For the $$P - V$$ diagram given for an ideal gas

Out of the following which one correctly represents the $$T - P$$ diagram?

Solution & Explanation

1. Analyze the Given $$P - V$$ Diagram

From the provided graph, we are given the equation governing the thermodynamic state transition from point 1 to point 2:

$$P = \frac{\text{Constant}}{V} \implies P \cdot V = \text{Constant}$$

This relationship ($$P \cdot V = C$$) defines an isothermal process, which means the temperature ($$T$$) of the ideal gas remains perfectly constant throughout the entire expansion/compression path.

2. Apply the Ideal Gas Equation

The ideal gas equation state relation is given by:

$$P \cdot V = n \cdot R \cdot T$$

Looking closely at the graph from point 1 to point 2:

• The volume ($$V$$) increases along the horizontal axis.
• As volume increases under an isothermal condition, the pressure ($$P$$) must simultaneously decrease along the vertical axis ($$P_2 < P_1$$).

3. Construct the Corresponding $$T - P$$ Diagram

To translate this path onto a Temperature vs. Pressure ($$T - P$$) coordinate plane:

• Since temperature is constant ($$T = \text{Constant}$$), the state transition line must be perfectly horizontal, perpendicular to the vertical temperature axis.
• Since the pressure decreases moving from state 1 to state 2 ($$P_1 \rightarrow P_2$$), the path must track horizontally from a higher pressure value to a lower pressure value (moving from right to left).

Therefore, on a standard $$T - P$$ plot where $$T$$ is on the vertical axis and $$P$$ is on the horizontal axis, the process is represented by a flat horizontal line starting at a high pressure point 1 and moving leftward to a lower pressure point 2.

Concept Check: An isothermal line always maps out horizontally on any graph where temperature acts as the dependent vertical metric, where its tracking arrow direction is purely dictated by whether the system is undergoing compression or expansion.

Correct Option Key: Option A (Horizontal line where T remains constant as pressure drops)