Consider the 1H19F molecule. The bond length of 1H19F is 91.7 pm. The wavenumber

Question

Consider the 1H19F molecule. The bond length of 1H19F is 91.7 pm. The wavenumber of light associated with a transition from the ground vibrational state to the first excited state is 1159.18 cm^-1.

a. Calculate the energy, in J, of the lowest energy rotational state of 1H19F.

b. Calculate the rotational frequency, in rads/s, of 1H19F in the ground state.

c. Considering only rotational energy levels, calculate the largest wavelength of light, in m, that can be absorbed by the 1H19F molecule. Assume it is a rigid rotator.

d. What values of the quantum number l will give a projection of the angular momentum of 1H19F on the z axis of 2hbar? Explain.

e. Calculate the vibrational frequency, in rads/s, of 1H19F in the ground state.

f. Will the vibrational frequency of 2H19F be greater than or less than the vibrational frequency of 1H19F? Explain. Assume the bond force constant does not change.

g. At what values of the vibrational quantum number will the probability of finding the bond of 1H19F at its equilibrium bond length be zero? Explain.

Explanation / Answer

Reduced mass ? = mHmF/(mH + mF)

= product of atomic masses/sum of atomic masses x atomic mass unit

= (1 x 19)/(1 + 19) x 1.661 x 10-27 kg = 1.578 x 10-27 kg

Equilibrium bond length re = 91.7 pm = 91.7 x 10-12 m

Moment of inertia I = ?re2

= 1.578 x 10-27 x (91.7 x 10-12)2 = 1.327 x 10-47 kg m2

Rotational constant B = h2/8?2I where h is Planck constant

= (6.626 x 10-34)2/(8 x 3.1422 x 1.327 x 10-47) = 4.190 x 10-22 J

Rotational energy levels EJ = BJ(J + 1) where quantum number J = 0, 1, 2, ...

Ground rotational state energy level (J = 0),

E0 = 4.190 x 10-22 x 0 x (0 + 1) = 0 J

First excited rotational state energy level (J = 1),

E1 = 4.190 x 10-22 x 1 x (1 + 1) = 8.380 x 10-22 J

Photon energy = hf = hc/? = energy level difference (E1 - E0)

where c is the speed of light, f is frequency and ? is wavelength

6.626 x 10-34 x f = (6.626 x 10-34 x 2.998 x 108)/? = 8.380 x 10-22 - 0

Frequency f = 1.26 x 1012 Hz

Wavelength ? = 2.37 x 10-4 m

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