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 heated at constant pressure, we can follow these steps:

### Step 1: Understand the relationship between heat, internal energy, and work done.
When an ideal gas is heated at constant pressure, the heat added to the system (ΔQ) is used to increase both the internal energy (ΔU) and to do work (W) on the surroundings. The first law of thermodynamics states:
\[ \Delta Q = \Delta U + W \]

### Step 2: Calculate the work done at constant pressure.
The work done by the gas during expansion at constant pressure can be expressed as:
\[ W = P \Delta V \]
For an ideal gas, we can also express this in terms of the number of moles (N) and the change in temperature (ΔT):
\[ W = N R \Delta T \]
where R is the universal gas constant.

### Step 3: Calculate the change in internal energy.
For an ideal diatomic gas, the change in internal energy (ΔU) is given by:
\[ \Delta U = N C_v \Delta T \]
where \( C_v \) is the molar heat capacity at constant volume. For a diatomic gas, \( C_v = \frac{5}{2} R \).

### Step 4: Calculate the heat added at constant pressure.
The heat added at constant pressure is given by:
\[ \Delta Q = N C_p \Delta T \]
where \( C_p \) is the molar heat capacity at constant pressure. For a diatomic gas, \( C_p = \frac{7}{2} R \).

### Step 5: Find the fraction of heat energy that increases internal energy.
Now we need to find the fraction of the heat energy supplied that increases the internal energy:
\[ \text{Fraction} = \frac{\Delta U}{\Delta Q} \]
Substituting the expressions for ΔU and ΔQ:
\[ \text{Fraction} = \frac{N C_v \Delta T}{N C_p \Delta T} \]
The \( N \) and \( \Delta T \) terms cancel out:
\[ \text{Fraction} = \frac{C_v}{C_p} \]

### Step 6: Substitute the values of \( C_v \) and \( C_p \).
Using the values for a diatomic gas:
- \( C_v = \frac{5}{2} R \)
- \( C_p = \frac{7}{2} R \)

Thus,
\[ \text{Fraction} = \frac{\frac{5}{2} R}{\frac{7}{2} R} = \frac{5}{7} \]

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