Ideal gas law

As explained in Wikipedia, The ideal gas law 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 Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law and Avogadro's Law.^[1] The ideal gas law is often written as

${\displaystyle PV=nRT}$,

where:

• ${\displaystyle P}$ is the pressure of the gas,
• ${\displaystyle V}$ is the volume of the gas,
• ${\displaystyle n}$ is the amount of substance of gas (in moles),
• ${\displaystyle R}$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
• ${\displaystyle T}$ is the absolute temperature of the gas.

It can also be derived from the microscopic kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

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 form

The most frequently introduced form is

${\displaystyle PV=nRT,}$

where:

• ${\displaystyle P}$ is the pressure of the gas,
• ${\displaystyle V}$ is the volume of the gas,
• ${\displaystyle n}$ is the amount of substance of gas (also known as number of moles),
• ${\displaystyle R}$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
• ${\displaystyle T}$ is the absolute temperature of the gas.

In SI units, P is measured in pascals, V is measured in cubic metres, n is measured in moles, and T in kelvins (the Kelvin scale is a shifted Celsius scale, where 0.00 K = −273.15 °C, the lowest possible temperature). R has the value 8.314 J/(K·mol) ≈ 2 cal/(K·mol), or 0.08206 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 the total mass of the gas (m) (in grams) divided by the molar mass (M) (in grams per mole):

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

By replacing n with m/M and subsequently introducing density ρ = m/V, we get:

${\displaystyle PV={\frac {m}{M}}RT,}$

${\displaystyle P=\rho {\frac {R}{M}}T.}$

Defining the specific gas constant R_specific(r) as the ratio R/M,

${\displaystyle P=\rho R_{\text{specific}}T.}$

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

${\displaystyle Pv=R_{\text{specific}}T.}$

It is common, especially in engineering 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 ${\displaystyle {\bar {R}}}$ 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 referred to.

Statistical mechanics

In statistical mechanics the following molecular equation is derived from first principles

${\displaystyle PV=Nk_{\text{B}}T,}$

where P is the absolute pressure of the gas, N is the number of molecules in the given volume V (the number density is given by the ratio n = N/V), k_B is the Boltzmann constant relating temperature and energy, and T is the absolute temperature.

K_B =R/N_A

The number density contrasts to the other formulation, which uses n, the number of moles and V, the volume. This relation implies that R = N_Ak_B, where N_A is Avogadro's constant, and the consistency of this result with experiment is a good check on the principles of statistical mechanics. In extreme conditions the principles of statistical mechanics may break down as some of the assumptions relating a real life example to an ideal gas become untrue.

From this we notice that for a gas with an average particle mass of μ times the atomic mass constant m_u (i.e., the mass is μ u) the number of molecules will be given by

${\displaystyle N={\frac {m}{\mu m_{\text{u}}}},}$

and since ρ = m/V = nμm_u, we find that the ideal gas law can be rewritten as

${\displaystyle P={\frac {1}{V}}{\frac {m}{\mu m_{\text{u}}}}k_{B}T={\frac {k_{B}}{\mu m_{\text{u}}}}\rho T.}$

In SI units, P is measured in pascals, V in cubic metres, Y is a dimensionless number, and T in measured kelvins. k_B has the value 1.38·10⁻²³ J/K in SI units.

Energy associated with a gas

According to the assumptions of the kinetic theory of gases, we assumed that there are no inter molecular attractions between the molecules of an ideal gas its potential energy is zero. hence, all the energy possessed by the gas is kinetic energy.

${\displaystyle E={\frac {3}{2}}RT}$

This is the kinetic energy of one mole of a gas.

Energy of gas	Mathematical formula
energy associated with one mole of a gas	${\displaystyle E={\frac {3}{2}}RT}$
energy associated with one gram of a gas	${\displaystyle E={\frac {3}{2}}rT}$
energy associated with one molecule of a gas	${\displaystyle E={\frac {3}{2}}k_{B}T}$

Applications to thermodynamic processes

The table below essentially simplifies the ideal gas equation for a particular processes, thus making this 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 subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (P, V, T, S, or H) is constant throughout the process.

For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios (which are listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).

