(a) Calculate the work W_L that would be required to liquefy one mole of a monatomic ideal gas if the liquefier operated reversibly. Assume that the gas is supplied at room temperature T_o, and tinder the same pressure p_o at which the liquefied gas is removed, typically I atmosphere, Let T_b be the boiling temperature of the as at this pressure and ?H the latent heat of vaporization. Show that under these conditions

To derive (11) assume that the gas is first cooled at fixed pressure p₀ from T₀ to T_b, by means of a reversible refrigerator that operates between the fixed upper temperature T_h = T₀ and a variable lower temperature equal to the gas temperature. Initially T₁ = T₀. and at the end T_t = T_b. After reaching T_b the refrigerator extracts the latent heat of vaporization at the fixed lower temperature T_b.

(b) Insert T₀ = 300K and values for T_b and ?H characteristic of helium. Re-express the result as kilowatt-hours per liter of liquid helium. Actual helium liquefiers consume S to 10kWh, liter.

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