In a process, the pressure of a gas is directly proportional to its temperature. Choose the correct options: A: The proc…

1. Condition:

   • If pressure \(P\propto T\), this implies: \(P=kT\ \text{(where }k\text{ is constant)}\mathrm{.}\)
   • This relationship is only valid if the volume \(V\) is constant, as per the ideal gas law:
   • \(PV=nRT\ \text{  }⟹\text{  }\ P\propto T\ \text{(if }V\text{ is constant)}\mathrm{.}\).
   • Thus, the process is isochoric (the statement \(A\) is correct).

2. Work Done:

   • In an isochoric process, the volume remains constant (\(\Delta V=0\)), so the work done: \(W=P\Delta V=0.\) Hence, \(B\) is also correct.

3. Internal Energy:

   • The internal energy \(U\) of an ideal gas depends on temperature: \(\Delta U=n{C}_{v}\Delta T\mathrm{.}\). Since temperature increases, internal energy also increases. Thus, \(C\) is correct.

4. Conclusion:

   • All statements \(A\), \(B\), and \(C\) are correct.

Answer: (3) \(A,B,\) and \(C\) are correct.