When an ideal gas is heated at constant pressure, the fraction of the heat energy supplied whicn 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 to an ideal gas at constant pressure that increases its internal energy, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concepts**:
- For an ideal gas, the internal energy change (\( \Delta U \)) is related to the heat added at constant volume (\( Q_1 \)) and at constant pressure (\( Q_2 \)).
- The specific heat at constant volume is denoted as \( C_v \) and at constant pressure as \( C_p \).

2. **Heat at Constant Volume**:
- The heat absorbed by the gas at constant volume is given by the formula:
\[ Q_1 = C_v \Delta T \]
- Here, \( \Delta T \) is the change in temperature.

3. **Heat at Constant Pressure**:
- The heat absorbed by the gas at constant pressure is given by the formula:
\[ Q_2 = C_p \Delta T \]

4. **Finding the Ratio**:
- To find the fraction of heat energy that increases the internal energy, we need to find the ratio of \( Q_1 \) to \( Q_2 \):
\[ \frac{Q_1}{Q_2} = \frac{C_v \Delta T}{C_p \Delta T} \]
- The \( \Delta T \) terms cancel out:
\[ \frac{Q_1}{Q_2} = \frac{C_v}{C_p} \]

5. **Using the Relation Between \( C_p \) and \( C_v \)**:
- For an ideal gas, the ratio of specific heats is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
- Therefore, we can express \( \frac{C_v}{C_p} \) as:
\[ \frac{C_v}{C_p} = \frac{1}{\gamma} \]

6. **Substituting the Value of \( \gamma \)**:
- For diatomic gases, \( \gamma \) is typically \( \frac{7}{5} \). Thus:
\[ \frac{C_v}{C_p} = \frac{1}{\frac{7}{5}} = \frac{5}{7} \]

7. **Conclusion**:
- The fraction of the heat energy supplied that increases the internal energy of the gas is:
\[ \frac{Q_1}{Q_2} = \frac{5}{7} \]

### Final Answer:
The fraction of the heat energy supplied which increases the internal energy of the gas is \( \frac{5}{7} \).