One mole of an ideal monatomic gas is taken through a thermodynamic cycle ABCDA as shown in the figure. The process AB is a part of a parabola whose vertex is at the origin.
$(C_{V} = \frac{3}{2} R)$ The total external work done on the gas in the whole process ABCDA is

Question

One mole of an ideal monatomic gas is taken through a thermodynamic cycle ABCDA as shown in the figure. The process AB is a part of a parabola whose vertex is at the origin.

$(C_{V} = \frac{3}{2} R)$ The total external work done on the gas in the whole process ABCDA is

TestHub · Accepted Answer

W = W A B ​ + W BC ​ + W C D ​ + W D A ​ W A B ​ ​ = ∫ V A ​ V B ​ ​ P d V = ∫ V A ​ V B ​ ​ V 2 C ​ d V = − V C ​ ​ V A ​ V B ​ ​ = P V ∣ A B ​ = P A ​ V A ​ − P B ​ V B ​ = − P V ​ W BC ​ W C D ​ W D A ​ ∴ W ​ = P B ​ ( V C ​ − V B ​ ) = 4 P [ 6 V ​ − 2 V ​ ] = − 3 4 ​ P V = 0 = P A ​ ( V A ​ − V D ​ ) = P A ​ V A ​ − P A ​ V D ​ = P V − P 6 V ​ = 6 5 P V ​ = − P V − 3 4 ​ P V + 6 5 ​ P V = − 2 3 ​ P V ​ Therefore, the work done on the gas is 2 3 ​ P V .

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