Animated Solution for Physics - Thermodynamics: Two moles of ideal helium gas are in a rubber balloon at $30^{\circ }\text{C}$. 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 $35^{\circ }\text{C}$. The amount of heat required in raising the temperature is nearly (take $R=8.31\text{ J/mol-K}$)

This problem is a beautiful example of how physical constraints dictate the thermodynamic process a system undergoes. Let's break down the language of the question to uncover the hidden physics.

The problem states that the helium gas is enclosed in a rubber balloon that is "fully expandable and can be assumed to require no energy in its expansion."

What does this phrase actually mean? When a balloon expands, it usually has to do work against two things: the atmospheric pressure outside, and the elastic tension of the rubber itself. By stating that the expansion requires no energy (meaning no extra energy to stretch the rubber), the problem implies that the pressure inside the balloon is solely determined by, and always equal to, the constant atmospheric pressure outside.

Because the pressure remains constant throughout the heating process, we are dealing with an isobaric process.

For an isobaric process, the heat $Q$ required to raise the temperature of $n$ moles of a gas by $\Delta T$ is given by:

where $C_p$ is the molar heat capacity of the gas at constant pressure.

We are given that the gas is Helium (He). Helium is a noble gas, which means it exists as single atoms; it is a monoatomic gas. For any ideal monoatomic gas, the molar heat capacity at constant pressure is:

Now, let's gather our known variables: Number of moles, $n=2$ Initial temperature, $T_1=30^{\circ }\text{C}$ * Final temperature, $T_2=35^{\circ }\text{C}$

The change in temperature is $\Delta T=35^{\circ }\text{C}-30^{\circ }\text{C}=5^{\circ }\text{C}$. Remember that a change of $5^{\circ }\text{C}$ is exactly equivalent to a change of $5\text{ K}$.

Substituting these values into our master equation:

The $2$ in the numerator and denominator cancel out perfectly:

Finally, we substitute the given value of the universal gas constant, $R=8.31\text{ J/mol-K}$:

Rounding this to the nearest integer, we get $208\text{ J}$. This matches option (d) perfectly.