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. **Identify the Variables**:
- Let \( N \) be the number of moles of the gas.
- Let \( C_p \) be the molar heat capacity at constant pressure.
- Let \( C_v \) be the molar heat capacity at constant volume.
- Let \( \Delta T \) be the change in temperature.

2. **Determine the Heat Supplied (Q)**:
The heat supplied to the gas at constant pressure can be expressed using the formula:
\[ Q = N C_p \Delta T \]
For a diatomic gas, the value of \( C_p \) is:
\[ C_p = \frac{7R}{2} \]
Therefore, we can write:
\[ Q = N \left(\frac{7R}{2}\right) \Delta T \]

3. **Determine the Change in Internal Energy (\( \Delta U \))**:
The change in internal energy for the gas can be expressed as:
\[ \Delta U = N C_v \Delta T \]
For a diatomic gas, the value of \( C_v \) is:
\[ C_v = \frac{5R}{2} \]
Thus, we can write:
\[ \Delta U = N \left(\frac{5R}{2}\right) \Delta T \]

4. **Calculate the Fraction of Heat Supplied that Increases Internal Energy**:
To find the fraction of the heat supplied that goes into increasing the internal energy, we take the ratio of \( \Delta U \) to \( Q \):
\[ \text{Fraction} = \frac{\Delta U}{Q} = \frac{N \left(\frac{5R}{2}\right) \Delta T}{N \left(\frac{7R}{2}\right) \Delta T} \]
Simplifying this expression, we find:
\[ \text{Fraction} = \frac{\frac{5R}{2}}{\frac{7R}{2}} = \frac{5}{7} \]

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