The energy of a vibrating molecule is quantized much like the energy of an elect

Agaiepsh

Agaiepsh

Answered question

2021-11-15

The energy of a vibrating molecule is quantized much like the energy of an electron in the hydrogen atom. The energy levels of a vibrating molecule are given by the equation: En=(n+12hv) where n is a quantum number with possible values of 1, 2, ... and v is the frequency of vibration. The vibration frequency of HCI is 8.85×1013s1 What minimum energy is required to excite a vibration in HCl? What wavelength of light is required to excite this vibration?

Answer & Explanation

Linda Tincher

Linda Tincher

Beginner2021-11-16Added 14 answers

Step 1
A vibrating molecule's energy is En=(n+12)hv when the quantum number is n; h=6.626×1034js and v is the vibration frequency.
Given that the vibration frequency of HCl is 8.85×1013s
Find the wavelength of light that can trigger this vibration and the minimal energy needed to cause HCl to vibrate.
Solution: The difference between the energies of the two lowest quantum numbers should represent the minimal energy needed to cause HCl to vibrate. Use E=hcλ to find the vibration's corresponding wavelength where c=3.00×108ms
Step 2
Since n should be a positive number in a quantum system, the transition that corresponds to the minimal amount of energy needed to for HCl to vibrate is n=1n=2. Calculate E1  and  E2
E1=(n+12)hv=(1+12)6.626×1034js8.85×1013s=8.796015×1020j
E2=(n+12)hv=(2+12)6.626×1034js8.85×1013s=1.466×1019j
Step 3
Calculate
E12=EfEi
E12=E2E1=1.466×1019j8.796×1020j=5.864×1020J
Step 4
Calculate λ using E=hcλ
λ12(6.626×1034Js)(3.00×108m/s)5.864×1020J=3.3898×106m1×109nm1m=3390nm
E12=5.864×1020j
λ12=3390nm

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