Problem Set 4 Solutions - Thermal Physics | PHYS 224, Assignments of Thermal Physics

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224 problem set 4 solutions

224 problem set 4 solutions Problem 2.10. Here's an Excel spreadsheet showing the case Ny = 200, Ng = 100, and Grotai = 100: Two Einstein Solids MA =|200 N_B =/100 _g_total =|100 aA | Omega A! a B Omega_total ° 1{ 100] 5&+58) 4.5276458 i 200, ge, 2e+58) 4. 55E+60 1.46495 2] 20100] 96| 1€+58) 2.286E462 1.26495 3[_1E+06} 97| 6E+57|7.659E+63|] > a] 7e+07| 96| sE157|1.924£,05|| 5 'E+95 5] 3E+09) 95| 16+57| 3. 864E+66) = BE+94 6[ tert] 94] 7e+5e, 64044671] 2 geioy 7] 3E+12) 93) 3E+56] 9.265E+68) s al 7E+13| 92] 26+56|1.161E+70/] g 4&+94 o[_2e+15| 91] #6455] 7 209e+74 2E+94 0] 4€+16| 90] 46485] 1.294E470 0 Tif 7E+17| 89] 26455] 1.17679 12|_iE+19| 88) 6e+54) 9.792E+73 O20 40 60 80 100 13] 26+20) 87] 46+54) 7. 515E+74 energy in solid A T4] Ses21| 86) 2e+54] 5 348E+75 15] 4E+22 65] 8E+53] 3.547E+76 16] 6E+23) 64] 46+59/2.202E+77| | 17) 7E+24 83] 2E+53] 1.284E+78 total_multiplicity =/1.6814E+96 (To save space I’ve shown only the first 18 of the 10] rows.) The most probable macrostate is g4 = 67, gg = 33, with probability 1.23x 10"/1.68x 10° = 7.3% (not terribly large). The least probable macrostate is ¢, = 0, gg = 100, with probability 4.53 x 1058/1.68 x 10% = 2.7 x 10°" (tiny!). Problem 2.17. To simplify 9 in the limit q < N (with both g and N large), we can start by repeating exactly the same steps as in equations 2.17 and 2.18 to obtain InQ = (N+ q)ln(N + 9) — ging- Nin. Now expand the first logarithm in the limit q < N: =mnfn(14£)) = 9) cipva dt In(N +4) = In[N (14 <) =InN +In{1+ x) win + 2 Plugging this result into the previous equation and canceling the NlnN terms, we obtain 2 N Inks qinN +44 Wrang=gin= +4 where in the last step I've dropped the g?/N since it’s much smaller than the others. Exponentiating now gives Qa clMN/Det — (“yer = (2y. q q Since the original formula for 9 is symmetrical under the interchange of q and N (after the initial approximation in equation 2.17), we could have obtained this result simply by interchanging g and N in equation 2.21.