UCSF 137 Lecture 5: UCSF 137 - Lecture 5 (1)

The vibrational tequency and rotational constant fr 12C160 are 2170cm" and 1.93 13 cm-1, respectively, a. Calculate the frequencies of the first ten lines in the R and P branches of the rotational b. Calculate the Boltzmann distribution arm for the first ten rotational levels within the n = 0 3. and its average bond length (R) is 112.8 pm. vibrational spectrum of 12cl6O using Excel. vibrational level at 100 K, where no corresponds to the state.a= 0,J= 0, In other words, calculate the Boltzmann distribution for: (n = 0, J= 1) jn = 0, J=0), (n = 0, J = 2)jn = 0, J = 0), (n = 0, J = 10)jn = 0,J = 0). Using your results from (a) and (b), make a rough sketch of the spectrum. c.

All of the following sets of quantum numbers are allowed EXCEPT n = 5, e = 3, m_e = 3 n = 4, e = 2, m_e = -2 n = 1, e = 0, m_e = 0 n = 1. e = 0, m_e = 1 n = 6, e = 4, m_e = 4

For the n times n matrix A = [0 1 0 ... 0 0 0 0 1. .. 0 0. .. .. .. .. .. .. .. .. . 0 0 0 000 1 0 0 0 0. .. 0 1 -a_0 -a_1 -a_2. .. -a_n - 2 -a_n - 1] a. Use mathematical induction o n to Show that the characteristic polynomial of A is lambda^n + a_n - 1 lambda^n - 1 + a_n - 2 lambda^n - 2 +. .. + a_2 lambda^2 + a_1 lambda + a_0 b. If lambda_i is an eigenvalue of A, show that a corresponding eigenvector is given by upsilon_i = [1 lambda_i lambda^2_i lambda^n - 1_i] c. Show that the geometric multiplicity of each distinct eigenvalue is one.