Consider the diatomic molecules listed below, and think of them as harmonic osci
ID: 895438 • Letter: C
Question
Consider the diatomic molecules listed below, and think of them as harmonic oscillators. Calculate the force constants in N/m of these oscillators based on the vibrational transition energies (v = 0 rightarrow v = 1) listed: Consider a proton in a box of length a = 0.1 nm, with the minimum of its potential energy curve at V = 0. Compare this to a proton in a harmonic oscillator potential with a force constant K = 1kN/m. The bottom of the harmonic oscillator potential has an offset of V_0 against the bottom of the box potential. Sketch the potential energy curves to show that you understand the situation. If the ground states of the harmonic oscillator and the particle in the box are to be at the same energy, what value must V_0 have (in eV)?Explanation / Answer
(i)HF
h*c = 1.986E-25 J.m = 1.986E-16 erg.cm
1/lambda = 4141 cm-1
E(erg) = h*c*(1/lambda) = 1.986E-16*4141
mu = (19*1/(19+1))*1.66E-24 g
hb = 1.05E-27 cm2*g*s-2
E = hb*w0 = hb*(k/mu)^0.5
k = mu*(E/hb)^2 = (19*1/(19+1))*1.66E-24*(1.986E-16*4141/1.05E-27)^2
= 1.577E-24*3.577E22*(4141)^2 = 9.67E5 dyne/cm = 967 N/m
(ii)HCl
h*c = 1.986E-25 J.m = 1.986E-16 erg.cm
1/lambda = 2989 cm-1
E(erg) = h*c*(1/lambda) = 1.986E-16*4141
mu = (35*1/(35+1))*1.66E-24 g
hb = 1.05E-27 cm2*g*s-2
E = hb*w0 = hb*(k/mu)^0.5
k = mu*(E/hb)^2 = (35*1/(35+1))*1.66E-24*(1.986E-16*2989/1.05E-27)^2
= 1.614E-24*3.577E22*(2989)^2 = 5.16E5 dyne/cm = 516 N/m
(iii)HBr
h*c = 1.986E-25 J.m = 1.986E-16 erg.cm
1/lambda = 2649 cm-1
E(erg) = h*c*(1/lambda) = 1.986E-16*4141
mu = (81*1/(81+1))*1.66E-24 g
hb = 1.05E-27 cm2*g*s-2
E = hb*w0 = hb*(k/mu)^0.5
k = mu*(E/hb)^2 = (81*1/(81+1))*1.66E-24*(1.986E-16*2649/1.05E-27)^2
= 1.64E-24*3.577E22*(2649)^2 = 4.12E5 dyne/cm = 412 N/m
(iv)HI
h*c = 1.986E-25 J.m = 1.986E-16 erg.cm
1/lambda = 2310 cm-1
E(erg) = h*c*(1/lambda) = 1.986E-16*4141
mu = (127*1/(127+1))*1.66E-24 g
hb = 1.05E-27 cm2*g*s-2
E = hb*w0 = hb*(k/mu)^0.5
k = mu*(E/hb)^2 = (127*1/(127+1))*1.66E-24*(1.986E-16*2310/1.05E-27)^2
= 1.647E-24*3.577E22*(2310)^2 = 3.14E5 dyne/cm = 314 N/m