An ideal monatomic gas is initially at pressure P₀ and volume V₀ . It undergoes an adiabatic expansion to a volume of 8V₀ . Subsequently, the gas is subjected to an isobaric compression until its volume returns to V₀ . The total work done by the gas in these two processes is:

Step-by-step solution

For an ideal monatomic gas, the ratio of specific heats is = 5 3 . Process 1: Adiabatic expansion from (P₀, V₀) to (P₁, 8V₀) . Using the adiabatic equation P₀ V₀^ = P₁ V₁^ : P₁ = P₀ ( V₀ 8V₀ )^ 5/3 = P₀ ( 1 2^3 )^ 5/3 = P₀ 32 The work done in the adiabatic process is: W₁ = P₀ V₀ - P₁ V₁ - 1 = P₀ V₀ - ( P₀ 32 )(8V₀) 5 3 - 1 W₁ = P₀ V₀ (1 - 1 4 ) 2 3 = 3 4 P₀ V₀ 2 3 = 9 8 P₀ V₀ Process 2: Isobaric compression from (P₁, 8V₀) to (P₁, V₀) . The work done in the isobaric process is: W₂ = P₁ (V_f - V_i) = P₀ 32 (V₀ - 8V₀)