If a gas is constrained by sufficient force, it cannot change volume
therefore \({dV}=0\)
So, \[{dU}={dq}+{dw}={dq}-{PdV}={dq}\]
Since, \({dq}={CdT}\)\[{dU}={CdT}\]
This gives us an important definition for the constant volume heat
capacity \[{C_{V}}\equiv \left ( \frac{\delta
U}{\delta T} \right )_{V}\]
Note:\(q=\int_{T_1}^{T_2}{nC_{V}dT}\)
Example 3.4
Consider \(1.00mol\) of an ideal gas
with \(C_V=\frac{3}{2}R\) that
undergoes a temperature change from \(125K\) to \(255K\) at a constant volume of \(10.0L\). Calculate \(\Delta U\), \(q\), and \(w\) for this change.