March 30,2005 Name:______________________________ PHY 3513/Kumar Mid-term Exam II You are allowed a formula sheet, a calculator and pencil and paper. The formula sheet must not contain anything other than the formulae. The exam has to be finished within the allotted time of 50 minutes. 1. An inventor claims to have developed an engine that takes in 108 J at a temperature of 440 K, rejects 4×107 J at a temperature of 200K and delivers 15 kilowatt-hours of mechanical work. Would you advise investing money to put this engine on the market? Why? There are two checks on the validity of the claim. Does it satisfy the first law of thermo? i.e. is w<QH -Qc? Here w is the work done and QH and Qc are respectively the heats extracted form the hot end and delivered at the cold end. The inequality is there to allow for some non-reversibility. QH -Qc = 6×107J The work done is 15×103×3600 = 5.4×107<6×107 Æ all OK. The second test is whether the engine efficiency does not exceed the Carnot (1-Tc/Th)? The engine efficiency η = w/Qh = 0.54. The Carnot efficiency is ηc = 1-200/440 = 0.545 Thus the engine efficiency doe not exceed the Carnot efficiency. 2. Two equal quantities of water, each of mass m and at temperature T1 and T2 are adiabatically mixed together, the pressure remaining constant. a. Show that the entropy change of the universe (system + environment) is ⎛ T + T2 ⎞ ⎟ ΔS = 2mc p ln⎜ 1 ⎜2 TT ⎟ 1 2 ⎠ ⎝ Where cp is the specific heat of water at constant pressure. b. Show that ΔS > 0 for any finite temperatures T1 and T2. 3. The Gibbs function of a certain gas is 1 1 G = nRT ln P + A + BP + CP 2 + DP 3 2 3 Where A, B, C and D are constants. Find the equation of state of this gas.