An ideal gas is expanding such that PT² = constant. The coefficient of volume expansion of the gas is 

We are given that the ideal gas is expanding in such a way that PT² = constant, where P is the pressure, T is the temperature, and the product of pressure and square of temperature remains constant.

Step 1: Express the Relationship between Pressure and Temperature

From the given relation PT² = constant, we can express the pressure P in terms of temperature as:

P = K/T²

Where K is a constant.

Step 2: Use the Ideal Gas Law

The Ideal Gas Law states:

PV = nRT

Here, P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature. Substituting the expression for pressure P = K/T² into the Ideal Gas Law:

(K/T²)V = nRT

Step 3: Simplify the Expression

We can now solve for the volume V:

V = (nRTK) / T²

As you can see, the volume V is inversely proportional to the square of the temperature T:

V ∝ 1/T

Step 4: Relating Volume Expansion to Temperature

The coefficient of volume expansion γ is defined as:

γ = (1/V) * (dV/dT)

Taking the derivative of the volume expression with respect to temperature:

dV/dT = - (2nRK) / T³

Thus, the coefficient of volume expansion is:

γ = (1/V) * (dV/dT) = 2 / T

Conclusion

So, the coefficient of volume expansion γ for an ideal gas expanding such that PT² = constant is given by:

γ = 2 / T

This result shows that the coefficient of volume expansion decreases as the temperature increases for an ideal gas under the given condition that PT² = constant.