Which of the following graphs represent the behaviour of an ideal gas? Symbols have their usual meaning.

We need to identify which graph correctly represents the behavior of an ideal gas based on the state variables: Pressure ($$P$$), Volume ($$V$$), and Absolute Temperature ($$T$$).

1. Establish the Governing Ideal Gas Equation

The behavior of an ideal gas containing $$n$$ moles is universally governed by the ideal gas law equation:

$$PV = nRT$$

Where $$R$$ is the universal gas constant. For a fixed amount of gas ($$n$$ is constant), both $$n$$ and $$R$$ are constants. Thus, we can write:

$$PV \propto T \implies PV = k T$$

Where $$k = nR$$ acts as a positive proportionality constant.

2. Analyze the Graph Variables and Equation of a Line

Let's map our ideal gas relation to the standard mathematical equation of a straight line passing through the origin ($$y = mx$$):

• y-axis variable ($$y$$): $$PV$$
• x-axis variable ($$x$$): $$T$$
• Slope ($$m$$): $$k = nR$$ (which is a constant, positive value)

Because the equation reduces to the form $$y = mx$$ with a positive slope ($$m > 0$$) and a zero y-intercept ($$c = 0$$), the graph of $$PV$$ versus $$T$$ must be a straight line pointing upward that extrapolates directly through the origin.

3. Evaluate the Options Shown in the Graph

• Graph A: Shows a straight line with a negative slope, implying $$PV$$ decreases as $$T$$ increases. This is incorrect.
• Graph B: Shows a straight line with a positive slope starting near the origin, indicating that $$PV$$ is directly proportional to $$T$$. This perfectly matches $$PV = k T$$.

Final Answer: Option B