1000 Solved Problems in Modern Physics

264 4 Thermodynamics and Statistical Physics P−= dAdt 2 λ ∫∞ 0 νdn ∫∞ 0 e−r/λdr ∫π/ 2 0 sinθcosθ ( mu+rcosθ dmu dz ) dθ The fact ...

4.3 Solutions 265 4.18 1eV=kT T= 1 eV k = 1. 6 × 10 −^19 J 1. 38 × 10 −^23 J/K = 11 ,594 K 4.19 (a) For a perfect gas at tempera ...

266 4 Thermodynamics and Statistical Physics From the law of equipartition of energy we have dEt dT = 3 2 k; dE ′ dT = β 2 k (2) ...

4.3 Solutions 267 Therefore, dU=Tds−PdV (6) whereUis the internal energy,Qthe heat absorbed,Wthe work done by the system,Sthe en ...

268 4 Thermodynamics and Statistical Physics { ∂ ∂y ( ∂U ∂x ) y } x = ( ∂T ∂y ) x ( ∂S ∂x ) y +T { ∂ ∂y ( ∂S ∂x ) y } x (15) − ( ...

4.3 Solutions 269 (b) Let the temperature and pressure be independent variables. Putx=Tand y=Pin (20). ( ∂T ∂x ) y = ( ∂P ∂y ) x ...

270 4 Thermodynamics and Statistical Physics δQ=Ldm (4) Ifν 1 andν 2 are the specific volumes (volumes per unit mass) of the liq ...

4.3 Solutions 271 u= T 3 ∂u ∂T − u 3 or du u + 4 dT T = 0 Integrating, lnu=4lnT+lna=lnaT^4 where lnais the constant of integrati ...

272 4 Thermodynamics and Statistical Physics ( ∂P ∂T ) V = R V (3) Re-writing (1) PV+ a V =RT DifferentiatingVwith respect toT, ...

4.3 Solutions 273 Use (2) and (3) in (4) CP−CV=−T ( ∂P ∂V ) T ( ∂V ∂T ) 2 P (5) CP−CV=−T ( ∂V ∂P ) T ( ∂P ∂T ) 2 V (6) Equation ...

274 4 Thermodynamics and Statistical Physics 4.31 TakingTandPas independent variables S=f(T,P) dS= ( ∂S ∂T ) P dT+ ( ∂S ∂P ) T d ...

4.3 Solutions 275 whereHis the enthalpy ∴TΔS+VΔP= 0 But by Problem 4.31 TΔS=CPΔT−T ( ∂V ∂T ) P ΔP ∴CPΔT+ [ V−T ( ∂V ∂T ) P ] ΔP= ...

276 4 Thermodynamics and Statistical Physics =(V−b)+ 2 a RT V^3 (V−b)^3 T ( ∂V ∂T ) P −V= 2 a RT −b (∴bV) Using this in the exp ...

4.3 Solutions 277 ES ET = (∂P/∂V)S (∂P/∂V)T = (∂P/∂V)S ( ∂T ∂V ) S (∂P/∂S)T ( ∂S ∂V ) T = (∂T/∂V)S ( ∂S ∂P ) T (∂T/∂P)S ( ∂S ∂V ...

278 4 Thermodynamics and Statistical Physics 4.39 By Maxwell’s first equation ( ∂S ∂V ) T = ( ∂P ∂T ) V (1) dS= dU+PdV T (2) usi ...

4.3 Solutions 279 4.43 (a) Use the relation dU=Tds−PdV (1) Here, dV=0(∵V=constant) and U=aV T^4 (2) dU= 4 aV T^3 dT=Tds ( ds dT ...

280 4 Thermodynamics and Statistical Physics 4.46 For stationary waves, in thex-direction kxa=nxπ ornx=kxa/π dnx=(a/π)dkx Simila ...

4.3 Solutions 281 4.49 p(EJ)=(2j+1)e −J(J+1)^2 2 I 0 kT The factor ^2 2 I 0 k = (1. 055 × 10 −^34 )^2 2 × 4. 64 × 10 −^48 × 1. ...

282 4 Thermodynamics and Statistical Physics LetP(E) be the probability function which gives the probability of the state at the ...

4.3 Solutions 283 orΔQ=TΔS=kTlnΔW =(1. 38 × 10 −^23 )(300) ln 10^8 = 7. 626 × 10 −^20 J= 0 .477 eV 4.58 The Gaussian (normal) di ...