Thermal Physics - Problem Set 8 Solutions | PHYS 224, Assignments of Thermal Physics

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224 problem set 8 solutions

224 problem set 8 solutions Problem 4.21, [Stirling engine.) (a) The cycle consists of two isothermal processes (at T), and T.) connected by constant- volume processes: Ph 1 T;, isotherm 4 2 T. isotherm 3 fie Mi V; (b) To calculate the efficiency we need to know the net work done and the total heat input for one cycle. The work done by the gas during the power stroke is v, ¥, 2 2 NAT) Vg Wo= | Paves [ 2O*k WV = NRT, In =, vi vy OV Yi 4 and the work done by the gas during the compression stroke is similarly Vi 7 , * NKT. Vo Wa = ——dV =—-NkT.In—, ‘na [ V d i In vs a so the net work done per cycle is W = Wit Wu = NAT, - 1) ng. 1 Meanwhile the heat input occurs during the power stroke and during the transfer to the hot cylinder. Because the power stroke is isothermal, the energy of the gas doesn’t change during this step and therefore, by the first law, Qi = Wee = NETy In 2, ¥ During the transfer to the hot cylinder there is no work done, so by the first law and the equipartition theorem, the heat input is Qa =U, - Uy where f is the number of degrees of freedom per molecule. Thus the total heat input Is r V; rep 1 Qn = @utQu = NAT In + Even, — 1) yw 2 Ewacn, -T.), Now the efficiency is defined as ¢ = W/Q,. It’s algebraically simpler to compute the reciprocal, 1 Qs, — NkT)In(¥e/Vi) 4 ENK(T, — Te) f e Ww NK(I — T:) in(Va/Vi) ~ 2In(¥a/Vi) The first term is just the reciprocal of the Carnot efficiency, ec = 1 — (L./Ty), so we “s te ean write Loa . f e ec 2in(¥s/V)