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Question 5 The rotational energy levels of a diatomic molecule are given by E,= BJ(J+1) with B the rotational constant equal to 8.02 cm Each level is (2) +1)-times degenerate. (wavenumber units) in the present case (a) Calculate the energy (in wavenumber units) and the statistical weight (degeneracy) of the levels with J =0,1,2. Sketch your results on an energy level diagram. (4 marks) (b) The characteristic rotational temperature is defined as where k, is the Boltzmann constant. Calculate the value of , in kelvin. (2 marks) (c) (0) Use the Maxwell-Boltzmann distribution to calculate the ratios T = 20K where N, N, and N, are the populations of the levels with / 0,7 = 1 and J=2, respectively. (3 marks) (1) Find an approximation for the rotational partition function at T = 20 K if it is given by the following expansion and hence, calculate the partial pressures P. (due to molecules in N) and P. (due to molecules in N) of a gas composed of a molecules per unit volume (number density) if the total pressure of the gas, considered to be ideal, ispank, 1 Pa. (8 marks) –9x5)- GS- (d) Calculate an expression for the value (1 ) of the most populated rotational level of population N, ). Estimate at T = 20 K and T = 2000 K. Hint: use the Maxwell- Boltzmann distribution to express the ratic in terms of the rotational partition function (N is the total population) and then solve the equation O for J. (8 marks)
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