A

Step 1: Since the cylinder is divided into two parts by a frictionless piston and is in equilibrium, the pressures on both sides of the piston must be equal at equilibrium: P1 = P2.
Step 2: Let V1 be the volume of the gas in the left part and V2 be the volume in the right part after the piston has moved. Since the total volume V0 is constant, we have:
$$ V1 + V2 = V0 $$
Step 3: Using the ideal gas law, we express the pressures in terms of volumes:
$$ P1 = \frac{n_1RT}{V1} $$
$$ P2 = \frac{n_2RT}{V2} $$
where n1 and n2 are the number of moles of gas in the left and right parts, respectively, R is the universal gas constant, and T is the temperature (assumed to be constant).
Step 4: Since P1 = P2 at equilibrium, we set the equations equal to each other:
$$ \frac{n_1RT}{V1} = \frac{n_2RT}{V2} $$
This simplifies to
$$ \frac{n_1}{V1} = \frac{n_2}{V2} $$
Step 5: Let n1 = P1V1/RT and n2 = P2V2/RT. Assuming both sides contain the same gas type and are at the same temperature, we see from the equilibrium condition that:
$$ P1V1 = P2V2 $$
Step 6: Solving these equations gives us the relationship between the volumes:
If we denote V1 = \frac{P2}{P1 + P2} \times V0 and V2 = \frac{P1}{P1 + P2} \times V0, we find the equilibrium volumes in terms of the total volume and pressures:
$$ V1 = \frac{P2}{P1 + P2} \times V0 $$
$$ V2 = \frac{P1}{P1 + P2} \times V0 $$
Step 7: Hence, the volumes of the two parts at equilibrium are determined based on the initial pressures and total volume.