Question

the frequency of molecular collision in the liquid phase is about 1 x 10¹³ s^-1. Ignoring all other mechanisms contributing to linewidth, calculate the width (in Hz) of vibrational transitions if

a) Every collision is effective in deactivating the molecule vibrationally

b) that one collision in 40 is effective

Answer 1

(a): According to Heisenberg's uncertainity pronciple:

E * t = h / (4)

Since all collisions are effective in decactivating the molecule vibrationally,  t = 1 / = 1 / (1*10¹³ s^-1) = 1*10^-13 s

=> E *  1*10^-13 s = 6.626*10^-34 J.s / (4*3.14)

=> E = 5.28*10^-22 J

Width of vibrational transition = 5.28*10^-22 J * (1 cm^-1 / 1.986*10^-23 J) = 26.6 cm^-1  

(b) When 1/40 of the collision are effective, t will be 40 times larger than (a).

Since E is inversely proportional to t, E will decrease by 40 times.

Hence Width of vibrational transition will also decrease by 40 times = 26.6 cm^-1 / 40 = 0.665 cm^-1 (Answer)