Henry's law in 1803. It stated that:

At a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.

Henry's law is commonly expressed as:[1] [2][3][4]

(4)     ${\displaystyle p=kc\,}$
${\displaystyle p}$ where: is the partial pressure of the solute gas above the liquid solution is the Henry's law constant in units such as L·atm/mol, atm/(mol fraction) or Pa·m3/mol is the concentration of the solute in the solution

Henry's Law is sometimes written as:

(5)     ${\displaystyle p={\frac {c}{k^{*}}}}$

where ${\displaystyle k^{*}}$ is also referred to as the Henry's law constant. As seen by comparing equations (4) and (5), ${\displaystyle k^{*}}$ is the reciprocal of ${\displaystyle k}$. Since both may be referred to as the Henry's law constant, readers of the technical literature must be quite careful to note which version of the Henry's law equation is being used.

Henry's law is an approximation that only applies for dilute, ideal solutions and for solutions where the liquid solvent does not react chemically with the gas being dissolved.

## Equilibrium constants of reactions involving gases

The Law of Mass Action, formulated in 1864 by Cater Guldberg and Peter Waage of Norway,[5] states that:[6][7][8][9]

The rate of a chemical reaction is proportional to the concentration of the reacting substances.

That law makes it possible to obtain the equilibrium reaction constant for reversible reactions involving gas reactants and gas products given the partial pressures of the reactant and product gases. As an example, for the following generalized reaction:

(6)     ${\displaystyle w\,A+x\,B\leftrightarrow y\,C+z\,D}$

the equilibrium constant of the reaction would be:

(7)     ${\displaystyle K_{p}={\frac {p_{C}^{y}\;p_{D}^{z}}{p_{A}^{w}\;p_{B}^{x}}}}$
${\displaystyle K_{p}}$ where: =  the equilibrium constant of the reaction =  moles of gas reactant ${\displaystyle A}$ =  moles of gas reactant ${\displaystyle B}$ =  moles of gas product ${\displaystyle C}$ =  moles of gas product ${\displaystyle D}$ =  the partial pressure of ${\displaystyle C}$ raised to the power of ${\displaystyle y}$ =  the partial pressure of ${\displaystyle D}$ raised to the power of ${\displaystyle z}$ =  the partial pressure of ${\displaystyle A}$ raised to the power of ${\displaystyle w}$ =  the partial pressure of ${\displaystyle B}$ raised to the power of ${\displaystyle x}$

When the Law of Mass Action is expressed using partial pressures, as in equations (6) and (7) above, the equilibrium reaction constant is denoted by ${\displaystyle K_{p}}$. When expressed using concentrations (such as mole/m3 ) rather than partial pressures, the equilibrium reaction constant is denoted by ${\displaystyle K_{c}}$. The relationship between ${\displaystyle K_{p}}$ and ${\displaystyle K_{c}}$ is:[10]

(8)     ${\displaystyle K_{p}=K_{c}(R\,T)^{(y+z-w-x)}}$

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

For reversible reactions, changes in the total pressure, temperature, or reactant concentrations will shift the equilibrium position so as to favor either the right or left side of the reaction in accordance with Le Chatelier's Principle. However, the reaction kinetics may either oppose or enhance the equilibrium shift. In some cases, the reaction kinetics may be the overriding factor to consider.

## Partial pressure in diving breathing gases

In recreational diving and professional diving the richness of individual component gases of breathing gases is expressed by partial pressure.

Using diving terminology, partial pressure is calculated as:

partial pressure = (total absolute pressure) × (volume fraction of gas component)

For the component gas "i":

ppi = P × Fi

For example, at 50 metres (164 feet), the total absolute pressure is approximately 6 bar (600 kPa) and the partial pressures of the main components of air, oxygen 21% by volume and nitrogen 79% by volume are:

ppN2 = 6 bar x 0.79 = 4.7 bar absolute
ppO2 = 6 bar x 0.21 = 1.3 bar absolute
ppi where: = partial pressure of gas component i  = ${\displaystyle p_{i}}$ in the terms used in this article = total pressure = ${\displaystyle p_{t}}$ in the terms used in this article = volume fraction of gas component i  =  mole fraction, ${\displaystyle x_{i}}$, in the terms used in this article = partial pressure of nitrogen  = ${\displaystyle p_{{\mathrm {N} }_{2}}}$ in the terms used in this article = partial pressure of oxygen  = ${\displaystyle p_{{\mathrm {O} }_{2}}}$ in the terms used in this article

The minimum safe lower limit for the partial pressures of oxygen in a gas mixture is 0.16 bar (16 kPa) absolute. Hypoxia and sudden unconsciousness becomes a problem with an oxygen partial pressure of less than 0.16 bar absolute. The NOAA Diving Manual recommends a maximum single exposure of 45 minutes at 1.6 bar absolute, of 120 minutes at 1.5 bar absolute, of 150 minutes at 1.4 bar absolute, of 180 minutes at 1.3 bar absolute, and of 210 minutes at 1.2 bar absolute. Oxygen toxicity, involving convulsions, becomes a risk when these oxygen partial pressures and exposures are exceeded. The partial pressure of oxygen determines the maximum operating depth of a gas mixture.

Nitrogen narcosis is a problem with gas mixes containing nitrogen. A typical planned maximum partial pressure of nitrogen for technical diving is 3.5 bar absolute, based on an equivalent air depth of 35 meters (115 feet).

## References

1. University of Delaware physical chemistry lecture
2. Robert G. Mortimer (2000). Physical Chemistry, Second Edition. Academic Press. ISBN 0-12-508345-9.
3. Green, Don W. and Perry, Robert H. (deceased) (1997). Perry's Chemical Engineers' Handbook, 6th Edition. McGraw-Hill. ISBN 0-07-049479-7.  (See page 14-9)
4. Online Introductory Chemistry: Solubility of gases in liquids
5. E.W. Lund (1965). "Guldberg and Waage and the Law of Mass Action". J. Chem. Ed 42: 548-550.
6. A.V. Jones, M. Clement, A. Higton and E. Golding (1999). Access to Chemistry, 1st Edition. Royal Society of Chemistry. ISBN 0-85404-564-3.
7. Mass Action Law
8. Michael Clugston and Rosalind Flemming (2000). Advanced Chemistry, 1st Edition. Oxford University Press. ISBN 0-19-914633-0.
9. E.N. Ramsden (2000). A-Level Chemistry, 4th Edition. Nelson Thornes. ISBN 0-7487-5299-4.
10. The Law of Mass Action From the website of Loyola University of Chicago. (Click through the slides from sld011.htm to sld024.htm)