Question

Propane gas ${\mathit{C}}_{\mathbf{3}}{\mathit{H}}_{\mathbf{8}}$ behaves like an ideal gas with $\mathit{\gamma }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{127}$. Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.

Short answer

$$\begin{gathered} C_P=73.81J/mol.K \\ {C}_V=65.5J/mol.K \end{gathered}$$

Step-by-step solution

The equations involved

The relations between the molar heat capacities at constant volume and constant pressure are ${\mathit{C}}_{\mathbf{P}}\mathbf{-}{\mathit{C}}_{\mathbf{V}}\mathbf{=}\mathit{R}$and$\frac{{\mathbf{C}}_{\mathbf{P}}}{{\mathbf{C}}_{\mathbf{V}}}\mathbf{=}\mathit{\gamma }$.

Use the two equations to find the heat capacities

$\gamma =1.127$Given that . We know that $C_P=C_V+R$.

$$\begin{gathered} \frac{C_V+R}{C_V}=\gamma \\ 1+\frac{R}{C_V}=\gamma \\ {C}_V=\frac{R}{\gamma -1} \\ {C}_V=\frac{8.314}{1.127-1} \\ {C}_V=65.5J/mol.K \end{gathered}$$

Now,$$\begin{gathered} C_P=\gamma C_V \\ {C}_P=1.127\times 65.5 \\ {C}_P=73.81J/mol.K \end{gathered}$$