Choose the correct option for graphical representation of Boyle’s law, which shows a graph of pressure vs. volume of a gas at different temperatures :

The correct option is Option 3 (corresponding to the graph with image filename Screenshot 2024-03-22 091520-1711079683745.png).

Step-by-Step Derivation and Explanation:

1. Statement of Boyle's Law:
Boyle's law states that at a constant temperature ($T$), the pressure ($P$) of a fixed mass of gas (constant number of moles, $n$) is inversely proportional to its volume ($V$).
Mathematically, this relationship is expressed as:
$P\propto \frac{1}{V}$
Multiplying both sides by $V$, we get:
$PV=k$
where $k$ is a constant.

2. Nature of the Graph of P vs. V:
A plot of pressure ($P$) on the y-axis versus volume ($V$) on the x-axis for a relationship of the form $yx=\mathrm{constant}$ yields a curve known as a rectangular hyperbola. These constant-temperature curves on a P-V diagram are called isotherms.

3. Effect of Temperature on the Isotherms:
To determine how these curves behave at different temperatures, we look at the ideal gas equation:
$PV=nRT$
Here, $R$ is the universal gas constant and $n$ is the number of moles of gas. Comparing this with Boyle's law constant:
$k=nRT$
This shows that the constant $k$ is directly proportional to the temperature ($T$). As temperature increases, the product $PV$ increases.

4. Position of the Curves:
For a fixed volume ($V$), a higher temperature leads to a higher pressure ($P$). Thus, as temperature increases:
$T_3>T_2>T_1$
the isotherm corresponding to the highest temperature ($T$₃) shifts furthest away from the origin (highest on the plot), while the isotherm for the lowest temperature ($T$₁) lies closest to the origin.
The correct graph shows three rectangular hyperbolas representing $T_3$, $T_2$, and $T_1$, where the outermost curve is labeled $T_3$, the middle curve is $T_2$, and the innermost curve is $T_1$.