Question

Two vessels A and B are connected via stopcock. Vessel A is filled with a gas at a certain pressure. The entire assembly is immersed in water and allowed to come to thermal equilibrium with water. After opening the stopcock the gas from vessel A expands into vessel B and no change in temperature is observed in the thermometer. Which of the following statement is true?

• A. \( dw = 0 \)
• B. \( dq = 0 \)
• C. \( du = 0 \)
• D. The pressure in the vessel B before opening the stopcock is zero

Hint:

In a free expansion, there is no work done, no heat transfer, and no change in internal energy, as the process is adiabatic and occurs without a temperature change.

Answer 1

The Correct Option is D

Step 1: Analyze the nature of the expansion process.

The problem describes a gas from vessel A expanding to fill vessel B. This is a classic setup for a free expansion. A free expansion is defined as an expansion against zero external pressure. For the gas in A to expand without doing any work, vessel B must be empty, meaning it must be a vacuum. If vessel B contained another gas or was at a non-zero pressure, the expanding gas would have to do work to push against it.

Therefore, the condition that vessel B is evacuated is implicit in the description of the process. This means the pressure in vessel B before opening the stopcock is zero. This statement corresponds directly to option (4).

Step 2: Evaluate the work done (\( dw \)).

Since the process is a free expansion into a vacuum, the external pressure \( P_{ext} \) is zero. The work done on the system is calculated as:

\[ dw = -P_{ext}dV \]

Since \( P_{ext} = 0 \), the work done is:

\[ dw = 0 \]

This shows that statement (1), \( dw \neq 0 \), is false.

Step 3: Analyze the temperature and internal energy (\( dU \)).

The problem states that "no change in temperature is observed in the thermometer" placed in the water bath. Since the gas is in thermal equilibrium with the water, this means the overall process for the gas is isothermal (\( dT = 0 \)).

• If we assume the gas is ideal, its internal energy depends only on temperature. Since \( dT = 0 \), the change in internal energy is also zero (\( dU = 0 \)). This would make statement (3), \( dU \neq 0 \), false.
• If the gas is real, its internal energy also depends on volume. During expansion, the average distance between molecules changes, which alters their potential energy. Thus, even for an isothermal process, \( dU \neq 0 \) for a real gas.

Since the problem does not specify if the gas is ideal or real, we cannot definitively conclude that statement (3) is true in all cases.

Step 4: Analyze the heat exchange (\( dq \)).

Using the First Law of Thermodynamics, \( dU = dq + dw \). We already know \( dw = 0 \).

• For an ideal gas, we found \( dU = 0 \). Therefore, \( 0 = dq + 0 \), which implies \( dq = 0 \). In this case, statement (2), \( dq \neq 0 \), would be false.
• For a real gas, we found \( dU \neq 0 \). Therefore, \( dq = dU \neq 0 \). In this case, statement (2) would be true.

Similar to \( dU \), the truth of statement (2) depends on whether the gas is ideal or real.

Step 5: Conclude the most accurate statement.

We have analyzed all four options:

• (1) \( dw \neq 0 \) is definitively false.
• (2) \( dq \neq 0 \) is true only for a real gas.
• (3) \( dU \neq 0 \) is true only for a real gas.
• (4) The pressure in vessel B before opening the stopcock is zero. This is the defining condition for the process to be a free expansion, which is the physical situation described.

Since a single-choice question must have one unambiguously correct answer, and the truth of statements (2) and (3) is conditional on the nature of the gas (ideal vs. real), the most universally true statement that characterizes the setup is (4).

The statement that the pressure in the vessel B before opening the stopcock is zero is the correct answer.

Answer 2

Since there is no change in temperature after the stopcock is opened, it is a free expansion, implying: \[ w = 0, \quad q = 0, \quad \Delta U = 0 \] Thus, the correct answer is option (4). The pressure in vessel B before opening the stopcock is zero.