[[Thermodynamics MOC]]
# Ideal gas law
The **ideal gas law** relates the [[Macroscopic state variables of an ideal gas|macroscopic state parameters]][^1] of an [[Ideal gas]] in [[Thermal equilibrium]].
It comes in two forms, the molar form
$$
\begin{align*}
pV = \nu RT
\end{align*}
$$
where $\nu: \pu{mol}$ is number of moles
and $R = \pu{8.314 J.K-1}$ is the **universal gas constant**;
and the particular form
$$
\begin{align*}
pV = NkT
\end{align*}
$$
where $N$ is the number of particles
and $k = \pu{1.381E-23 J K-1}$ is **Boltzmann's constant**.
These two proportionality constants are related by Avogadro's number $N_A$
$$
\begin{align*}
N_A = \frac{N}{n} = \frac{R}{k} = \pu{6.02E23 mol-1}
\end{align*}
$$
[^1]: For this reason, it is sometimes called the **ideal gas equation of state**

## Simplified form
When the number of gas particles is constant the law can be simplified to
$$
\begin{align*}
\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}
\end{align*}
$$
### Derivation
Originally, the ideal gas law was discovered empirically.
However, a very similar equation can be derived microscopically from first principles of newtonian physics[^2022]:
$$
\begin{align*}
pV = \frac{2}{3} N \left( \frac{1}{2} m \overline{v^2} \right)
\end{align*}
$$
where $\sqrt{\overline{v^2}}$ is the [[Thermal speed]].
This demonstrates the [[Relationship between Kinetic Energy and Temperature]].

[^2022]: 2022\. [[Sources/@grassoHeatThermodynamicsLecture2022|Heat and thermodynamics lecture notes 2022]], pp. 23–26

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