Two moles of ideal helium gas are in a rubber balloon at `30^@C.` The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to `35^@C.` The amount of heat required in raising the temperature is nearly (take R
`=8.31 J//mol.K`)

Options

A. 62 J
B. 104 J
C. 124 J
D. 208 J

Explanation

To find the amount of heat required to raise the temperature of two moles of ideal helium gas from \(30^\circ C\) to \(35^\circ C\), we can use the formula for heat transfer in an isobaric process:

### Step-by-Step Solution:

1. **Identify the Process**:
The gas is in a rubber balloon, which can expand freely. Therefore, the process is isobaric (constant pressure).

2. **Use the Heat Transfer Formula**:
The amount of heat \(Q\) required in an isobaric process is given by:
\[ Q = n C_p \Delta T \]
where:
- \(n\) = number of moles of gas
- \(C_p\) = molar heat capacity at constant pressure
- \(\Delta T\) = change in temperature

3. **Calculate the Change in Temperature**:
The initial temperature \(T_i = 30^\circ C\) and the final temperature \(T_f = 35^\circ C\).
\[ \Delta T = T_f - T_i = 35^\circ C - 30^\circ C = 5^\circ C \]

4. **Determine \(C_p\) for Helium**:
Helium is a monatomic ideal gas. For a monatomic ideal gas, the molar heat capacity at constant pressure \(C_p\) is given by:
\[ C_p = \frac{5}{2} R \]
where \(R = 8.31 \, J/(mol \cdot K)\).

5. **Substitute Values**:
Now substituting \(C_p\) into the heat transfer formula:
\[ C_p = \frac{5}{2} \times 8.31 \, J/(mol \cdot K) = 20.775 \, J/(mol \cdot K) \]

6. **Calculate \(Q\)**:
Now substituting \(n = 2\) moles, \(C_p\), and \(\Delta T\) into the heat transfer formula:
\[ Q = 2 \times 20.775 \, J/(mol \cdot K) \times 5 \, K \]
\[ Q = 2 \times 20.775 \times 5 = 207.75 \, J \]

7. **Final Result**:
The amount of heat required to raise the temperature is approximately:
\[ Q \approx 208 \, J \]