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. 153 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 154, show that the
Helmholtz function for rotation is PM = —3NkTe20wﬂ. (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. 155 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 \fg7 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. 159 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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 Winter '10
 Dr.asdas

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