The equilibrium dissociation energy of a diatomic molecule is 4.75 eV and its stretching frequency corresponds to 0.5 eV. The minimum energy required to dissociate the molecule in eV is
4.50
Dissociation Energy Concepts
The energy required to break apart a diatomic molecule can be described in different ways. Two important terms are equilibrium dissociation energy and spectroscopic dissociation energy.
- Equilibrium Dissociation Energy (\(D_e\)): This is the energy difference between the minimum of the potential energy curve (where the atoms are at their equilibrium separation) and the energy of the separated atoms. It represents the depth of the potential well.
- Spectroscopic Dissociation Energy (\(D_0\)): This is the energy difference between the ground vibrational state (the lowest energy level the molecule can occupy) and the energy of the separated atoms. This is the minimum energy actually required to dissociate the molecule starting from its most stable state.
The molecule is always vibrating, even in its lowest energy state (ground vibrational state, \(v=0\)). This minimum vibrational energy is called the zero-point energy.
The energy levels of a simple harmonic oscillator are given by \(E_v = (v + 1/2)\hbar\omega\), where \(v\) is the vibrational quantum number (0, 1, 2, ...), \(\hbar\) is the reduced Planck constant, and \(\omega\) is the angular frequency related to the stretching frequency.
For the ground vibrational state (\(v=0\)), the zero-point energy is \(E_0 = (0 + 1/2)\hbar\omega = \frac{1}{2}\hbar\omega\).
The relationship between the equilibrium dissociation energy \(D_e\) and the spectroscopic dissociation energy \(D_0\) is given by:
\[ D_e = D_0 + E_0 \] \[ D_e = D_0 + \frac{1}{2}\hbar\omega \]The stretching frequency of the molecule corresponds to an energy of 0.5 eV. This value represents \(\hbar\omega\).
Given values:
- Equilibrium dissociation energy, \(D_e = 4.75\) eV
- Energy corresponding to stretching frequency, \(\hbar\omega = 0.5\) eV
Calculating Minimum Dissociation Energy
We need to find the minimum energy required to dissociate the molecule, which is the spectroscopic dissociation energy \(D_0\).
First, calculate the zero-point energy (\(E_0\)):
\[ E_0 = \frac{1}{2}\hbar\omega \] \[ E_0 = \frac{1}{2} \times 0.5 \text{ eV} \] \[ E_0 = 0.25 \text{ eV} \]Now, use the relationship \(D_e = D_0 + E_0\) to find \(D_0\):
\[ D_0 = D_e - E_0 \]Substitute the given values:
\[ D_0 = 4.75 \text{ eV} - 0.25 \text{ eV} \] \[ D_0 = 4.50 \text{ eV} \]Thus, the minimum energy required to dissociate the diatomic molecule from its ground vibrational state is 4.50 eV.
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