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Assignment 6 - “£2(a For a system of localized...

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Unformatted text preview: “£2 (a) For a system of localized distinguishable oscillators, Boltzmann statistics applies. Show that the entropy S is given by N S = —k;Nj m(#). (b) Substitute the Boltzmann distribution in the previous result to show that U S=?+Nkan. (c) Using the expressions derived in the text for U and Y; prove that 0/1" 6’” — 1 S=Nk — ln(1 — 6—9/7) , where 0 = hv/ k. Examine the behavior of S as T approaches zero. 15-3 Consider 1000 diatomic molecules at a temperature Gym/2. (a) Find the number in each of the three lowest vibrational states. (b) Find the vibrational energy of the system. J54 (a) In the low temperature approximation of Section 15-4, show that the Helmholtz function for rotation is PM = —3NkTe-20wfl. (b) Use the reciprocal relation S = — (BF/6T)V to find the entropy Srot in the same approximation. Note that S —> 0 as T—>O, in agreement with the third law. 15-5 As an alternative evaluation of Zrot to that given in the text, assume that for T >> 9m: the numbers I in the sum of Equation (15.13) are large compared with unity and replace the summation by integration with respect to I. Show that Zrot = T/erot - £45 Use the data of Table 15.1 to determine re, the equilibrium distance between the nuclei, for (a) an H2 molecule; (h) a CO molecule. W 7 V 217mk 3”TS/2]} — + S Nk{2 lniN< hz ) 29,0. ’ \/ if the atoms of the diatomic molecule are identical. 5 \fg-7 Consider a diatomic gas near room temperature. Show that the entropy is -8 For a kilomole of nitrogen (N2) at standard temperature and pressure, compute (a) the internal energy U; (b) the Helmholtz function F; and (c) the entropy S. 15-9 Using the relation 8 1n Z P = NkT , ( 3V )T show that the equation of state of a diatomic gas is the same as that of a monatomic gas. ...
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