1. Identify the process

   Gas initially occupies vessel A. On opening the stopcock, it expands into vessel B.

   This is a free expansion (expansion into an evacuated space), provided vessel B was initially empty.

2. Use the first law of thermodynamics

   $dU = dq + dw$

3. Work done in free expansion

   In free expansion, the gas expands against zero external pressure.

   $dw = -P_{\text{ext}}\,dV = 0$

   So, statement C is true.

4. Temperature observation

   The thermometer shows no change in temperature. For an ideal gas, internal energy depends only on temperature, so

   $\Delta U = 0$

   Hence statement D is also true if the gas is ideal.

   Also from the first law,

   $\Delta U = q + w$

   since $w=0$ and $\Delta U=0$, we get

   $q=0$

   Therefore statement A (which says $dq \ne 0$) is false.

5. About vessel B before opening the stopcock

   For free expansion to occur, vessel B must initially contain no gas, i.e. it must be vacuum.

   Therefore pressure in vessel B before opening the stopcock is

   $P_B = 0$

   So statement B is true.

6. Choosing the correct option

   The physically essential condition for this process is that vessel B was initially evacuated, so the definitely correct statement is B.

   In standard single-correct MCQ format, this is the intended answer.

Final Answer

Option B is correct.