Question

The force constants for F2 and I2 are 470. and 172 Nm−1, respectively. The atomic masses in amu are as follows: 19 F - 18.9984, 127 I - 126.9045.

1) Calculate the ratio of the vibrational state populations n1/n0 for F2 at 310 K

2) Calculate the ratio of the vibrational state populations n1/n0 for F2 at 900 K

3) Calculate the ratio of the vibrational state populations n2/n0 for F2 at 310 K .

4) Calculate the ratio of the vibrational state populations n2/n0 for F2 at 900 K .

5) Calculate the ratio of the vibrational state populations n1/n0 for I2 at 310 K

6) Calculate the ratio of the vibrational state populations n1/n0 for I2 at 900 K

7) Calculate the ratio of the vibrational state populations n2/n0 for I2 at 310 K .

8) Calculate the ratio of the vibrational state populations n2/n0 for I2 at 900 K .

Answer 1

For F2,

reduced mass for F2 (u1) = [(18.9984)^2/(18.9984 + 18.9984)]/6.023 x 10^26 = 1.58 x 10^-26 kg

vibrational frequency for F2 (v1) = [1/2pi.sq.rt.(k/u1)]

k = 470 N.m-1

v1 = (1/2 x 3.14) x sq.rt.(470/1.58 x 10^-26) = 2.746 x 10^13 s-1

Population analysis

1) n1/no = e^-(hv1/Kb.T)

T = 310 K

Kb = Bolzman constant

h = planck's constant

n1/no = e^-(6.626 x 10^-34 x 2.746 x 10^13/1.38 x 10^-23 x 310) = 0.0142

2) at T = 900 K

n1/no = e^-(6.626 x 10^-34 x 2.746 x 10^13/1.38 x 10^-23 x 900) = 0.2311

3) at T = 310 K

n2/no = e^-(2 x 6.626 x 10^-34 x 2.746 x 10^13/1.38 x 10^-23 x 310) = 0.0002

4) at T = 900 K

n2/no = e^-(2 x 6.626 x 10^-34 x 2.746 x 10^13/1.38 x 10^-23 x 900) = 0.0534

For I2

reduced mass for I2 (u2) = [(126.9045)^2/(126.9045 + 126.9045)]/6.023 x 10^26 = 1.05 x 10^-25 kg

vibrational frequency for F2 (v1) = [1/2pi.sq.rt.(k/u1)]

k = 172 N.m-1

v1 = (1/2 x 3.14) x sq.rt.(172/1.05 x 10^-25) = 6.445 x 10^12 s-1

Population analysis

5) n1/no = e^-(hv1/Kb.T)

T = 310 K

Kb = Bolzman constant

h = planck's constant

n1/no = e^-(6.626 x 10^-34 x 6.445 x 10^12/1.38 x 10^-23 x 310) = 0.3685

6) at T = 900 K

n1/no = e^-(6.626 x 10^-34 x 6.445 x 10^12/1.38 x 10^-23 x 900) = 0.7090

7) at T = 310 K

n2/no = e^-(2 x 6.626 x 10^-34 x 6.445 x 10^12/1.38 x 10^-23 x 310) = 0.1358

8) at T = 900 K

n2/no = e^-(2 x 6.626 x 10^-34 x 6.445 x 10^12/1.38 x 10^-23 x 900) = 0.5027