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# Working with partition function and entropy

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1. Given the rotational temperature of HCl is 15.02 K,
a) what is the ratio of the number of particles in the J= 14 to the J=4 level at a temperature of 1900 K?
b) What J level has the highest proportion of particles at the temperature 1900 K?
c) What is the energy, E, of rotation at this temperature?
d) What is the entropy, S, of rotation at this temperature?

https://brainmass.com/chemistry/physical-chemistry/working-partition-function-entropy-9338

## SOLUTION This solution is FREE courtesy of BrainMass!

The rotational energy of a linear molecule (neglecting such things as centrifugal distortion) is given by BJ(J+1) and each J level is 2J+1 degenerate.

The rotational temperature is defined as,

T = B/k ......(1)

where k is the Boltzmann constant and B is the rotational constant given by

B = h/(8pi^2I c) cm^-1 ........(2)

From (1), B = kT

B = 1.38*10^-23*15.02 = 20.73*10^-23 J

= (20.73*10^-23/hc) cm^-1

= 20.73*10^-23/19.89*10^-24

= 1.042 * 10 = 10.42 cm^-1

(a) Nj/N_0 = (2J+1) exp{-BhcJ(j+1)/kT}

BhcJ(j+1)/kT = (20.73*10^-23 )J(J+1)/kT

number of molecules at J= 4 is given by

N4 = (2*4+1) N_0 exp{-(20.73*10^-23 )4(4+1)/kT}

= 9 N_0 exp{-20.73*10^-23 )20/kT}
= 9 N_0 exp{-414.6*10^-23/kT}

where k T = 1.38*10^-23*1900 = 2622 *10^-23 J

Similarly,
N14 = (2*14+1) N_0 exp{-(20.73*10^-23 )14(14+1)/kT}

= 29 N-0 exp{-20.73*10^-23*14*15/kT}

= 29 N-0 exp{-4353.3*10^-23/kT}

Now, N14/N4

= [29 N-0 exp{-4353.3*10^-23/kT}]/9 N_0 exp{-414.6*10^-23/kT}

= [29 exp{-4353.3*10^-23/kT}]/9 exp{-414.6*10^-23/kT}

= (29/9) [exp{-1.66}/exp{-0.158}]

= 3.22 exp(0.158)/exp(1.66)

b)maximum population is for

J = sqrt[kT/2hcB] - (1/2)

= sqrt(2622*10^-23/2*20.73*10^-23) - ()1/2

= sqrt63.24 - 1/2

= 7.45

J value is 7

(c) E = B J(J+1)

= 20.73*10^-23 (7)(8)

= 1160.88 * 10^ -23 J

= 583.52 cm^-1

(d)For a linear molecule, the rotational partition function is

q = (1/sigma)*[T/y]

where T is the temperature and y is the rotational temp.

sigma = 1 for HCl

q = 1900/15.02 = 126.5

The rotational contribution to the entropy S = R(ln q + 1)

= 8.3143* (4.84+1)

= 48.55

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