Which one is the correct option for the two different thermodynamic processes?

We need to identify the correct graphical representations for two different thermodynamic processes—isothermal and adiabatic—across different state variables ($$P-V$$, $$V-T$$, and $$P-T$$ diagrams).

1. Core Thermodynamic Principles

For an ideal gas undergoing isothermal and adiabatic expansions:

• Isothermal Process: Temperature remains constant ($$T = \text{constant}$$).
  • Governing equation: $$PV = \text{constant}$$
• Adiabatic Process: No heat exchange occurs ($$Q = 0$$).
  • Governing equation: $$PV^\gamma = \text{constant}$$ (where $$\gamma > 1$$ is the adiabatic index)

2. Analyzing the Slopes on a $$P-V$$ Diagram (Graph a)

Differentiating both governing equations with respect to volume ($$V$$) gives the slopes of the curves on a pressure-volume graph:

• Isothermal slope: $$\frac{dP}{dV} = -\frac{P}{V}$$
• Adiabatic slope: $$\frac{dP}{dV} = -\gamma \frac{P}{V}$$

Since $$\gamma > 1$$, the adiabatic curve is steeper than the isothermal curve during expansion. In graph (a), the label pointing to the steeper curve is incorrectly labeled as isothermal, making graph (a) incorrect.

3. Analyzing the $$V-T$$ Diagram (Graph c)

• Isothermal: Since $$T$$ is constant, the curve must be a straight vertical line parallel to the volume axis.
• Adiabatic: From $$PV^\gamma = \text{constant}$$ and the ideal gas law ($$P = \frac{nRT}{V}$$), we get:

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

  As volume ($$V$$) increases during an expansion, temperature ($$T$$) must decrease. This produces a downward-sloping curve. Graph (c) correctly displays both of these behaviors.

4. Analyzing the $$P-T$$ Diagram (Graph d)

• Isothermal: Since $$T$$ is constant, the curve must be a straight vertical line parallel to the pressure axis.
• Adiabatic: From $$PV^\gamma = \text{constant}$$ and the ideal gas law ($$V = \frac{nRT}{P}$$), we get:

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

  Since $$\gamma > 1$$, as temperature ($$T$$) decreases during an expansion, pressure ($$P$$) must also decrease. This forms a curved path dropping toward the origin. Graph (d) correctly displays both of these behaviors.

Conclusion

Graphs (c) and (d) accurately represent the characteristics of the isothermal and adiabatic processes under their respective thermodynamic variables.

Correct Answer: (c) and (d) 
HENCE ; Option B would be the right answer