175. ABSORPTION OF RADIATION(6) – MORSE POTENTIAL(2)

After learning the concept of Morse’s potential in the last post, let us start this post by getting to know the man behind this concept and his great mentor!

PHILIP Mc CORD MORSE
Philip Mc Cord Morse

Morse potential is named after an American physicist Philip Mc Cord Morse. He was a professor of physics at the Massachusetts Institute of Technology (MIT) and was closely associated with US defense operations. He earned his Ph.D. from Princeton University and then went to Munich for further research under Arnold Sommerfield!

As we know from post 26, Arnold Sommerfield is a scientist with a world record of being nominated 84 times (maximum number of times in history) for the Nobel Prize. Unfortunately, he never received it!

However, many of his students like Werner Heisenberg, Peter Debye, Linus Pauling won their respective Nobel prizes under his guidance.Such was the genius of this scientist, who proved to be a great mentor to many fellows.Philip Morse was another such student of his, who outshone many other physicists of that era, by his extra-ordinary work.


Let us now continue the discussion on Morse’s potential. In the previous post, we discussed how Morse potential correctly describes the vibrational interaction between atoms in a molecule. In this post, we take that discussion further and study how to make sense of the concept mathematically.

Mathematically, the Morse potential can be calculated by the following formula-

E(r) = De {1-e-a(r-re)}2 , where ,

E(r) ⇒ Potential energy
De ⇒ Bond dissociation energy at equilibrium / Depth of the potential well
r ⇒ Displacement
re ⇒ Equilibrium bond distance/internuclear distance
a ⇒ molecular parameter unique to an individual molecule

The equation looks very complicated but by plugging in some extreme values, we can study how it points to the facts we learned in the last post.

We will study two cases with this equation –

CASE 1THE EQUILIBRIUM POSITION

At the equilibrium position, the molecule is considered to be at rest i.e it is still with no vibrations. At this point, r=re.

∴ r- re = 0.

Plugging this value in the above equation-

E(r) = De {1-e-a(r-re)}2
E(r) = De {1-e-a(0)}2
E(r) = De {1-e0}2
E(r) = De {1-1}2
E(r) = De {0}2
E(r) = 0

Thus, the energy of the system is zero. This is exactly what we call zero-point energy. Thus, this equation correctly predicts the energy at equilibrium.

CASE 2 WHEN ‘r’ IS TENDING TO INFINITY.

Let us assume that the displacement r is very large and is tending to infinity. This means that the two atoms are very far apart from each other. Thus, r = ∞.

∴ r- re = ∞.
Plugging this value in the above equation again,

E(r) = De {1-e-a(r-re)}2
E(r) = De {1-e-a(∞)}2
E(r) = De {1-0}2
E(r) = De {1}2
E(r) = De

At infinity, when the two atoms are very far apart from each other, the energy equals the dissociation energy (De). This means that the molecule dissociates. Thus, again this equation correctly points to the facts we studied in the previous post.


In the next post, we continue discussing the vibrational levels and the corresponding transitions. Till then, Be a perpetual student of life and keep learning…

Good Day!

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