When an ideal diatomic gas is heated at constant pressure, the fraction of the heat energy supplied, which increases the internal energy of the gas, is

Options

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

Explanation

To solve the problem of finding the fraction of heat energy supplied that increases the internal energy of an ideal diatomic gas when heated at constant pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Variables**:
- Let \( \Delta U \) be the change in internal energy.
- Let \( \Delta Q \) be the heat energy supplied.
- \( n \) is the number of moles of the gas.
- \( C_v \) is the molar heat capacity at constant volume.
- \( C_p \) is the molar heat capacity at constant pressure.
- \( \gamma \) (gamma) is the ratio of \( C_p \) to \( C_v \), defined as \( \gamma = \frac{C_p}{C_v} \).

2. **Relate Internal Energy and Heat**:
- The change in internal energy for an ideal gas can be expressed as:
\[ \Delta U = n C_v \Delta T \]
- The heat supplied at constant pressure is given by:
\[ \Delta Q = n C_p \Delta T \]

3. **Find the Fraction**:
- The fraction \( f \) of the heat energy that increases the internal energy is given by:
\[ f = \frac{\Delta U}{\Delta Q} \]
- Substituting the expressions for \( \Delta U \) and \( \Delta Q \):
\[ f = \frac{n C_v \Delta T}{n C_p \Delta T} \]
- Here, \( n \) and \( \Delta T \) cancel out:
\[ f = \frac{C_v}{C_p} \]

4. **Use the Relation of \( C_v \) and \( C_p \)**:
- For a diatomic ideal gas, the value of \( \gamma \) is:
\[ \gamma = \frac{C_p}{C_v} = \frac{7}{5} \]
- Therefore, we can express \( C_v \) in terms of \( C_p \):
\[ C_v = \frac{C_p}{\gamma} = \frac{C_p}{\frac{7}{5}} = \frac{5}{7} C_p \]

5. **Substitute Back**:
- Now substituting \( C_v \) back into the fraction:
\[ f = \frac{C_v}{C_p} = \frac{\frac{5}{7} C_p}{C_p} = \frac{5}{7} \]

6. **Final Result**:
- Thus, the fraction of the heat energy supplied that increases the internal energy of the gas is:
\[ f = \frac{5}{7} \]

### Conclusion:
The fraction of the heat energy supplied that increases the internal energy of the gas when an ideal diatomic gas is heated at constant pressure is \( \frac{5}{7} \).