When the volume of a gas is changed from 11.5 cm3 to ? cm the temperature will change from 415 k to 200 k

Assuming that the gas acts like an ideal gas, we can calculate for the final volume using the ideal gas law:

PV = nRT

Where P = pressure, V = volume, n = number of moles, R = gas constant, and T = temperature

Assuming that P, n, and R are constant throughout the process, we can define another constant K:

V / T = K               where K = nR / P

Equating the initial and final states:

Vi / Ti = Vf / Tf

Substituting the given values:

11.5 cm^3 / 415 K = Vf / 200 K

Vf = 5.54 cm^3