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?

Question

For reaction $A \rightleftharpoons B$, $\Delta G^{\circ} = 105 - 35 \log T$. Find the transition temperature (in $^{\circ} C$) of the above reaction at 1 bar.

Answer 1

Step 1: Determine the expansion type.

The scenario involves gas expanding from vessel A to fill vessel B, characteristic of a free expansion. This is defined as expansion against zero external pressure. For the gas to expand without performing work, vessel B must be a vacuum. If vessel B contained another substance or had a non-zero pressure, the expanding gas would expend energy doing work.

The problem implies vessel B is evacuated, meaning its initial pressure is zero. This aligns with option (4).

Step 2: Calculate work done ($ dw $).

As it is a free expansion into a vacuum, the external pressure $ P_{ext} $ is zero. Work done on the system is given by:

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

With $ P_{ext} = 0 $, the work done is:

\[ dw = 0 \]

This negates statement (1), $ dw eq 0 $.

Step 3: Examine temperature and internal energy ($ dU $).

The observation of no temperature change from a thermometer in a water bath indicates the process is isothermal ($ dT = 0 $) for the gas, as it is in thermal equilibrium with the bath.

For an ideal gas, internal energy is solely a function of temperature. With $ dT = 0 $, $ dU = 0 $. Thus, statement (3), $ dU eq 0 $, is false for an ideal gas.
For a real gas, internal energy is dependent on both temperature and volume. Expansion alters molecular spacing and potential energy. Consequently, even in an isothermal process, $ dU eq 0 $ for a real gas.

Without knowing if the gas is ideal or real, statement (3) cannot be universally confirmed.

Step 4: Analyze heat exchange ($ dq $).

Applying the First Law of Thermodynamics, $ dU = dq + dw $, and knowing $ dw = 0 $:

For an ideal gas, $ dU = 0 $. Substituting, $ 0 = dq + 0 $, which yields $ dq = 0 $. Statement (2), $ dq eq 0 $, is false in this scenario.
For a real gas, $ dU eq 0 $. Therefore, $ dq = dU eq 0 $. Statement (2) holds true for a real gas.

The validity of statement (2) is conditional on the gas type.

Step 5: Identify the most accurate statement.

Evaluating all options:

(1) $ dw eq 0 $ is definitively false.
(2) $ dq eq 0 $ is true only for a real gas.
(3) $ dU eq 0 $ is true only for a real gas.
(4) The pressure in vessel B prior to opening the stopcock is zero. This is the fundamental condition defining the free expansion process described.

Given that a single correct answer is required, and statements (2) and (3) are contingent on the gas being real, statement (4) is the only universally accurate description of the physical setup.

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

Show Hint

The Correct Option is D

Solution and Explanation

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