Ideal gas law

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 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:
where, and are the pressure, volume and temperature respectively; is the amount of substance; and is the ideal gas constant.
It can also be derived from the microscopic kinetic theory, as was achieved 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:where:

is the absolute pressure of the gas,
is the volume of the gas,
is the amount of substance of gas,
is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
is the Boltzmann constant,
is the Avogadro constant,
is the absolute temperature of the gas,
is the number of particles 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. R has for value 8.314 J/ = 1.989 ≈ 2 cal/, or 0.0821 L⋅atm/.

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, is equal to total mass of the gas divided by the molar mass, M :
By replacing n with m/''M and subsequently introducing density ρ'' = m/''V, we get:
Defining the specific gas constant R''_specific as the ratio R/''M,
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
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 or 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 is derived from first principles:
where is the absolute pressure of the gas, is the number density of the molecules, is the absolute temperature, and is the Boltzmann constant relating temperature and energy, given by:
where is the Avogadro constant.
The form can be further simplified by defining the kinetic energy corresponding to the temperature:
so the ideal gas law is more simply expressed as:
From this we notice that for a gas of mass, with an average particle mass of times the atomic mass constant,, the number of molecules will be given by
and since, we find that the ideal gas law can be rewritten as
In SI units, is measured in pascals, in cubic metres, in kelvins, and in SI units.

Combined gas law

Combining the laws of Charles, Boyle, and Gay-Lussac gives the combined gas law, which can take the same functional form as the ideal gas law. This form does not specify the number of moles, and the ratio of to is simply taken as a constant:
where is the pressure of the gas, is the volume of the gas, is the absolute temperature of the gas, and is a constant. More commonly, when comparing the same substance under two different sets of conditions, the law is written as:

Energy associated with a gas

According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas.
This corresponds to the kinetic energy of n moles of a monoatomic gas having 3 degrees of freedom: x, y, z. The table here below gives this relationship for different amounts of a monoatomic gas.

Energy of a monoatomic gas	Mathematical expression
Energy associated with one mole
Energy associated with one gram
Energy associated with one atom

Applications to thermodynamic processes

The table below essentially simplifies the ideal gas equation for a particular process, making the equation easier to solve using numerical methods.
A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by a subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties is constant throughout the process.
For a given thermodynamic process, in order to specify the extent of a particular process, one of the properties ratios must be specified. Also, the property for which the ratio is known must be distinct from the property held constant in the previous column.
In the final three columns, the properties at state 2 can be calculated from the properties at state 1 using the equations listed.

Process	Constant	Known ratio or delta	p₂	V₂	T₂
Isobaric process	Pressure	V₂/V₁	p₂ = p₁	V₂ = V₁	T₂ = T₁
Isobaric process	Pressure	T₂/T₁	p₂ = p₁	V₂ = V₁	T₂ = T₁
Isochoric process	Volume	p₂/p₁	p₂ = p₁	V₂ = V₁	T₂ = T₁
Isochoric process	Volume	T₂/T₁	p₂ = p₁	V₂ = V₁	T₂ = T₁
Isothermal process	Temperature	p₂/p₁	p₂ = p₁	V₂ = V₁	T₂ = T₁
Isothermal process	Temperature	V₂/V₁	p₂ = p₁	V₂ = V₁	T₂ = T₁
Isentropic process		p₂/p₁	p₂ = p₁	V₂ = V₁	T₂ = T₁^/γ
Isentropic process		V₂/V₁	p₂ = p₁^−γ	V₂ = V₁	T₂ = T₁
Isentropic process		T₂/T₁	p₂ = p₁^γ/	V₂ = V₁^1/	T₂ = T₁
Polytropic process	P Vⁿ	p₂/p₁	p₂ = p₁	V₂ = V₁	T₂ = T₁^/n
Polytropic process	P Vⁿ	V₂/V₁	p₂ = p₁⁻ⁿ	V₂ = V₁	T₂ = T₁
Polytropic process	P Vⁿ	T₂/T₁	p₂ = p₁^n/	V₂ = V₁^1/	T₂ = T₁
Isenthalpic process		p₂ − p₁	p₂ = p₁ +		T₂ = T₁ + μ_JT
Isenthalpic process		T₂ − T₁	p₂ = p₁ + /μ_JT		T₂ = T₁ +

a. In an isentropic process, system entropy is constant. Under these conditions, p₁V₁^γ = p₂V₂^γ, where γ is defined as the heat capacity ratio, which is constant for a calorifically perfect gas. The value used for γ is typically 1.4 for diatomic gases like nitrogen and oxygen,. Also γ is typically 1.6 for mono atomic gases like the noble gases helium, and argon. In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.
b. In an isenthalpic process, system enthalpy is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gases, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the Joule–Thomson effect. For reference, the Joule–Thomson coefficient μ_JT for air at room temperature and sea level is 0.22 °C/bar.