1 Show that if

f(T) = Sexp(-ei/kT).

then

[{Se_iexp(-e_i/kT)}/Sexp(-e_i/kT)]= kT² d lnf(T)/dT

2. A molecule absorbs at a vibrational wavenumber of 500 cm^-1 for a molecule at 298 K. What would the value of q_vib be at 298?

3. NMR signal is proportional to f₊ - f_- or the fraction in the lower energy state - the fraction in the upper energy state. Calculate this quantity for H atoms in a 200 MHz NMR at 298.

4. Derive the approximate partition function of rotational states of a linear molecular gas (g_J= 2J+1 and e_J = B*J*(J+1) ) by integration of the appropriate equation. Be sure to give the details of any change in variables.

5. Derive the equation for the partition function of a harmonic oscillator (g_n=1, e_n = nhno ).

6. q_rot = kT/B for a linear molecule. Show that

kT² d ln(q_rot)/dT = kT.

7. Set up the equation that must be integrated for calculating the partition function for the translational motion in the x direction. You need not do the integration. The right answer should include the limits of the integration, the variable that is being integrated over, and the differential of that variable.

8. The answer to the integration in #7 is

qx = L/h sqr(2pmkT)

What is the total translational partition function in all three dimensions?

 9. Calculate (using MathCad Program) q_rot for a molecule with B = 10 cm^-1 at -150 ^oC by direct summation.