One mole of a monatomic ideal gas is taken on the path A B C D A, as shown in Figure. All paths are reversible.

A B is a reversible isothermal expansion of the gas.

B C is a reversible adiabatic expansion of the gas.

C D is a reversible isothermal compression of the gas.

D A is a reversible adiabatic compression of the gas.

Figure: A cycle for one mole of a monatomic ideal gas with two isothermal paths and two adiabatic paths. The area enclosed by ABCD is the work done by the gas during the cycle.

a. Derive expressions for U, q, and w during each step in terms of V_a , V_b , V_c , V_d , t₁, t₂, and R. Determine the sign of each.

b. Determine the values of (w_i), (q_i), and (U_i) in terms of V_a , V_b , V_c , V_d , t₁, t₂, and R. Determine the sign of each.