Question

A pure rotational spectrum of CO is taken using microwave spectroscopy. The data for the absorption frequencies for each rotational transition are given in the excel file on Blackboard called "Exam 4 Q3." Use the data to determine the following for CO: The rotational constant in a) cm1 b) The rotational distortion constant in cm1 c) The bond length of CO 7 Freq, MHzJ 115271.2 230538 345796 461040.8 576267.9 691473.1 806651.7 921799.7 0 1036912 1 1151985 2 1267015 3 1381995 4 1496923 5 1611794 6 1841346 7 1956018 8 2070616 9 2185135 0 2299570 20 1 2413917 21 2 2528172 22 3 2756388 24 4 2870339 5 2984181 26 6 3097909 27 7 3438365 8 3551592 31 9 3777636 0 3890443 1 4115606 36 2 4340138 38 10 12 13 14 16 12 13 15 16 17 18 18 19 21 23 24 25 25 2 30 30 32 34 35 37

Answer 1

a]. rotational constant B and and rotation energy E(J) , J is rotationa energy level

we get a formula for rotational energy level

E(J) = B J (J+1) --------(1)

from rotational spectra between two rotational levels j^' to j^{​​​​''}

∆E/ hc = {B j^{​​​​​​'}​​​​​ ( j^​​' +1)} - { B j^{​​​​​​''} ( j^{​​​​​​''}​ + 1) } cm-1 ----(2)

------- .for 1st experiment

here j^{​​​​​​'} = 1 and j^{​​​​​​''} = 0 put in eqn 2

∆E / hc = B [ 1(1+1) - 0] = B [ 2]

2B = hv / hc = v /c here v is frequency in hertz or per second and c is the velocity of light in cm /second = 3x10¹⁰ cm/second

rotational constant B = v/2c = 115271.2 Mhz / 2 x 3x10¹⁰

B = 115271.2 x 106 per second  / 6 x 10¹⁰ cm per second

B = 19211.86 x 10^-4 = 1.921186 cm-1 --------------- answer a

b] rotational distortion constant D = B^{​​​​​​3} / 4 [v(in cm-1)]²

v (in cm-1) = 115271.2 x 10⁶ per second / 3 x 10¹⁰ cm/second

= 38423.73 x 10^-4 = 3.842373 cm-1

now D = 4 x ( 1.921186)³ / (3.842373)² = 7.091 / 58.982 = 0.012

Distortion constant D = 0.012 cm^{​​​​​-1} ------------ answer b

c] bond length of CO will be determined by

B = h / 8 π^{​​​​​​2} I C where h is planks constant , I is moment of inertia , c is velocity of light also Inertia I = u.r² where u is reduced mass , r is bond length

u.r² = I = h / 8π² c B

bond length r = [ h / 8 π^{​​​​​​2} B C U]^1/2

reduced mass for CO is u = 1.13826 x10^{​​​​-27}​x10-27

on solving we get bond length r = 1.131 x 10 ^-10 = 1.131 A° ------ ans c