Identification of Thermal Operations in a Cyclic Thermodynamic Process

To determine the sequence of thermal operations (heating or cooling) for the complete cycle of state transformations $K \rightarrow L \rightarrow M \rightarrow N \rightarrow K$, we analyze the temperature change $\Delta T$ for each process using the ideal gas equation of state: $PV = nRT \implies T = \frac{PV}{nR}$

Let the coordinates in the $P-V$ diagram be:

• State $K$: $(V_1, P_2)$
• State $L$: $(V_2, P_2)$
• State $M$: $(V_2, P_1)$
• State $N$: $(V_1, P_1)$

where $V_2 > V_1$ and $P_2 > P_1$.

1. Process $K \rightarrow L$ (Isobaric Expansion):

   • The pressure remains constant at $P_2$, while the volume increases from $V_1$ to $V_2$ ($V_2 > V_1$).
   • The temperature changes from $T_K = \frac{P_2 V_1}{nR}$ to $T_L = \frac{P_2 V_2}{nR}$.
   • Since $T_L > T_K$, the temperature increases. Thus, this step requires Heating.

2. Process $L \rightarrow M$ (Isochoric Depressurization):

   • The volume remains constant at $V_2$, while the pressure decreases from $P_2$ to $P_1$ ($P_2 > P_1$).
   • The temperature changes from $T_L = \frac{P_2 V_2}{nR}$ to $T_M = \frac{P_1 V_2}{nR}$.
   • Since $T_M < T_L$, the temperature decreases. Thus, this step requires Cooling.

3. Process $M \rightarrow N$ (Isobaric Compression):

   • The pressure remains constant at $P_1$, while the volume decreases from $V_2$ to $V_1$ ($V_2 > V_1$).
   • The temperature changes from $T_M = \frac{P_1 V_2}{nR}$ to $T_N = \frac{P_1 V_1}{nR}$.
   • Since $T_N < T_M$, the temperature decreases. Thus, this step requires Cooling.

4. Process $N \rightarrow K$ (Isochoric Pressurization):

   • The volume remains constant at $V_1$, while the pressure increases from $P_1$ to $P_2$ ($P_2 > P_1$).
   • The temperature changes from $T_N = \frac{P_1 V_1}{nR}$ to $T_K = \frac{P_2 V_1}{nR}$.
   • Since $T_K > T_N$, the temperature increases. Thus, this step requires Heating.

Therefore, the sequential operations enabling this transformation are Heating, cooling, cooling, heating.

Correct Option: C