Editing Ideal gas law (section)

==== Statistical mechanics ====
{{Main|Statistical mechanics}}

Let '''q''' = (''q''<sub>x</sub>, ''q''<sub>y</sub>, ''q''<sub>z</sub>) and '''p''' = (''p''<sub>x</sub>, ''p''<sub>y</sub>, ''p''<sub>z</sub>) 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 (two times) the time-averaged kinetic energy of the particle is:
: <math>\begin{align}
\langle \mathbf{q} \cdot \mathbf{F} \rangle
&= \left\langle q_{x} \frac{dp_{x}}{dt} \right\rangle +
\left\langle q_{y} \frac{dp_{y}}{dt} \right\rangle +
\left\langle q_{z} \frac{dp_{z}}{dt} \right\rangle\\
&=-\left\langle q_{x} \frac{\partial H}{\partial q_x} \right\rangle -
\left\langle q_{y} \frac{\partial H}{\partial q_y} \right\rangle -
\left\langle q_{z} \frac{\partial H}{\partial q_z} \right\rangle = -3k_\text{B} T,
\end{align}</math>
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
: <math>3Nk_{\rm B} T = - \left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle.</math>

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
: <math>-\left\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\right\rangle = P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S},</math>
where d'''S''' is the infinitesimal area element along the walls of the container. Since the [[divergence]] of the position vector '''q''' is
: <math>
\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,
</math>
the [[divergence theorem]] implies that
: <math>P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S}
= P \int_{\text{volume}} \left( \nabla \cdot \mathbf{q} \right) dV
= 3PV,</math>
where ''dV'' is an infinitesimal volume within the container and ''V'' is the total volume of the container.

Putting these equalities together yields
: <math>3 N k_\text{B} T = -\left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle = 3PV,</math>
which immediately implies the ideal gas law for ''N'' particles:
: <math>PV = Nk_{\rm B} T = nRT,</math>
where ''n'' = ''N''/''N''<sub>A</sub> is the number of [[mole (unit)|moles]] of gas and ''R'' = ''N''<sub>A</sub>''k''<sub>B</sub> is the [[gas constant]].