# Boltzmann Constant

$PV=\frac{1}{3}Nm\bar{c}^2$ $PV=nRT$ $PV=NkT$

Where $P$ is pressure $V$ is volume $N$ is number of particles $n$ is number of moles $m$ is mass $\bar{c}$ is the root mean squared of velocity $k$ is Boltzmann's constant and $T$ is temperature

## Finding k

$PV=nRT=NkT$ $nR=Nk$ $N=nN_A$ where $N_A$ is Avogadro's constant. $\frac{nR}{N}=k$ $\frac{R}{N_A}=k$ $R$ and $N_A$ are both constants, Units: $\frac{Jmol^{-1}K^{-1}}{mol{-1}}=JK^{-1}$

$k=1.38\times10^{-23}$

## Derivation of $E_k=\frac{3}{2}kT$

$PV=\frac{1}{3}Nm\bar{c}^2$ $PV=nRT$ $nRT=\frac{1}{3}Nm\bar{c}^2$ $\frac{2}{3}N(\frac{1}{2}m\bar{c}^2)=nRT$ $\frac{1}{2}m\bar{c}^2=\frac{3}{2}\frac{nRT}{N}$ Or Average KE = $\frac{3}{2}kT$ $E_k=\frac{3}{2}kT;\dot{.\hspace{.095in}.}\hspace{.05in} E_k \propto T$