In this problem we will treat the bond vibrations of diatomic molecules as the m

Question

In this problem we will treat the bond vibrations of diatomic molecules as the motions of quantized linear harmonic oscillators a For H Cl the bond force constant is A 480Nm t. Calculate the reduced mass of HCl i.e. calculate mum and calculate the bond mur m HCI vibration frequency v HCI b) For a harmonic oscillator, on average the kinetic energy K and the potential energy V are equal. This means that for every quantum number n: (Kn) (V) where E, h (n +t) Using the fact that (K.)-(v)- calculate (x and p for HC1 c) Calculate the vibrational partition function for HCl at T-300K. Also, calculate the vibrational heat capacity for HCl at this temperature. d) 127 has a much smaller bond vibrational frequency than 1H35Cl. For 127 I2 14 Repeat the calculations in part c for I2 and explain the differences in the results.

Explanation / Answer

a) reduced mass of 1H35Cl,

mu(HCl) = mH.mCl/mH+mCl/6.022 x 10^23 x 10^3 = 1x35/1+35/6.022 x 10^26 = 1.6266 x 10^-27 kg

Now, to calculate vibration frequency v(HCl) = 1/2pi.sq.rt.(K(HCl)/mu(HCl))

Given, K(HCl) = 480 N.m-1

Feed the value,

v(HCl) = 1/2x3.1416.sq.rt.(480/1.6266x10^-27) = 8.65 x 10^13 cm-1

c) Vibrational partition function z(HCl) would be,

z(HCl) = e^(-hv/2KT)/1-e^(-hv/KT)

Feed the values,

z(HCl) = e^(-6.626 x 10^-34 x 8.65 x 10^13)/2 x 4 x 300)/1-e^(-6.626 x 10^-34 x 8.65 x 10^13)/4 x 300)

           = 1

d) Vibrational partition function for I2

z(I2) = e^(-6.626 x 10^-34 x 8.65 x 10^13/14)/2 x 4 x 300)/1-e^(-6.626 x 10^-34 x 8.65 x 10^13/14)/4 x 300)

        = 1

Both have almost the same vibrational partition function values.

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