Gaseous phase is a rarefied state of matter. In this state the interaction between molecules is often weak. If a gas is dilute, that is, if the density is low, we can completely ignore inter-molecular interactions without losing much of physics. The model gas with no interaction between molecules is called the ideal gas.
The thermodynamic state of a given amount of “ideal” gas, say containing \(N\) molecules, is completely determined by its temperature \(T\text{,}\) pressure \(p\text{,}\) and volume \(V\text{,}\) where temperature is in kelvin scale. These thermodynamic variables are not all independent, but are related by an equation of state.
\begin{equation}
p V = N k_B T,\tag{21.1}
\end{equation}
where \(k_B\) is called Boltzmann’s constant that has the following value.
\begin{equation}
k_B = 1.38 \times 10^{-23} \text{ J}\,\text{K}^{-1}.\tag{21.2}
\end{equation}
In Chemistry, one writes the equation of state in another form. In place of number of molecules \(N \text{,}\) we can defines number of moles \(n\) by diving \(N \) by Avogadro number \(N_A\text{.}\)
\begin{equation}
n = \dfrac{N}{N_A},\tag{21.3}
\end{equation}
where \(N_A\) is the number of moecules in one mole.
\begin{equation}
N_A = 6.022\times 10^{23},\tag{21.4}
\end{equation}
and in place of \(k_B\text{,}\) we define another constant, called the universal gas constant, \(R\text{,}\) by
\begin{equation}
R = N_A k_B = 8.31\text{ J}\,\text{K}^{-1}.\tag{21.5}
\end{equation}
Often the unit of \(R \) is given as \(\text{ J}\,\text{K}^{-1}\, \text{mol}^{-1}\) to reflect the fact that \(R \) is often multiplied by \(n \text{,}\) the number of moles. It is not necessary to do that as long as we rememeber the relation of \(R \) and \(n\) to the total number of molecules \(N\) and the Boltzmann constant \(k_B\text{.}\)
\begin{equation*}
n R = Nk_B.
\end{equation*}
In terms of
\(n \) and
\(R\text{,}\) the equation of state
(21.1) takes the following form.
\begin{equation*}
p V = N k_B T = \left(\dfrac{N}{N_A}\right) \left( N_Ak_B\right) T = n R T.
\end{equation*}
That is,
\begin{equation}
pV = n R T.\tag{21.6}
\end{equation}
