Question

The harmonic vibrational frequency of a diatomic molecule is 2000 cm⁻¹ . Its zero point energy is eV.
[Given: Planck’s constant = 6.62x10⁻³⁴ J s; 1 eV = 1.6x10⁻¹⁹ J] 
(round off to two decimal places)

Answer 1

Correct Answer: 0.12 - 0.13

The zero-point energy of a harmonic oscillator is given by the formula: E₀ = (1/2)hν, where h is Planck's constant, and ν is the frequency in Hz. First, we need to convert the frequency from cm⁻¹ to Hz: 

1 cm⁻¹ = 100 cm = 100 m⁻¹, hence, the frequency ν = 2000 cm⁻¹ × 3.00 × 10¹⁰ cm/s = 6.00 × 10¹³ Hz.

Now, calculate the zero-point energy in Joules:

E₀ = (1/2) × 6.62 × 10⁻³⁴ J s × 6.00 × 10¹³ Hz = 1.986 × 10⁻²⁰ J.

Convert the zero-point energy to electron volts (eV):

E₀ (eV) = 1.986 × 10⁻²⁰ J ÷ 1.6 × 10⁻¹⁹ J/eV ≈ 0.1241 eV.

The computed zero-point energy is 0.12 eV when rounded to two decimal places,  [0.12, 0.13].