# 2.7: The Ideal Gas Constant and Boltzmann's Constant

Having developed the ideal gas equation and analyzed experimental results for a variety of gases, we will have found the value of R. It is useful to have R expressed using a number of different energy units. Frequently useful values are

\begin{aligned} R & = 8.314 \text{ Pa m}^{3} \text{ K}^{-1} \text{ mol}^{-1} \\ ~ & = 8.314 \text{ J K}^{-1} \text{ mol}^{-1} \\ ~ & = 0.08314 \text{ L bar K}^{-1} \text{ mol}^{-1} \\ ~ & = 1.987 \text{ cal K}^{-1} \text{ mol}^{-1} \\ ~ & = 0.08205 \text{ L atm K}^{-1} \text{ mol}^{-1} \end{aligned}

We also need the gas constant expressed per molecule rather than per mole. Since there is Avogadro’s number of molecules per mole, we can divide any of the values above by $$\overline{N}$$ to get $$R$$ on a per-molecule basis. Traditionally, however, this constant is given a different name; it is Boltzmann’s constant, usually given the symbol $$k$$.

$k={R}/{\overline{N}}=1.381\times {10}^{-23}\ \mathrm{J}\ {\mathrm{K}}^{-1}\ {\mathrm{molecule}}^{-1}$

This means that we can also write the ideal gas equation as $$PV=nRT=n\overline{N}kT$$. Because the number of molecules in the sample, $$N$$, is $$N=n\overline{N}$$, we have

$PV=NkT.$