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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?

    © BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf
    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

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf>
    https://brainmass.com/chemistry/physical-chemistry/working-partition-function-entropy-9338

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