SQUARE45

Let

\mathcal{S}

be a system of

N

non-interacting particles of mass

m

confined to a volume

V

at temperature

T

. The system's Hamiltonian is

\mathcal{H} = \sum_{i=1}^{N} \frac{\vec{p}_i^2}{2m}

. The pressure

P

is defined by the average momentum flux tensor

\langle \Pi \rangle

exerted on the container walls. For an ideal gas, the equation of state is derived from the partition function

Z

Z = \frac{1}{N!}\left(\frac{V}{h^3}\right)^N \left(\frac{2\pi m k_B T}{h^2}\right)^{3N/2}

where

k_B

is the Boltzmann constant and

h

is Planck's constant. The internal energy

U

is related to

Z

U = -\left(\frac{\partial \ln Z}{\partial \beta}\right)_{V, N}

, where

\beta = 1/(k_B T)

. By the equipartition theorem and the ideal gas law, the pressure

P

is given by the thermodynamic relation:

P = \frac{N k_B T}{V} = \frac{n R T}{V}

where

n = N/N_A

is the number density and

R = N_A k_B

is the universal gas constant.

Pressure of an Ideal Gas