Ideal gas law
= Ideal gas law
{title2=$PV=Nk_BT=n_{\rm mol}RT$}
{wiki}
The <ideal gas law> is the <equation of state> relating the absolute <pressure> $P$, <volume (thermodynamics)> $V$, particle count $N$, and absolute <thermodynamic temperature> $T$ of a classical <ideal gas>:
$$
PV=Nk_BT=n_{\rm mol}RT,\qquad
N=n_{\rm mol}N_{\mathrm A},\qquad R=N_{\mathrm A}k_B.
$$
Here $k_B$ is the <Boltzmann constant>, $N_{\mathrm A}$ is the <Avogadro constant>, $R$ is the <molar gas constant>, and $n_{\rm mol}$ is the <amount of substance> in <moles>. The distinct symbol $n_{\rm mol}$ avoids confusing <amount of substance> with the <number density> $n=N/V$, measured in $\mathrm{m^{-3}}$: the same law can be written $P=nk_BT$. Use absolute <pressure>, relative to a <vacuum>, rather than an unconverted <gauge pressure>. <University Physics Volume 2 by Samuel J. Ling and others/2.1 Molecular Model of an Ideal Gas>[OpenStax University Physics Volume 2, section 2.1: particle-count and molar forms]{ref}
The exact <SI> definitions of the <Avogadro constant> and <Boltzmann constant> give
$$
R=(6.02214076\times10^{23})(1.380649\times10^{-23})
\,\mathrm{J\,mol^{-1}\,K^{-1}}
=8.31446261815324\,\mathrm{J\,mol^{-1}\,K^{-1}}.
$$
This numerical value is exact; the applicability of the <ideal gas> model to a <real gas> is an approximation. https://www.bipm.org/en/si-base-units/mole[BIPM, exact Avogadro constant in the mole definition]{ref} https://www.bipm.org/en/si-base-units/kelvin[BIPM, exact Boltzmann constant in the kelvin definition]{ref}
For $n_{\rm mol}=1\,\mathrm{mol}$, $T=300\,\mathrm K$, and $V=0.025\,\mathrm{m^3}$, direct substitution gives
$$
RT=2494.338785445972\,\mathrm{J\,mol^{-1}},\qquad
P=\frac{n_{\rm mol}RT}{V}
=99773.55141783888\,\mathrm{Pa}
\approx99.774\,\mathrm{kPa}.
$$
At fixed <temperature> and particle count, doubling the <volume (thermodynamics)> to $0.050\,\mathrm{m^3}$ halves the <pressure> to approximately $49.887\,\mathrm{kPa}$: this is <Boyle's law>. At fixed <volume (thermodynamics)> and particle count, doubling the <temperature> from $300\,\mathrm K$ to $600\,\mathrm K$ doubles the <pressure> to approximately $199.547\,\mathrm{kPa}$. These ratios use <kelvins>, not readings in the <degree Celsius> scale: $300\,\mathrm K$ is $26.85\,{}^\circ\mathrm C$, and $600\,\mathrm K$ is $326.85\,{}^\circ\mathrm C$.
\Image[/ideal-gas-state-slices.png]
{title=Ideal gas law: pressure versus volume at fixed temperatures, and pressure versus absolute temperature at fixed volume}
The curved slices show <Boyle's law> at three <temperatures>; the straight slice shows proportionality to absolute <temperature> at one fixed <volume (thermodynamics)>. They are relations among equilibrium states, rather than a prescribed process history. Compression could hold the <temperature> fixed through <heat transfer>, or raise it during <adiabatic compression>. To calculate a process's <heat> or <work>, supplement the <ideal gas law> with the process constraint, the <first law of thermodynamics>, and the relevant <internal energy> or <heat capacity> information. The graph's extrapolation toward $T=0$ is mathematical; it does not assert that a <real gas> remains classical or gaseous there.
For a stellar <mixture> with fixed <mean molecular weight>, the <ideal gas law> can be written $P=\mathcal R\rho T/\mu$, where $\rho$ is <mass density> and $\mathcal R$ is the <stellar gas constant>. It assumes negligible interaction corrections and a <nondegenerate gas>; excitation or <ionization> can change the particle count and thermodynamic response.
In his 1834 memoir on the motive power of <heat>, Émile Clapeyron combined gas laws into a relation between <pressure>, <volume (thermodynamics)>, and <temperature> while analysing <heat engines> (printed page 164). The equation above uses modern <SI> notation and absolute <thermodynamic temperature>; it is not a transcription of his historical temperature convention. https://www3.nd.edu/~powers/ame.20231/clapeyron1834.pdf[Clapeyron, Mémoire sur la puissance motrice de la chaleur (1834), page 164]{ref}