Question

Vibrational spectrum of H-CI: theory Calculate the vibrational amplitude of a state: al amplitude of a H-Cl molecule in its ground state, and in the first excited a) Describe in words or sketches what needs to be done. b) Create a symbolic expression for the vibrational amplitude (this would be the first step of calculating a numerical value). c) Almost all crystals have a positive coefficient of thermal expansion. In other words, they expands when you heat them. Use the inter-atomic potential energy curve to explain qualitatively why crystals expand with temperature.

Answer 1

Vibrational spectrum of Hace a) Beceride We can freal Hcl molecule as by a spring. two particles attached Borroroso@ Beth particles are osilaling. Therefore, it is not a quantum oscilator which consists of a single particle attached to a spring However, we can model the diatomic molecule as a quantem oscilator if we use the reduced mase e of the molecule. The energy level of the molecule is associated with different vibrational a level. Every vibrational level has different vibrational amplitude. We can reprecent this by mouse potential . curve. The amplitude is increasing as we go higher ribrational level. val

b) Expressions ; Energy (0+12) ha, 0, 0, 1, 2,.... to / h where, wis vibrational frequency. 2 2 h kalore constant Me reduced mas. As vibrational amplitude Again, E = Ź KA² => A= √2E for, vro, EzEo, Azdo vsi, EzE, , APA, Location MH+Mce Calculation : Mace Max Mch x t = 196x 10-27 kg KHCl = 480 Nout Es = *(6.626x10-16 eVis) * = 0.18 ev. E = (1+2) h2 -3% hal - 3x(0.18) = 0.54 eV 480 N.ml les 16x10-27 kg A ren 2x (0)x(+60-1) 5. - = lolx 10-11 m 480 N.mt x. 10.54) × (1.6x10-125 = 109810-11 m 480 Nml

As we increase the temparkatur more no. of particles goes to higher vibrational level. Therefore the vibrational amplitude of the system increas (we can see from the potential curve) Therefore the crystal expand with increase in temparature.