Question

In: Physics

The vibrations of a bi-atomic molecule AB can be approximated as a harmonic oscillator with the...

The vibrations of a bi-atomic molecule AB can be approximated as a harmonic oscillator with the potential energy of type V (x) = (kx^2)/2, x representing the spatial deviation of the relative position between those 2 molecules towards equilibrium. Calculate the vibrational energy of the H2 molecule knowing that k = 510N/m and the mass of a hydrogen atom is m0 = 1.66 ∗ 10−27 kg

Solutions

Expert Solution

Ans. The vibrations of a diatomic molecule AB can be approximated as a harmonic oscillator.

The potential energy of this oscillator,   ................... (1)

Where, x represents the spatial deviation of the relative position between those molecules towards equilibrium and k is force constant.

Equation (1) is a consequence of Hooke's law, which states that the force between two co-linked objects is directly proportional to the displacement of the objects from their equilibrium position.

So, (negative sign indicates that the restoring force is opposite in direction to the displacement)

( is reduced mass and is equal to   )

(where, ) ....................... (2)

From the eigenvalues of equation (2), we can formulate vibrational energy as,

,   .................. (3)

where, h is Planck's constant and f is harmonic frequency,  

Now, for H2 molecule, mass of hydrogen atom,   kg.

So,  

kg.

Also, provided, k = 510 N/m

Hz.

Now, for residual vibrational energy or zero-point energy,  

So,  

J

J

eV

eV

Hence, the vibrational energy of H2 molecule is   J or   eV. (Ans.)

N.B. ::  The following problem is basically belongs to the topic 'vibrational and rotational spectra' in atomic physics. To solve these kinds of problems, you need to understand the key concept of the topic. Also, a strong knowledge in mathematical background like differential equation, calculus etc. is required.

If you are able to grasp these knowledge and practise in a regular basis, you can achieve your target and boost your level up.


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