To calculate the maximum work that can be extracted from the gas, we can use the concept of Helmholtz free energy A in statistical mechanics. The Helmholtz free energy is defined as:A = U - TSwhere U is the internal energy, T is the temperature, and S is the entropy of the system. The maximum work that can be extracted from the gas is equal to the decrease in Helmholtz free energy:W_max = A = A_initial - A_finalIn the canonical ensemble, the partition function Q is related to the Helmholtz free energy by:A = -kT ln Q where k is the Boltzmann constant. For an ideal gas, the partition function can be written as:Q = q^N / N!where q is the single-particle partition function, and N is the number of particles or moles of gas .The single-particle partition function for an ideal gas is given by:q = 2m kT / h^2 ^3/2 * Vwhere m is the mass of a single gas particle, h is the Planck constant, and V is the volume of the container.Now, we can calculate the initial and final Helmholtz free energies using the given number of moles of gas n , pressure P , and temperature T . The initial state is the gas in the container, and the final state is when the gas has expanded isothermally and reversibly against the pressure P.For the initial state, the Helmholtz free energy is:A_initial = -kT ln Q_initial For the final state, the gas has expanded against the pressure P, so the volume has increased. The final volume can be calculated using the ideal gas law:PV = nRTV_final = nRT / PThe final Helmholtz free energy is:A_final = -kT ln Q_final Now, we can calculate the maximum work that can be extracted from the gas:W_max = A = A_initial - A_finalBy substituting the expressions for A_initial and A_final, and using the given values for the number of moles of gas, pressure, and temperature, the student can calculate the maximum work that can be extracted from the gas.