Concept explainers
All molecules undergo vibrational motions.
Where n is a quantum number
Interpretation:
The first three vibrational energy levels for
l is to be drawn. The energy required for the transition of molecule from ground state to first excited state is to be determined, and “the reason for lowest vibrational energy in ground state is not zero, but it is equivalent to
” is to be justified by using Heisenberg Principle.
Concept introduction:
The energy of a photon can be expressed as follows:
Here, E is the energy of photon,
Heisenberg uncertainty principle explains that the product of uncertainty in position and momentum of particle cannot be less than
Here,
denotes uncertainty in position,
denotes uncertainty in momentum, and
denotes Planck’s constant
Answer to Problem 112AP
Solution:
a)
(b)
(c) Consider the diatomic molecule
This is disallowed by the Heisenberg uncertainty principle.
Explanation of Solution
a) Plot the first three vibrational energy levels for HCl
The vibrational energy of diatomic molecules is given as
The first three vibrational energy levels n are
For
For
For
The vibrational energy for the first three energy levels is
b) The vibrational energy required to excite HCl molecule from ground state to first exited state.
The vibrational energy for the ground state and first excited state are as follows:
For the transition from ground state
Now, substitute the value
and
in the above equation
c) Justify the prediction “that the lowest vibrational energy in the ground state is not zero, but it is equivalent to
Consider the diatomic molecule
As the two atoms are bonded to each other, the uncertainty in position, that is,
Thus,
So, this is disallowed by the Heisenberg uncertainty principle.
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