Interpret the figure via the option patterns

The given options indicate that processes $A$ and $B$ must correspond to standard thermodynamic processes of an ideal gas.

The useful standard relations are:

• Isothermal process: $T=\text{constant}, \qquad PV=\text{constant}$

• Adiabatic process: $PV^\gamma=\text{constant}$

Also, using the ideal gas law, $PV=nRT$

Derive the adiabatic relation involving $P$ and $T$

For an adiabatic process, $PV^\gamma = k$

From ideal gas law, $V=\frac{nRT}{P}$

Substitute into adiabatic equation: $P\left(\frac{nRT}{P}\right)^\gamma = k$

$P\cdot \frac{(nR)^\gamma T^\gamma}{P^\gamma}=k$

$\frac{T^\gamma}{P^{\gamma-1}}=\text{constant}$

Rearranging, $\frac{P^{\gamma-1}}{T^\gamma}=\text{constant}$

So both of the following are valid equivalent forms for an adiabatic process: $\frac{T^\gamma}{P^{\gamma-1}}=k$ and $\frac{P^{\gamma-1}}{T^\gamma}=k$

Identify which process is isothermal and which is adiabatic

From the options:

• Option B says:

  • for process $B$: $\frac{P^{\gamma-1}}{T^\gamma}=k$
  • for process $A$: $T=k$

  That means $B$ is adiabatic and $A$ is isothermal.

• Option D says:

  • for process $B$: $PV^{\gamma}=k$
  • for process $A$: $PV=k$

  Again, this means $B$ is adiabatic and $A$ is isothermal.

These two options are fully consistent with each other.

Check the incorrect options

• Option A: says $PV=k$ for both $A$ and $B$. This would make both processes isothermal, which is not correct if one of them is adiabatic.

• Option C: says adiabatic relation for $A$ and isothermal relation for $B$. This reverses the identification, so it is incorrect.

Final answer

Therefore, the correct statements are: $\boxed{B, D}$