When gas is given heat `DeltaQ`, a part of heat energy is utilized into work done W by gas and the remaining part is utilized to change in internal energy. An ideal diatomic gas is heated at constant pressure, the ratio of the internal energy change to heat energy supplied, is

Options

A. `(3)/(7)`
B. `(2)/(5)`
C. `(5)/(7)`
D. `(3)/(5)`

Answer

To solve the problem, we need to find the ratio of the change in internal energy (\(\Delta U\)) to the heat energy supplied (\(\Delta Q\)) when an ideal diatomic gas is heated at constant pressure.

### Step-by-Step Solution:

1. **Identify the properties of the ideal diatomic gas:**
- For a diatomic gas, the degrees of freedom \(f = 5\).
- The molar specific heat at constant volume \(C_V\) is given by:
\[ C_V = \frac{f}{2} R = \frac{5}{2} R \]
- The molar specific heat at constant pressure \(C_P\) is:
\[ C_P = C_V + R = \frac{5}{2} R + R = \frac{7}{2} R \]

2. **Calculate the change in internal energy (\(\Delta U\)):**
- The change in internal energy for an ideal gas is given by:
\[ \Delta U = n C_V \Delta T \]
- Substituting \(C_V\):
\[ \Delta U = n \left(\frac{5}{2} R\right) \Delta T \]

3. **Calculate the heat energy supplied (\(\Delta Q\)):**
- The heat supplied at constant pressure is given by:
\[ \Delta Q = n C_P \Delta T \]
- Substituting \(C_P\):
\[ \Delta Q = n \left(\frac{7}{2} R\right) \Delta T \]

4. **Find the ratio of internal energy change to heat energy supplied:**
- The ratio \(\frac{\Delta U}{\Delta Q}\) is:
\[ \frac{\Delta U}{\Delta Q} = \frac{n \left(\frac{5}{2} R\right) \Delta T}{n \left(\frac{7}{2} R\right) \Delta T} \]
- The \(n\), \(R\), and \(\Delta T\) terms cancel out:
\[ \frac{\Delta U}{\Delta Q} = \frac{5/2}{7/2} = \frac{5}{7} \]

5. **Final Result:**
- The ratio of the change in internal energy to the heat energy supplied is:
\[ \frac{\Delta U}{\Delta Q} = \frac{5}{7} \]