# Ideal gas law

Isotherms of an ideal gas. The curved lines represent the relationship between pressure (on the vertical, y-axis) and volume (on the horizontal, x-axis) for an ideal gas at different temperatures: lines which are further 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 is the equation of state of a hypothetical ideal gas. It is a good approximation to 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 Boyle's law and Charles's law.[1] The ideal gas law is often written as:

$pV=NRT\,$

where the letters denote pressure, volume, amount, the ideal gas constant, and temperature of the gas, respectively.

Actually, the most conceptual and pure form of this law is:

$p=nT\,$

where n denotes the number density. The ideal gas constant, being a dimensional constant, has in fact been eliminated. The law in this form is also readily derivable from kinetic theory, as was achieved (apparently independently) by August Krönig in 1856[2] and Rudolf Clausius in 1857.[3]

## Equation

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: in the SI system of units, Kelvin.[4]

### Common form

The most frequently introduced form is

$PV=NRT\,$

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)

R is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant.

T is the temperature of the gas

In SI units, P is measured in pascals, V in cubic metres, N in moles, and T in Kelvin (The Kelvin scale is a shifted Celsius scale where 0.00 Kelvin = -273.15 degrees Celsius, the lowest possible temperature). R has the value 8.314 J·K−1·mol−1 or 0.08206 L·atm·mol−1·K−1or ≈2 calories if using pressure in standard atmospheres (atm) instead of pascals, and volume in litres instead of cubic metres.

### 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 mass (m) (in grams) divided by the molar mass (M) (in grams per mole):

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

By replacing N with m / M:

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

and subsequently introducing density ρ = m/V, we get:

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

Defining the specific gas constant Rspecific as the ratio R/M,

$\ P = \rho R_{\rm specific}T$

Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as

$\ Pv = R_{\rm specific}T$

This form of the ideal gas law is very useful in the study of the thermodynamics of a fixed ideal gas (being a compound or a mixture of ideal gases), because it links its pressure, density (or specific volume), and temperature in a unique equation. 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 R0 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]

### General form

The most general (holding also for inhomogenous systems) and the most simple form of the ideal gas law is local:

$\ P = n T$

where P is the absolute pressure of the gas; n is the number density; T is the absolute temperature. In the framework of SI they can be respectively measured in Pascals, in molecules per cubic meter, and in joules. Explicitly:

$\ P[Pa] = n[1/m^3] T[J]$

Notably with the statistical mechanics this equation has become no more a principle but has been derived from first principles.

If the temperature was in kelvin, the measure in joules could be simply obtained by multiplication with the Boltzmann constant:

$T[J] = k_B[J/K] T[K]$

so we could introduce this dimensional constant in the simple above equation:

$\ P[Pa] = n[1/m^3] k_B[J/K] T[K]$

Furthermore, (only) in an homogeneous system the local number density is equal to the average number density:

$n[1/m^3]= \frac{N[]}{V[m^3]}$

So we can further complicate the equation to approach the empirical form:

$\ P[Pa] V[m^3] = k_B[J/K] N[ ] T[K]$

and the amount of substance can be expressed in moles rather in simple molecules by dividing for the Avogadro's number:

$N [mol] = \frac{N[ ]}{N_A[1/mol]}$

arriving to:

$\ P[Pa] V[m^3] = N_A[1/mol] k_B[J/K] N[mol] T[K]$

Finally, by comparing with the empirical form:

$\ P[Pa] V[m^3] = R[J/mol K] N[mol] T[K]$

, the ideal gas constant turns actually to be simply a dimensional constant:

$R[J/mol K]= N_A[1/mol]\, k_B[J/K]$

On the other hand in an inhomogeneous system, the local form is the only one applicable and the tentative to forcedly employ the empirical formula brings to misleading results. This is why in numerical simulations, ususally featuring inhomogenous systems, the empirical formula of ideal gas law is completely useless.

## 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, or S) 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 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
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)

^ a. In an isentropic process, system entropy (S) is constant. Under these conditions, P1 V1γ = P2 V2γ, 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 monatomic 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.

## 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 intermolecular 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.

## Theoretical derivations

Since the end of the XIX century, ideal gas law is no more considered an empirical law nor a physical principle but it has been derivated as a consequence of the kinetic motion of gas particles under some special assumptions. The discipline that arrived to this result was the kinetic theory of gases, now become a branch of statistical mechanics.

### Kinetic theory

Main article: 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

Main article: Statistical mechanics

Let q = (qx, qy, qz) and p = (px, py, pz) 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:

\begin{align} \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{align}

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

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

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

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

which immediately implies the ideal gas law for N particles:

$P= n T$

where n is the number density.