Question

A balloon containing an ideal gas is initially kept in an evacuated and insulated room. If the balloon ruptures and the gas fills the entire room, what is the correct statement at the end of the process?

• A. The internal energy of the gas decreases from its initial value, but the enthalpy remains constant
• B. The internal energy of the gas increases from its initial value, but the enthalpy remains constant
• C. Internal energy and enthalpy of the gas remain constant
• D. Internal energy and enthalpy of the gas increase

Hint:

Free expansion of an ideal gas is simultaneously isothermal ($T = \text{constant}$), isenuclic/isothermal-internal-energy ($U = \text{constant}$), and isenthalpic ($H = \text{constant}$). However, it is highly irreversible, so entropy increases ($\Delta S \gt 0$).

Answer 1

The Correct Option is C

Concept: The phenomenon of a gas expanding into an evacuated space is known as a

Free Expansion or an unresisted expansion. Let us apply the First Law of Thermodynamics to this system: \[ Q = \Delta U + W \] We define our system boundary to encompass the entire insulated room. Let's evaluate the conditions:

• Insulated Room: No thermal interactions occur across the system boundaries, meaning heat transfer is zero ($Q = 0$).

• Evacuated Space: The gas expands against a vacuum, meaning there is zero resisting pressure ($P_{\text{external}} = 0$). Since work is defined as $W = \int P_{\text{external}} \, dV$, the work performed by the system is zero ($W = 0$).
Substituting these boundary conditions into the First Law yields: \[ 0 = \Delta U + 0 \quad \Rightarrow \quad \Delta U = 0 \quad \Rightarrow \quad U_1 = U_2 \] Thus, the internal energy remains constant during a free expansion.

Step 1: Connect internal energy stability to ideal gas properties.
For an ideal gas, Joule's law states that internal energy is strictly a function of temperature alone, $U = f(T)$. Because $\Delta U = 0$ for this process, the temperature of the ideal gas must also remain unchanged: \[ T_1 = T_2 \]

Step 2: Evaluate the change in enthalpy (\(H\)).
Enthalpy is defined by the property relation: \[ H = U + PV \] Using the ideal gas equation of state, $PV = mRT$, we can substitute this directly into the enthalpy definition: \[ H = U + mRT \] Since both internal energy ($U$) and temperature ($T$) are constant for this ideal gas process, the product $mRT$ remains constant. Consequently, the enthalpy $H$ must also remain unchanged throughout the free expansion: \[ \Delta H = 0 \quad \Rightarrow \quad H_1 = H_2 \]

Step 3: Match results to the options.
Both the internal energy and the enthalpy of the ideal gas remain perfectly constant from the beginning to the end of this process. This matches Option (C).