Ideal Gas Law

An “ideal” gas is a gas that is made of very weakly interacting molecules that are very dilute.

Ideal gas molecules do not attract or repel each other. The only interaction between ideal gas molecules would be an elastic collision upon impact with each other or an elastic collision with the walls of the container.
Ideal gas molecules themselves take up no volume. The gas takes up volume since the molecules expand into a large region of space, but the Ideal gas molecules are approximated as point particles that have no volume in and of themselves.

Internal energy as the sum of kinetic energies:

U = \sum_{i = 1}^{N} K_{i} = \sum_{i = 1}^{N} \frac{1}{2} m | {\vec{v}}_{i} |^{2}

It is also true that:

U = \frac{d}{2} N k_{B} T

d - number of degrees of freedom per molecule, e.g. 2 for 2-dimensional molecules' translational kinetic energy, and 3 for 3-dimensional
More complex molecules:

Diatomic molecules at room temperature: d ≈ 5 (3 translational + 2 rotational), giving $U = \frac{5}{2} N k_{B} T$
Polyatomic molecules: d ≈ 6 (3 translational + 3 rotational), giving $U = 3 N k_{B} T$

Coarse graining force

F = \frac{Δ p}{Δ t}

as $Δ t \to 0$ , we take the instantaneous force:

F = \frac{d p}{d t}

k_{B} = 1.38 \times 10^{- 23} J \cdot K^{- 1} R = 8.31 J \cdot K^{- 1} \cdot {mol}^{- 1} = 0.082 L \cdot atm \cdot K^{- 1} \cdot {mol}^{- 1}