Two vertical cylinders are placed in a vacuum chamber and contain...

From the First Law of Thermodynamics, $Q = \Delta U + W$.

Since $Q_A = Q_B$ and $\Delta U_A = \Delta U_B$, the work done by both gases must be equal ($W_A = W_B$).

In a vacuum chamber, the external atmospheric pressure is zero. The pressure of the gas is purely due to the weight of the piston: $P = \dfrac{mg}{\pi r^2}$.

The work done by the gas during an upward displacement $x$ is $W = P \Delta V = \left(\dfrac{mg}{\pi r^2}\right) (\pi r^2 x) = mgx$.

Notice that the work done is simply the change in gravitational potential energy of the piston, which is independent of the piston's radius.

Equating the work done for both cylinders:
$m_A g x_A = m_B g x_B$
$m g x_A = (2m) g x_B$
$\dfrac{x_A}{x_B} = \dfrac{2}{1}$

The ratio of their displacements is $2 : 1$.

Answer: $2 : 1$