Partial pressure

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

Partial Pressure

In a mixture of ideal gases, each gas has a partial pressure which is the pressure which the gas would have if it alone occupied the volume. The total pressure of a gas mixture is the sum of the partial pressures of each individual gas in the mixture.

In chemistry, the partial pressure of a gas in a mixture of gases is defined as above. The partial pressure of a gas dissolved in a liquid is the partial pressure of that gas which would be generated in a gas phase in equilibrium with the liquid at the same temperature. The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules. Gases will always flow from a region of higher partial pressure to one of lower pressure; the larger this difference, the faster the flow.

Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases (i.e., liquid or solid). Most often the term is used to describe a liquid's tendency to evaporate. It is a measure of the tendency of molecules and atoms to escape from a liquid or a solid. A liquid's boiling point corresponds to the point where its vapor pressure is equal to the surrounding atmospheric pressure.

Gases dissolve, diffuse, and react according to their partial pressures, and not necessarily according to their concentrations in a gas mixture.

Dalton's law of partial pressures

Main article: Dalton's law

The pressure of an ideal gas in a mixture is equal to the pressure it would exert if it occupied the same volume alone at the same temperature. This is because ideal gas molecules are so far apart that they don't interfere with each other at all. Actual real-world gases come very close to this ideal.

A consequence of this is that the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the individual gases in the mixture as stated by Dalton's law. For example, given an ideal gas mixture of nitrogen (N₂), hydrogen (H₂) and ammonia (NH₃):

<math>P = P_{{\mathrm{N}}_2} + P_{{\mathrm{H}}_2} + P_{{\mathrm{NH}}_3}</math>

where:
<math>P \,</math>	= total pressure of the gas mixture
<math>P_{{\mathrm{N}}_2}</math>	= partial pressure of nitrogen (N2)
<math>P_{{\mathrm{H}}_2}</math>	= partial pressure of hydrogen (H2)
<math>P_{{\mathrm{NH}}_3}</math>	= partial pressure of ammonia (NH3)

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:

<math>x_{\mathrm{i}} = \frac{P_{\mathrm{i}}}{P} = \frac{n_{\mathrm{i}}}{n}</math>

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

<math>P_{\mathrm{i}} = x_{\mathrm{i}} \cdot P</math>

where:
<math>x_{\mathrm{i}}</math>	= mole fraction of any individual gas component in a gas mixture
<math>P_{\mathrm{i}}</math>	= partial pressure of any individual gas component in a gas mixture
<math>n_{\mathrm{i}}</math>	= moles of any individual gas component in a gas mixture
<math>n</math>	= total moles of the gas mixture
<math>P</math>	= pressure of the gas mixture

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

Equilibrium constants of reactions involving gas mixtures

It is possible to work out the equilibrium constant for a chemical reaction involving a mixture of gases given the partial pressure of each gas and the overall reaction formula. For a reversible reaction involving gas reactants and gas products, such as:

<math>a\,A + b\,B \leftrightarrow c\,C + d\,D</math>

the equilibrium constant of the reaction would be:

<math>K_P = \frac{P_C^c\, P_D^d} {P_A^a\, P_B^b}</math>

where:
<math>K_P</math>	=  the equilibrium constant of the reaction
<math>a</math>	=  coefficient of reactant <math>A</math>
<math>b</math>	=  coefficient of reactant <math>B</math>
<math>c</math>	=  coefficient of product <math>C</math>
<math>d</math>	=  coefficient of product <math>D</math>
<math>P_C^c</math>	=  the partial pressure of <math>C</math> raised to the power of <math>c</math>
<math>P_D^d</math>	=  the partial pressure of <math>D</math> raised to the power of <math>d</math>
<math>P_A^a</math>	=  the partial pressure of <math>A</math> raised to the power of <math>a</math>
<math>P_B^b</math>	=  the partial pressure of <math>B</math> raised to the power of <math>b</math>

For reversible reactions, changes in the total pressure, temperature or reactant concentrations will shift the equilibrium 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 over-riding factor to consider.

Henry's Law and the solubility of gases

Main article: Henry's Law

Gases will dissolve in liquids to an extent that is determined by the equilibrium between the undissolved gas and the gas that has dissolved in the liquid (called the solvent). The equilibrium constant for that equilibrium is:

(1)     <math>k = \frac {P_X}{C_X}</math>

where:
<math>k</math>	=  the equilibrium constant for the solvation process
<math>P_X</math>	=  partial pressure of gas <math>X</math> in equilibrium with a solution containing some of the gas
<math>C_X</math>	=  the concentration of gas <math>X</math> in the liquid solution

The form of the equilibrium constant shows that the concentration of a solute gas in a solution is directly proportional to the partial pressure of that gas above the solution. This statement is known as Henry's Law and the equilibrium constant <math>k</math> is quite often referred to as the Henry's Law constant.

Henry's Law is sometimes written as:

(2)     <math>k' = \frac {C_X}{P_X}</math>

where <math>k'</math> is also referred to as the Henry's Law constant. As can be seen by comparing equations (1) and (2) above, <math>k'</math> is the reciprocal of <math>k</math>. 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.

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 terms, partial pressure is calculated as:

partial pressure = total absolute pressure x volume fraction of gas component

For the component gas "i":

ppi = P x Fi

For example, at 50 metres (165 feet), the total absolute pressure is 6 bar (600 kPa) (i.e., 1 bar of atmospheric pressure + 5 bar of water pressure) 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

where:
ppi	= partial pressure of gas component i  = <math>P_{\mathrm{i}}</math> in the terms used in this article
P	= total pressure = <math>P</math> in the terms used in this article
Fi	= volume fraction of gas component i  =  mole fraction, <math>x_{\mathrm{i}}</math>, in the terms used in this article
ppN2	= partial pressure of nitrogen  = <math>P_{{\mathrm{N}}_2}</math> in the terms used in this article
ppO2	= partial pressure of oxygen  = <math>P_{{\mathrm{O}}_2}</math> 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 metres (115 feet).

Related Chapters

• Vapor and Vapor pressure
• Gas, Ideal gas and Ideal gas law
• Mole fraction and Mole (unit)
• Dalton's law
• Henry's law
• Breathing gas

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