Ideal gas law

Isotherms of an ideal gas. The curved lines represent the relationship between pressure (on the vertical axis) and volume (on the horizontal axis) for an ideal gas at different temperatures: lines that are farther away from the origin (that is, lines that are nearer to the top right-hand corner of the diagram) represent higher temperatures.

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

PV=nRT,

where $P$ , $V$ and $T$ are the pressure, volume and absolute temperature; $n$ is the number of moles of gas; and $R$ is the ideal gas constant. It is the same for all gases. It can also be derived from the microscopic kinetic theory, as was achieved (apparently independently) by August Krönig in 1856^[2] and Rudolf Clausius in 1857.^[3]

Equation

Molecular collisions within a closed container (the propane tank) are shown (right). The arrows represent the random motions and collisions of these molecules. The pressure and temperature of the gas are directly proportional: as the temperature is increased, the pressure of the propane increases by the same factor. A simple consequence of this proportionality is that on a hot summer day, the propane tank pressure will be elevated, and thus propane tanks must be rated to withstand such increases in pressure.

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.^[4]

Common forms

The most frequently introduced form is

PV=nRT=Nk_{\text{B}}T,

where:

$P$ is the pressure of the gas,
$V$ is the volume of the gas,
$n$ is the amount of substance of gas (also known as number of moles),
$N$ is the number of gas molecules (or the Avogadro constant times the amount of substance $N_{\text{A}}n$ ),
$R$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
$k_{\text{B}}$ is the Boltzmann constant
$T$ 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 total mass of the gas (m) (in grams) divided by the molar mass (M) (in grams per mole):

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

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

PV={\frac {m}{M}}RT

P={\frac {m}{V}}{\frac {RT}{M}}

P=\rho {\frac {R}{M}}T

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

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

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 ${\bar {R}}$ 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.^[5]

Statistical mechanics

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

P=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, in contrasts to the previous formulation in which n is the number of moles), T is the absolute temperature, and k_B is the Boltzmann constant relating temperature and energy, given by:

k_{\text{B}}={\frac {R}{N_{\text{A}}}}

where N_A is the Avogadro constant.

From this we notice that for a gas of mass m, 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

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

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

In SI units, P is measured in pascals, V in cubic metres, 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 intermolecular attractions between the molecules of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is kinetic energy.

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

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

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

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)
Isobaric process	Pressure	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)
	Volume	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
Isothermal process	Temperature	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^[b]	P₂ − P₁	P₂ = P₁ + (P₂ − P₁)		T₂ = T₁ + μ_JT(P₂ − P₁)
		T₂ − T₁	P₂ = P₁ + (T₂ − T₁)/μ_JT		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 (PV=nRT) 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

{\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

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 kinetic energy of the particle is:

{\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

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

-{\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

\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

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

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:

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.

References

↑ Clapeyron, E. (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l'École Polytechnique (in French). XIV: 153–90. Facsimile at the Bibliothèque nationale de France (pp. 153–90).
↑ Krönig, A. (1856). "Grundzüge einer Theorie der Gase". Annalen der Physik und Chemie (in German). 99 (10): 315–22. Bibcode:1856AnP...175..315K. doi:10.1002/andp.18561751008. Facsimile at the Bibliothèque nationale de France (pp. 315–22).
↑ Clausius, R. (1857). "Ueber die Art der Bewegung, welche wir Wärme nennen". Annalen der Physik und Chemie (in German). 176 (3): 353–79. Bibcode:1857AnP...176..353C. doi:10.1002/andp.18571760302. Facsimile at the Bibliothèque nationale de France (pp. 353–79).
↑ "Equation of State".
↑ Moran; Shapiro (2000). Fundamentals of Engineering Thermodynamics (4th ed.). Wiley. ISBN 0-471-31713-6.
↑ J. R. Roebuck (1926). "The Joule-Thomson Effect in Air". Proceedings of the National Academy of Sciences of the United States of America. 12 (1): 55–58. Bibcode:1926PNAS...12...55R. doi:10.1073/pnas.12.1.55. PMC 1084398. PMID 16576959.

External links

"Website giving credit to Benoît Paul Émile Clapeyron, (1799–1864) in 1834". Archived from the original on July 5, 2007.
Configuration integral (statistical mechanics) where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the Helmholtz free energy and the partition function, but without using the equipartition theorem, is provided. Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28.
pv=nrt calculator Engineering Units online calculator

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[1] Clapeyron, E. (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l'École Polytechnique (in French). XIV: 153–90. Facsimile at the Bibliothèque nationale de France (pp. 153–90).

[2] Krönig, A. (1856). "Grundzüge einer Theorie der Gase". Annalen der Physik und Chemie (in German). 99 (10): 315–22. Bibcode:1856AnP...175..315K. doi:10.1002/andp.18561751008. Facsimile at the Bibliothèque nationale de France (pp. 315–22).

[3] Clausius, R. (1857). "Ueber die Art der Bewegung, welche wir Wärme nennen". Annalen der Physik und Chemie (in German). 176 (3): 353–79. Bibcode:1857AnP...176..353C. doi:10.1002/andp.18571760302. Facsimile at the Bibliothèque nationale de France (pp. 353–79).

[Equation_of_State-4] "Equation of State".

[5] Moran; Shapiro (2000). Fundamentals of Engineering Thermodynamics (4th ed.). Wiley. ISBN 0-471-31713-6.

[6] J. R. Roebuck (1926). "The Joule-Thomson Effect in Air". Proceedings of the National Academy of Sciences of the United States of America. 12 (1): 55–58. Bibcode:1926PNAS...12...55R. doi:10.1073/pnas.12.1.55. PMC 1084398. PMID 16576959.

[a]

[b]

Mole concepts
Physical quantities	Mass concentration Molar concentration Molality Mass Volume Density Mole fraction Mass fraction Amount of substance Molar mass Atomic mass Particle number Avogadro constant Pressure Kelvin temperature Gas constant Molar volume Specific volume
Laws	Charles's law Boyle's law Gay-Lussac's law Combined gas law Ideal gas law

Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity condensed matter physics complex system chaos information theory Boltzmann machine

Ideal gas law

Equation

Common forms

Molar form

Statistical mechanics

Energy associated with a gas

Applications to thermodynamic processes

Deviations from ideal behavior of real gases

Derivations

Empirical

Theoretical

Kinetic theory

Statistical mechanics

See also

References

Further reading

External links