Heat Required for Expandable Balloon Containing Helium Gas

To find the heat required to raise the temperature of the helium gas inside the balloon, we analyze the thermodynamic process taking place:

1. Identify the Gas Properties

• Helium is a monoatomic ideal gas.
• For a monoatomic ideal gas, the molar heat capacity at constant volume ($C_v$) and constant pressure ($C_p$) are given by: $C_v = \frac{3}{2}R$ $C_p = C_v + R = \frac{5}{2}R$

2. Identify the Nature of the Process

• The balloon is fully expandable and requires no energy for its elastic expansion.
• As a result, the pressure inside the balloon remains equal to the constant external atmospheric pressure ($P = \text{constant}$).
• Therefore, the heating process occurs at constant pressure (an isobaric process).

3. Calculate the Heat Required ($Q$)

The heat required at constant pressure is given by: $Q = n C_p \Delta T$

Given data:

• Number of moles, $n = 2\text{ mol}$
• Universal gas constant, $R = 8.31\text{ J/(mol}\cdot\text{K)}$
• Temperature change, $\Delta T = 35^\circ\text{C} - 30^\circ\text{C} = 5\text{ K}$

Substituting the value of $C_p = \frac{5}{2}R$: $Q = n \left(\frac{5}{2} R\right) \Delta T$

$Q = 2 \times \frac{5}{2} \times 8.31 \times 5$

$Q = 25 \times 8.31\text{ J} = 207.75\text{ J} \approx 208\text{ J}$

Thus, the approximate amount of heat required is $208\text{ J}$.

Correct Option: D