The ideal gas law is the equation of state of an ideal gas. The state of an amount of gas is determined by its pressure, volume, and temperature. The equation has the form

$pV\; =\; nRT\; \backslash$

where

$p\; \backslash$ is the pressure,

$V\; \backslash$ is the volume,

$n\; \backslash$ is the number of moles of gas,

$R\; \backslash$ is the gas constant, and

$T\; \backslash$ is the temperature in kelvin or rankine.

The ideal gas law is most accurate for monoatomic gases and is favored at high temperatures and low pressures. It does not factor in the size of each gas molecule or the effects of intermolecular attraction. The more accurate Van der Waals equation takes these into consideration.

Alternate Forms

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Considering that the number of moles ($n\; \backslash$) could also be given in mass, sometimes you may wish to use an alternate form of the ideal gas law. This is particularly useful when asked for the ideal gas law approximation of a known gas. Consider that the number of moles ($n\; \backslash$) is equal to the mass ($m\; \backslash$) divided by the molar mass ($M\; \backslash$), such that:

$n\; =\; \{\backslash frac\{m\}\{M\}\}\; \backslash$

Therefore, replacing $n\; \backslash$ gives you:

$pV\; =\; \{\backslash frac\{mRT\}\{M\}\}\; \backslash$

In thermodynamics and physics, when something is referred to as specific, it simply means the value per unit mass. In the case of the gas constant, the specific gas constant ($r\; \backslash$) would be$R\; \backslash$divided by the molar mass of the gas you are working with:

$r\; =\; \{\backslash frac\{R\}\{M\}\}\; \backslash$ or $R\; =\; rM\; \backslash$ (where$r\; \backslash$is the specific gas constant)

Replacing$r\; \backslash$with these values into the above function yields:

$pV\; =\; mrT\; \backslash$ (the molar masses cancel) or $pV\; =\; nrMT\; \backslash$

Proof

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Empirical

The ideal gas law can be proved using Boyle, Charles and Gay-Lussac laws.

Consider a volume $v\_0$ of gas. Let its state be defined as:

$p\_0\; =\; 100\; \backslash \; \backslash mathrm\{kPa\}\; \backslash ,$

$t\_0\; =\; 290\; \backslash \; \backslash mathrm\{K\}$

If this gas undergoes an isobaric process, its final volume will be:

$v\text{'}\; =\; v\_0(1\; +\; \backslash alpha\; t)\; \backslash ,$

and its temperature will be $t$.

If it then undergoes an isothermal process:

$p\_0v\text{'}\; =\; pv\; \backslash ,$

So:

$pv\; =\; p\_0v\text{'}\; \backslash ,$;

$pv\; =\; p\_0v\_0(1\; +\; \backslash alpha\; t)\; \backslash ,$;

$pv\; =\; \{\backslash frac\{p\_0\; v\_0\}\{290\; \backslash \; \backslash mathrm\{K\}\}\}T$;

where $\{\backslash frac\{p\_0\; v\_0\}\{290\; \backslash \; \backslash mathrm\{K\}\}\}$ called $R$, is the universal gas constant. Using this notation we get:

$pv\; =\; RT\; \backslash ,$

And multiplying both sides of the equation by n (numbers of moles):

$pnv\; =\; nRT\; \backslash ,$

Using the symbol $V$ as a shorthand for $nv$ (volume of n moles) we get:

$pV\; =\; nRT\; \backslash ,$

Theoretical

The ideal gas law can also be derived from first principles using the kinetic theory of gases, if the molecules are assumed to be hard spheres.

See also

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• Equation of state
• Ideal gas
• Thermodynamics

Gas laws

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