Question

When \(70\,\text{J}\) of heat is supplied to a rigid diatomic gas at a constant pressure \(P\), the change in the volume of the gas is \(\Delta V\). If the same amount of heat is supplied to a monoatomic gas at the same constant pressure \(P\), then the change in the volume of the monoatomic gas is

• A. \(1.4\,\Delta V\)
• B. \(0.7\,\Delta V\)
• C. \(2.1\,\Delta V\)
• D. \(2.8\,\Delta V\)

Hint:

At constant pressure, \[ \boxed{\Delta V=\frac{QR}{PC_P}.} \] For ideal gases, \[ \boxed{ C_P=\frac{5R}{2}\ \text{(monoatomic)}, \qquad C_P=\frac{7R}{2}\ \text{(diatomic)}. } \]

Answer 1

The Correct Option is A

Step 1: Use the relation between heat supplied and volume change. At constant pressure, \[ Q=nC_P\Delta T. \] Also, \[ P\Delta V=nR\Delta T. \] Eliminating \(\Delta T\), \[ \Delta V=\frac{QR}{PC_P}. \] Thus, \[ \boxed{\Delta V\propto\frac{1}{C_P}} \] for the same heat supplied and the same pressure.

Step 2: Write the specific heats. For a diatomic gas, \[ C_P=\frac{7R}{2}. \] For a monoatomic gas, \[ C_P=\frac{5R}{2}. \]

Step 3: Find the ratio of volume changes. Therefore, \[ \frac{\Delta V_{\text{mono}}}{\Delta V_{\text{dia}}} = \frac{\dfrac{1}{5R/2}}{\dfrac{1}{7R/2}} = \frac75 = 1.4. \] Hence, \[ \Delta V_{\text{mono}} = 1.4\,\Delta V. \] Thus, \[ \boxed{\Delta V_{\text{mono}}=1.4\,\Delta V.} \] Therefore, the correct option is \(\boxed{(A)}\).