Subsection 10.5.4 Internal Energy of a Gas
A liquid freezes into solid gives heat energy, Zinc reacts with copper sulfate in a voltaic cell produces electric energy. A mixture of hydrogen and oxygen explodes to release mechanical energy. This production of energy is actually a conversion of stored energy of a system. The apparently stored energy of a system is called an internal energy. Real gas molecules posses both kinetic energy (due to motion of the molecules) and potential energy (due to intermolecular force of attraction) and hence the internal energy of the real gas is the sum of all of its internal kinetic and internal potential energy of the molecules. Ideal gas molecules are very far away from each other hence they do not have internal potential energy. The internal energy of an ideal is therefore solely due to the internal kinetic energy of all of its gas molecules and it is denoted by \(U.\) Therefore the internal energy of one molecule of a gas,
\begin{equation*}
\frac{1}{2}mv_{rms}^{2}= \frac{3}{2}k_{B}T
\end{equation*}
Hence, for N molecules of a gas, the internal energy is given by
\begin{equation*}
U= \frac{3}{2}Nk_{B}T
\end{equation*}
and n moles of a gas,
\begin{equation*}
U= \frac{3}{2}nRT
\end{equation*}
\(N=nN_{A}\) and \(N_{A}k_{B}=R\)
\begin{equation}
\therefore\quad U= \frac{3}{2}Nk_{B}T = \frac{3}{2}nRT = \frac{3}{2}pV \tag{10.5.11}
\end{equation}
It must be noted that the internal energy of an ideal gas is directly proportional to the absolute temperature, i.e., \(U \propto T.\)
