Ideal gas law

Isotherms of an ideal gas

The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834.

The state of an amount of gas is determined by its pressure, volume, and temperature according to the equation:

$\ pV=nRT$

where

$\ p$ is the absolute pressure [Pa],

$\ V$ is the volume [m³],

$\ n$ is the amount of substance of gas [mol],

$\ R$ is the gas constant [8.314 472 m³·Pa·K⁻¹·mol⁻¹],

$\ T$ is the temperature in kelvin [K].

The ideal gas constant (R) is dependent on what units are used in the formula. The value given above, 8.314472, is for the SI units of pascal cubic meters per mole per kelvin. Another value for R is 0.082057 L·atm·mol⁻¹·K⁻¹)

The ideal gas law is the most accurate for monatomic gases at high temperatures and low pressures because it does not factor in the size of each gas molecule nor the effects of intermolecular attraction. The more accurate Van der Waals equation takes these factors into consideration.

Ideal gas law mathematically follows from statistical mechanics of primitive identical particles (=particles without internal structure) when the only interaction between them is exchange of momentum and kinetic energy in elastic collisions.

Alternate forms

Considering that the number of moles ($n$) could also be given in mass, sometimes you may wish to use an alternate form of the ideal gas law. The number of moles ($n$) is equal to the mass ($m$) divided by the molar mass ($M$):

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

Then, replacing $n$ gives:

$\ pV={\frac {m}{M}}RT$

from where

$\ p=\rho {\frac {R}{M}}T$

particularly useful because it links pressure, density, and temperature in a unique formula independent from the quantity of the considered gas.

In statistical mechanics, and is often derived from first principles:

$\ pV=Nk_{b}T$

Here, $k_{b}$ is Boltzmann's constant, and $N$ is the actual number of molecules, in contrast to the other formulation, which uses $n$, the number of moles. This relation implies that $Nk_{b}=nR$, and the consistency of this result with experiment is a good check on the principles of statistical mechanics.

From here we can notice that for an average particle mass of $\mu$ times the atomic mass of Hydrogen,

$N={\frac {m}{\mu m_{H}}}$

and since $\rho =m/V$, we find that the ideal gas law can be re-written as:

$p={\frac {k_{b}}{\mu m_{H}}}\rho T$

Proof

Empirical

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

Consider an amount of gas. Let its initial state be defined as:

volume = $v_{0}$

pressure = $p_{0}$

temperature = $t_{0}$

If this gas now undergoes an isobaric process, its state will change:

volume: $v'=v_{0}(1+\alpha t_{0})\,$

pressure $p'=p_{0}\,$

temperature $t'=t_{0}(1+\alpha t_{0})\,$.

If it then undergoes an isothermal process:

$pv=p_{0}v'\,$

where

p = final pressure

v = final volume

T = final temperature (= t')

So:

$pv=p_{0}v'=p_{0}v_{0}(1+\alpha t_{0})={\frac {p_{0}v_{0}}{t_{0}}}T$;

where

${\frac {p_{0}v_{0}}{t_{0}}}$, termed $R$, is the universal gas constant.

Using this notation we get:

$pv=RT\,$

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

$pnv=nRT\,$

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

$pV=nRT\,$

Theoretical

The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief amongst which is that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume.

See also

• Ideal gas
• Equation of state
• Thermodynamics
• Van der Waals equation
• Boyle's Law
• Charles's Law
• Dalton's Law
• Amagat's law

References

• Davis and Masten Principles of Environmental Engineering and Science, McGraw-Hill Companies, Inc. New York (2002) ISBN 0-07-235053-9
• Website giving credit to Benoît Paul Émile Clapeyron, (1799-1864) in 1834
• Website containing Ideal Gas Law Calculator & a host of other scientific calculators, Rex Njoku & Dr. Anthony Steyermark -University of St.Thomas