Q.10 For an ideal gas at room temperature, choose the CORRECT representation(s) of
Boyle’s Law.
(P = Pressure, V = Volume, T = Temperature)
(A) only (i) (B) both (ii) and (iii) (C) only (iii) (D) both (iii) and (iv)

Correct representations of Boyle’s law in the given figure are options (iii) and (iv), so the correct answer is (D) both (iii) and (iv).​

Introduction

Boyle’s law graphical representation is a frequent exam question, especially in NEET, JEE and CSIR-type tests, because it tests both conceptual understanding of gas laws and the ability to interpret graphs. For an ideal gas at constant temperature (isothermal condition), pressure and volume are inversely related and their product $PV$ remains constant, which directly determines which graphs are correct.​

Core concept of Boyle’s law

• For a fixed mass of an ideal gas at constant temperature, pressure is inversely proportional to volume: $P\propto 1/V$.​

• Mathematically, this is written as $PV=k$, where $k$ is a constant for a given temperature and amount of gas.​

• Graphically:

  • A plot of $P$ versus $V$ is a rectangular hyperbola.​

  • A plot of $P$ versus $1/V$ is a straight line through the origin because $P=k(1/V)$.​

  • A plot of $PV$ versus any variable (e.g., $P$ or $T$) is a horizontal line at constant temperature, since $PV$ is constant.​

These facts allow evaluation of each option in the question.

Detailed analysis of each option

Option (i): P vs T (straight line)

In figure (i), pressure $P$ is plotted on the y‑axis and temperature $T$ on the x‑axis, with a straight line passing through the origin, indicating $P\propto T$. This relationship corresponds to Gay‑Lussac’s law (pressure–temperature law) at constant volume, not Boyle’s law. Since Boyle’s law keeps temperature constant and relates pressure and volume, a $P$ vs $T$ graph is not an appropriate representation, so option (i) is incorrect.​

Option (ii): P vs V (straight line)

In figure (ii), pressure $P$ is plotted against volume $V$ as a straight line through the origin, suggesting $P\propto V$. Boyle’s law states that $P$ is inversely proportional to $V$ at constant temperature, given by $P\propto 1/V$. Therefore, a correct $P$ vs $V$ plot should be a rectangular hyperbola, not a straight line, making option (ii) an incorrect representation.​

Option (iii): P vs 1/V (straight line)

In figure (iii), pressure $P$ is plotted against $1/V$ as a straight line through the origin. Using Boyle’s law, $P=k(1/V)$, which is a linear equation in the form $y=mx$ with $y=P$, $x=1/V$, and slope $k$. Hence, the correct graphical representation of Boyle’s law is a straight line when $P$ is plotted against $1/V$, so option (iii) is correct.​

Option (iv): PV vs T (horizontal line at room temperature)

In figure (iv), the product $PV$ is plotted against temperature $T$ and shown as a horizontal line. For Boyle’s law, at a fixed temperature and fixed amount of gas, $PV=k$ remains constant. Therefore, if the system is considered at a particular temperature (here, room temperature), a graph of $PV$ vs any variable while temperature is held constant will be a horizontal line, indicating constant $PV$, which is consistent with Boyle’s law, so option (iv) is also correct.​

Summary table: validity of each graph

Thus, the correct answer is (D) both (iii) and (iv).