(a) The Sackur-Tetrode equation for the entropy of an ideal gas, derived via a statistical mechanics approach, is V 4πmU

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(a) The Sackur-Tetrode equation for the entropy of an ideal gas, derived via a statistical mechanics approach, is V 4πmU [ { x + ( 45V/2²) 2/²} + 5/2] N 3Nh² S = Nk In where the variables have their usual meanings. (i) Consider adiabatic compression of an ideal gas at constant pressure. What is the heat flow AQ into the gas? (ii) For an adiabatic compression of N molecules of an ideal gas at constant pres- sure from volume V₁ to volume V₂, use the Sackur-Tetrode equation to show that AS = S₂ − S₁ = Nk ln ((V₂/V₁) (U₂/U₁)³/2). Calculate U₂ in terms of U₁ and other quantities using the first thermodynamic identity and then com- ment on whether AS is always negative? (iii) Compare your result in (b)(ii) to the classical expectation AS = S²dQ/T, where do is the differential for the heat input to the gas and T is the tempera- ture, and give a qualitative comment involving the physics in the two theoretical approaches for why differences might be expected.