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

Explanation

To find the amount of heat required to raise the temperature of the helium gas in the balloon, we can use the formula for heat transfer in an ideal gas:

\[ Q = n C_v \Delta T \]

Where:
- \( Q \) is the heat added,
- \( n \) is the number of moles of gas,
- \( C_v \) is the molar heat capacity at constant volume,
- \( \Delta T \) is the change in temperature.

### Step 1: Identify the values
- Number of moles, \( n = 2 \) moles
- Initial temperature, \( T_1 = 30^\circ C = 303 \, K \)
- Final temperature, \( T_2 = 35^\circ C = 308 \, K \)
- Change in temperature, \( \Delta T = T_2 - T_1 = 308 \, K - 303 \, K = 5 \, K \)

### Step 2: Determine the molar heat capacity \( C_v \)
For a monatomic ideal gas like helium, the molar heat capacity at constant volume \( C_v \) is given by:

\[ C_v = \frac{3}{2} R \]

Where \( R = 8.31 \, J/(mol \cdot K) \).

Calculating \( C_v \):

\[ C_v = \frac{3}{2} \times 8.31 \, J/(mol \cdot K) = 12.465 \, J/(mol \cdot K) \]

### Step 3: Calculate the heat \( Q \)
Now we can substitute the values into the heat transfer equation:

\[ Q = n C_v \Delta T \]

Substituting the known values:

\[ Q = 2 \, \text{moles} \times 12.465 \, J/(mol \cdot K) \times 5 \, K \]

Calculating \( Q \):

\[ Q = 2 \times 12.465 \times 5 = 124.65 \, J \]

### Conclusion
The amount of heat required to raise the temperature of the helium gas in the balloon from \( 30^\circ C \) to \( 35^\circ C \) is approximately \( 124.65 \, J \).

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