Introduction:
In an H2 gas process, the relationship between pressure (P) and volume (V) is given as PV^2 = constant. We need to determine the ratio of work done by the gas to the change in its internal energy.

Understanding PV^2 = constant:
The equation PV^2 = constant represents a specific type of process called an isobaric process, where the pressure remains constant while the volume changes. This equation implies an inverse relationship between pressure and volume. As the volume increases, the pressure decreases, and vice versa, as long as the product of PV^2 remains constant.

Work done by the gas:
Work (W) done by a gas is given by the equation W = PΔV, where P is the pressure and ΔV is the change in volume. In an isobaric process, the pressure remains constant, so the work done can be simplified to W = P(Vf - Vi), where Vf is the final volume and Vi is the initial volume.

Change in internal energy:
The change in internal energy (ΔU) of a gas is given by the equation ΔU = Q - W, where Q is the heat added to the system and W is the work done by the gas. Since PV^2 = constant, this implies that the process is adiabatic, meaning no heat is exchanged with the surroundings (Q = 0). Therefore, the change in internal energy simplifies to ΔU = -W.

Ratio of work done to change in internal energy:
To find the ratio of work done by the gas to the change in its internal energy, we can divide the work done by the change in internal energy: W/ΔU = (P(Vf - Vi))/(-W).

Simplifying the equation, we get W/ΔU = -P(Vf - Vi)/W.

Conclusion:
In an H2 gas process where PV^2 = constant, the ratio of work done by the gas to the change in its internal energy is given by -P(Vf - Vi)/W. This ratio indicates the relationship between the work done by the gas and the change in its internal energy.