In the final three columns, the properties (P, V, or T) 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(V2/V1)	T2 = T1(V2/V1)
		T2/T1	P2 = P1	V2 = V1(T2/T1)	T2 = T1(T2/T1)
Isochoric process (Isovolumetric process) (Isometric process)	Volume	P2/P1	P2 = P1(P2/P1)	V2 = V1	T2 = T1(P2/P1)
		T2/T1	P2 = P1(T2/T1)	V2 = V1	T2 = T1(T2/T1)
Isothermal process	Temperature	P2/P1	P2 = P1(P2/P1)	V2 = V1/(P2/P1)	T2 = T1
		V2/V1	P2 = P1/(V2/V1)	V2 = V1(V2/V1)	T2 = T1
Isentropic process (Reversible adiabatic process)	Entropy[a]	P2/P1	P2 = P1(P2/P1)	V2 = V1(P2/P1)(−1/γ)	T2 = T1(P2/P1)(γ − 1)/γ
		V2/V1	P2 = P1(V2/V1)−γ	V2 = V1(V2/V1)	T2 = T1(V2/V1)(1 − γ)
		T2/T1	P2 = P1(T2/T1)γ/(γ − 1)	V2 = V1(T2/T1)1/(1 − γ)	T2 = T1(T2/T1)
Polytropic process	P Vn	P2/P1	P2 = P1(P2/P1)	V2 = V1(P2/P1)(-1/n)	T2 = T1(P2/P1)(n − 1)/n
		V2/V1	P2 = P1(V2/V1)−n	V2 = V1(V2/V1)	T2 = T1(V2/V1)(1 − n)
		T2/T1	P2 = P1(T2/T1)n/(n − 1)	V2 = V1(T2/T1)1/(1 − n)	T2 = T1(T2/T1)
Isenthalpic process (Irreversible adiabatic process)	Enthalpy	P2 − P1	P2 = P1 + (P2 − P1)		T2 = T1 + μJT(P2 − P1)
		T2 − T1	P2 = P1 + (T2 − T1)/μJT		T2 = T1 + (T2 − T1)

^{^} a. In an isentropic process, system entropy (S) is constant. Under these conditions, P₁ V₁^γ = P₂ V₂^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 (N₂) and oxygen (O₂), (and air, which is 99% diatomic). Also γ is typically 1.6 for mono atomic gases like the noble gases helium (He), and argon (Ar). 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 (H) is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, 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.^[6]

Deviations from ideal behavior of real gases

The equation of state given here applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and inter molecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. More detailed equations of state, such as the van der Waals equation, account for deviations from ideality caused by molecular size and intermolecular forces.

A residual property is defined as the difference between a real gas property and an ideal gas property, both considered at the same pressure, temperature, and composition.

Derivations

Empirical

The ideal gas law can be derived from combining two empirical gas laws: the combined gas law and Avogadro's law. The combined gas law states that

${\displaystyle {\frac {PV}{T}}=C,}$

where C is a constant that is directly proportional to the amount of gas, n (Avogadro's law). The proportionality factor is the universal gas constant, R, i.e. C = nR.

Hence the ideal gas law is

${\displaystyle PV=nRT.}$

Theoretical

Kinetic theory

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 among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.

Statistical mechanics

Let q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time-averaged potential energy of the particle is:

${\displaystyle {\begin{aligned}\langle \mathbf {q} \cdot \mathbf {F} \rangle &={\Bigl \langle }q_{x}{\frac {dp_{x}}{dt}}{\Bigr \rangle }+{\Bigl \langle }q_{y}{\frac {dp_{y}}{dt}}{\Bigr \rangle }+{\Bigl \langle }q_{z}{\frac {dp_{z}}{dt}}{\Bigr \rangle }\\&=-{\Bigl \langle }q_{x}{\frac {\partial H}{\partial q_{x}}}{\Bigr \rangle }-{\Bigl \langle }q_{y}{\frac {\partial H}{\partial q_{y}}}{\Bigr \rangle }-{\Bigl \langle }q_{z}{\frac {\partial H}{\partial q_{z}}}{\Bigr \rangle }=-3k_{B}T,\end{aligned}}}$

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of N particles yields

${\displaystyle 3Nk_{B}T=-{\biggl \langle }\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}{\biggr \rangle }.}$

By Newton's third law and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure P of the gas. Hence

${\displaystyle -{\biggl \langle }\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}{\biggr \rangle }=P\oint _{\mathrm {surface} }\mathbf {q} \cdot d\mathbf {S} ,}$

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

${\displaystyle \nabla \cdot \mathbf {q} ={\frac {\partial q_{x}}{\partial q_{x}}}+{\frac {\partial q_{y}}{\partial q_{y}}}+{\frac {\partial q_{z}}{\partial q_{z}}}=3,}$

the divergence theorem implies that

${\displaystyle P\oint _{\mathrm {surface} }\mathbf {q} \cdot d\mathbf {S} =P\int _{\mathrm {volume} }\left(\nabla \cdot \mathbf {q} \right)dV=3PV,}$

where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields

${\displaystyle 3Nk_{B}T=-{\biggl \langle }\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}{\biggr \rangle }=3PV,}$

which immediately implies the ideal gas law for N particles:

${\displaystyle PV=Nk_{B}T=nRT,\,}$

where n = N/N_A is the number of moles of gas and R = N_Ak_B is the gas constant.