One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $(V-T)$ diagram. The correct statement(s) is/are: [$R$ is the gas constant]

Step 1: Identify the properties of the gas.

For $1$ mole of a monatomic ideal gas, the molar heat capacities at constant volume ($C_V$) and constant pressure ($C_P$) are:

$$C_V = \frac{3}{2}R$$ $$C_P = \frac{5}{2}R$$

Step 2: Analyze each process in the cycle.

Identify the nature of each transition in the $V-T$ diagram:

• For $1 \to 2$ and $3 \to 4$, the path is a straight line passing through the origin, meaning $V \propto T$. By the ideal gas law ($PV = nRT$), constant $\frac{V}{T}$ implies constant pressure. Thus, these processes are isobaric.
• For $2 \to 3$ and $4 \to 1$, the volume remains constant. Thus, these processes are isochoric.

Since there are no adiabatic processes in the cycle, statement (2) is false.

Step 3: Calculate the work done in each process.

Work done in the isochoric processes is zero:

$$W_{23} = W_{41} = 0$$

Work done in the isobaric processes is given by $W = nR\Delta T$:

$$W_{12} = R(2T_0 - T_0) = RT_0$$ $$W_{34} = R\left(\frac{T_0}{2} - T_0\right) = -\frac{1}{2}RT_0$$

The net work done in the cycle is:

$$W_{\text{net}} = W_{12} + W_{23} + W_{34} + W_{41}$$ $$W_{\text{net}} = RT_0 + 0 - \frac{1}{2}RT_0 + 0 = \frac{1}{2}RT_0$$

Thus, statement (1) is true.

Step 4: Calculate the heat transfer in each process.

For the isobaric processes, the heat transfer is $Q = nC_P\Delta T$:

$$Q_{1\to2} = \frac{5}{2}R(2T_0 - T_0) = \frac{5}{2}RT_0$$ $$Q_{3\to4} = \frac{5}{2}R\left(\frac{T_0}{2} - T_0\right) = -\frac{5}{4}RT_0$$

For the isochoric processes, the heat transfer is $Q = nC_V\Delta T$:

$$Q_{2\to3} = \frac{3}{2}R(T_0 - 2T_0) = -\frac{3}{2}RT_0$$ $$Q_{4\to1} = \frac{3}{2}R\left(T_0 - \frac{T_0}{2}\right) = \frac{3}{4}RT_0$$

Step 5: Evaluate the ratios of heat transfers.

Calculate the ratio for statement (3):

$$\frac{|Q_{1\to2}|}{|Q_{2\to3}|} = \frac{\frac{5}{2}RT_0}{\frac{3}{2}RT_0} = \frac{5}{3}$$

Thus, statement (3) is true.

Calculate the ratio for statement (4):

$$\frac{|Q_{1\to2}|}{|Q_{3\to4}|} = \frac{\frac{5}{2}RT_0}{\frac{5}{4}RT_0} = 2$$

Thus, statement (4) is false.

Answer: The correct statements are (1) and (3).