Thermodynamics, Statistical Physics, and Quantum Mechanics

96 Equating the amount ofheat required tomelt amass of ice with the heatavailable in themetal, we have This mass exceeds theamou ...

97 4.13 Maxwell–Boltzmann Averages (MIT) a) We may write the unnormalizedMaxwell–Boltzmanndistributionimme- diately as We would ...

98 The integral required here may befound by differentiating(S.4.13.7) once with respect to Using (S.4.13.8) in (S.4.13.3),where ...

99 d) The mean square speed of the atoms may befoundimmediately by recalling the equipartition theorem (see Problem 4.42) and us ...

100 We integratefrom 0 to since we only consider the atomsmoving toward the wall. On the other hand, bydefinition, theaverage ve ...

THERMODYNAMICS AND STATISTICAL PHYSICS 101 So is indeed higherthan theaverage energy of theatoms: c) From (b) weknow thateach at ...

(^102) SOLUTIONS For an estimate we can assumethat the adsorbed molecules are closely packed and that the number of adsorption s ...

THERMODYNAMICS AND STATISTICAL PHYSICS 103 where is the concentration of themolecules inside thevessel and is the concentration ...

(^104) SOLUTIONS where is thearea of the hole. Atequilibrium, so we have The mean free path is relatedto the volumeconcentration ...

THERMODYNAMICS AND STATISTICAL PHYSICS 105 The factor 2appears for the flowrate sincethere are two portalsfrom regionB.On the ot ...

106 SOLUTIONS 4.19 Rocket in Drag (Princeton) Use dimensionalanalysis toderive thedragforceFon the rocket: We thenhave So and Th ...

THERMODYNAMICS AND STATISTICAL PHYSICS 107 Therefore, c) At a height we have,from(S.4.19.3), where weused foruniform acceleratio ...

108 SOLUTIONS Substituting (S.4.20.3) into (S.4.20.2), we obtain Assuming that theacceleration ofgravity is constant, using the ...

THERMODYNAMICS AND STATISTICAL PHYSICS 109 The total internal energy isgiven by the integral of(S.4.21.3): Rearranging(S.4.21.5) ...

(^110) SOLUTIONS The ratio ofenergies from(S.4.21.10) and(S.4.21.7) is Finally, 4.22 Puncture (Moscow Phys-Tech) a) UseBernoulli ...

111 The temperature may befound from theequation foradiabats and the ideal gas law: Rewritinggives Substituting into (S.4.22.4) ...

(^112) SOLUTIONS Note that this estimateimplies that i.e.,that the gas would cool to absolute zero. This is, of course, nottrue; ...

113 On the other hand, at equilibrium, there is no net force acting on the piston (seeFigure S.4.23), and we have Substituting(S ...

114 SOLUTIONS where 0 and 1correspond to the initial and final equilibriumstates of the system, withsets ofparameters and respec ...

115 Therefore, Substituting(S.4.24.8) and(S.4.24.2) into(S.4.24.3), we obtain and The temperatureindeed has decreased. As for th ...