Thermodynamics, Statistical Physics, and Quantum Mechanics

176 SOLUTIONS where (S.4.68.9) defines the equation of state. To find quantum corrections to the classical equation of state (wh ...

THERMODYNAMICS AND STATISTICAL PHYSICS 177 where b) The condition for validity of this approximation is that the first correc- t ...

178 SOLUTIONS Therefore, for asystem consisting of identical particles, and we may write for where and From(S.4.69.3) we have an ...

THERMODYNAMICS AND STATISTICAL PHYSICS 179 Using theequation ofstate for aFermi gas (seeProblem 4.66), we have Now,using(S.4.69. ...

180 SOLUTIONS The energy of the Bose gas at is The free energy Fis since and So thepressure So we see that thepressure does not ...

THERMODYNAMICS AND STATISTICAL PHYSICS 181 Letting we rewrite(S.4.70.3) as (S.4.70.4)defines aparametric equation for thechemica ...

182 SOLUTIONS The condition for Bose condensation tooccur isthat, at someparticular temperature, thechemicalpotentialgoes tozero ...

THERMODYNAMICS AND STATISTICAL PHYSICS 183 The total radiation energy density over all frequencies is The integral with factor i ...

184 SOLUTIONS where is the Stefan–Boltzmannconstant. Only afraction of this flux reaches the Earth. In equilibrium this fraction ...

THERMODYNAMICS AND STATISTICAL PHYSICS 185 e) Let us estimate the radius of a particle for which the radiation force will equal ...

186 SOLUTIONS where is the energydensity. So, c) Using the equation of state for a photon gas and we have The entropySis then d) ...

THERMODYNAMICS AND STATISTICAL PHYSICS 187 Thisresult can be immediatelyobtained if weconsider the process as a “chemical” react ...

188 SOLUTIONS Disregarding the 1 in the denominator of(S.4.74.16) andexpanding the square rootwithrespect to the small parameter ...

THERMODYNAMICS AND STATISTICAL PHYSICS 189 where wehave used theintegral given in the problem.Finally, 4.75 D-Dimensional Blackb ...

190 SOLUTIONS But Therefore, We may then find the pressure P of the gas: The isothermal work done by the gas b) For a photon gas ...

THERMODYNAMICS AND STATISTICAL PHYSICS 191 4.77 Blackbody Radiation and Early Universe (Stony Brook) By definition the free ener ...

192 SOLUTIONS where we let Performing the integralgives Substituting theenergy for this mode werecover theentropy in the form 4. ...

THERMODYNAMICS AND STATISTICAL PHYSICS 193 Let us calculate the Helmholtz free energy F. For a Bose gas with the grand thermodyn ...

194 SOLUTIONS The entropy Sof the gas is given by The energyEmay now be found from Comparing(S.4.78.7) and (S.4.78.9) gives Note ...

THERMODYNAMICS AND STATISTICAL PHYSICS 195 For an inversesquare lawforce(gravitation), the averagekineticenergy and potential en ...