The pressure $$(P)$$ and temperature $$(T)$$ relationship of an ideal gas obeys the equation $$PT^2 =$$ constant. The volume expansion coefficient of the gas will be:

Given that $$PT^2 = \text{constant}$$ for an ideal gas, we need to find the volume expansion coefficient.

We start by noting that

From the ideal gas law: $$PV = nRT \implies P = \frac{nRT}{V}$$

Substituting into $$PT^2 = C$$:

$$\frac{nRT}{V} \cdot T^2 = C \implies \frac{nRT^3}{V} = C$$

$$V = \frac{nRT^3}{C}$$

Next,

The volume expansion coefficient is defined as:

$$\beta = \frac{1}{V}\frac{dV}{dT}$$

$$\frac{dV}{dT} = \frac{3nRT^2}{C}$$

$$\beta = \frac{1}{V} \cdot \frac{3nRT^2}{C} = \frac{C}{nRT^3} \cdot \frac{3nRT^2}{C} = \frac{3}{T}$$

The volume expansion coefficient is $$\frac{3}{T}$$.

The correct answer is Option 4: $$\frac{3}{T}$$.