Question

The molar-specific heat at a constant pressure of an ideal gas is$(7/2)R$. The ratio of specific heat at constant pressure to that at constant volume is:

• A. $\frac{5}{7}$
• B. $\frac{7}{5}$
• C. $\frac{8}{7}$
• D. $\frac{9}{7}$

Answer 1

The correct option is B

$\frac{7}{5}$

The explanation for the correct option

Step 1: Molar-specific heat at constant pressure and volume

• Molar-specific heat at constant pressure is denoted as ${\mathrm{C}}_{\mathrm{p}}$and it is given as $\frac{7}{2}R$.
• Molar-specific heat at constant volume is denoted as ${\mathrm{C}}_v$ and we have to find it.
• The relation connecting ${\mathrm{C}}_{\mathrm{p}}$and ${\mathrm{C}}_v$ is, ${\mathrm{C}}_{\mathrm{p}}-{\mathrm{C}}_{\mathrm{v}}=\mathrm{R}$
• By substituting the given value we get

$$\begin{gathered} {\mathrm{C}}_{\mathrm{p}}-{\mathrm{C}}_{\mathrm{v}}=\mathrm{R} \\ \frac{7\mathrm{R}}{2}-{\mathrm{C}}_{\mathrm{v}}=\mathrm{R} \\ {\mathrm{C}}_{\mathrm{v}}=\frac{7\mathrm{R}}{2}-\mathrm{R} \\ =\frac{5\mathrm{R}}{2} \end{gathered}$$

• Thus ${\mathrm{C}}_v$ is$\frac{7}{2}R$

Step 2: Ratio of specific heat at constant pressure to constant volume.

• Ratio of specific heat at constant pressure to constant volume is $\frac{C_p}{C_v}$
• Substituting the values we get,

$$\begin{gathered} \frac{C_p}{C_v}=\frac{{\displaystyle \frac{7R}{2}}}{{\displaystyle \frac{5R}{2}}} \\ =\frac{7}{5} \end{gathered}$$

The explanation for the incorrect options

Since, the correct ratio is $\frac{\mathbf{7}}{\mathbf{5}}$. Hence, options (a), (c) and (d) are incorrect.

Therefore, the correct answer is option (B) $\frac{\mathbf{7}}{\mathbf{5}}$