A cylinder contains 0.1 cubic meters of gas at atmospheric pressure. This gas is allowed to expand until its pressure drops to 7800 Pa. What is the gas volume after this expansion has taken place?

Is this answer right: 7800/10130 = 0.770

The final volume will depend uipon whether the expansion is isothermal (no temperature change) or adiabatic (no heat transfer with the surroundings).

In either case your answer is incorrect. You have just stated a pressure ratio, P2/P1.