Here's a breakdown of how to solve this problem:

Understanding the Concepts

• Ideal Gas: An ideal gas is a theoretical gas that follows the ideal gas law perfectly. The ideal gas law is a relationship between pressure ($$P$$), volume ($$V$$), temperature ($$T$$), and the number of moles ($$n$$) of a gas:

$$PV = nRT$$, where $$R$$ is the ideal gas constant.

• Heat Capacity: Heat capacity is the amount of heat energy required to raise the temperature of a substance by 1 degree Celsius (or 1 Kelvin). For an ideal gas, we can use the specific heat capacity at constant volume ($$C_v$$) or at constant pressure ($$C_p$$).


• Constant Volume Process: In a constant volume process, the volume of the gas remains constant. The heat required to raise the temperature is given by:

$$Q = nC_v\Delta T$$

• Constant Pressure Process: In a constant pressure process, the pressure of the gas remains constant. The heat required to raise the temperature is given by:

$$Q = nC_p\Delta T$$

• Molar Heat Capacities of Helium: Helium is a monatomic gas. For monatomic gases, we have:


• $$C_v = \frac{3}{2}R$$


• $$C_p = \frac{5}{2}R$$

Solving the Problem

1. Identify the Process: Since the balloon is fully expandable, the pressure remains constant. This is a constant pressure process.


2. Calculate the Heat: Use the formula for heat at constant pressure:

$$Q = nC_p\Delta T$$

1. Substitute the Values:

• $$n = 2 \text{ moles}$$


• $$C_p = \frac{5}{2}R = \frac{5}{2}(8.31 \text{ J/mol.K})$$


• $$\Delta T = 35^\circ C - 30^\circ C = 5 K$$

1. Solve for Q:

$$Q = (2 \text{ moles}) \times \left(\frac{5}{2} \times 8.31 \text{ J/mol.K}\right) \times (5 K) $$

$$Q = 207.75 \text{ J}$$

Answer:

The amount of heat required to raise the temperature of the helium gas is approximately 208 J. Therefore, the correct answer is Option D.