The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron and independently of him, Dmitry Mendeleev in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:
p V = n R T {\displaystyle pV=nRT}
where p, V, and T are respectively the pressure, volume, and temperature, n is the amount of substance, and R is the ideal gas constant. It can also be derived from the microscopic kinetic theory, as was achieved (independently) by August Krönig in 1856 and Rudolf Clausius in 1857.
Formulations
The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin.
Common forms The most frequently introduced forms are: p V = n R T = n k B N A T = N k B T {\displaystyle pV=nRT=nk_{\text{B}}N_{\text{A}}T=Nk_{\text{B}}T} where:
p {\displaystyle p} is the absolute pressure of the gas,
V {\displaystyle V} is the volume of the gas,
n {\displaystyle n} is the amount of substance of gas (also known as number of moles),
R {\displaystyle R} is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
k B {\displaystyle k_{\text{B}}} is the Boltzmann constant,
N A {\displaystyle N_{A}} is the Avogadro constant,
T {\displaystyle T} is the absolute temperature of the gas,
N {\displaystyle N} is the number of particles (usually atoms or molecules) of the gas. In SI units, p is measured in pascals, V is measured in cubic meters, n is measured in moles, and T in kelvins (the Kelvin scale is a shifted Celsius scale, where 0 K = −273.15 °C, the lowest possible temperature). R has for value 8.314 J/(mol·K) = 1.989 ≈ 2 cal/(mol·K), or 0.0821 L⋅atm/(mol⋅K).
Molar form How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount, n (in moles), is equal to total mass of the gas (m) (in kilograms) divided by the molar mass, M (in kilograms per mole):
n = m M . {\displaystyle n={\frac {m}{M}}.}
By replacing n with m/M and subsequently introducing density ρ = m/V, we get:
p V = m M R T {\displaystyle pV={\frac {m}{M}}RT}
p = m V R T M {\displaystyle p={\frac {m}{V}}{\frac {RT}{M}}}
p = ρ R M T {\displaystyle p=\rho {\frac {R}{M}}T}
Defining the specific gas constant Rspecific as the ratio R/M,
p = ρ R specific T . {\displaystyle p=\rho R_{\text{specific}}T.}
This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as
p v = R specific T . {\displaystyle pv=R_{\text{specific}}T.}
It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol R. In such cases, the universal gas constant is usually given a different symbol such as R ¯ {\displaystyle {\bar {R}}} or R ∗ {\displaystyle R^{*}} to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being used.
Statistical mechanics In statistical mechanics, the following molecular equation (i.e. the ideal gas law in its theoretical form) is derived from first principles:
p = n k B T , {\displaystyle p=nk_{\text{B}}T,}
where p is the absolute pressure of the gas, n is the number density of the molecules (given by the ratio n = N/V, in contrast to the previous formulation in which n is the number of moles), T is the absolute temperature, and kB is the Boltzmann constant relating temperature and energy, given by:
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