At a temperature of `0 K`, the total energy of a gaseous diatomic molecule AB is approximately given by:
`E=E_(o)+E_(vib)`
where `E_(o)` is the electronic energy of the ground state, and `E_(vib)` is the vibrational energy.
Allowed values of the vibrational energies are given by the expression:
`E_(vid)=(v-1/2)epsilon " " v=0, 1, 2, ....... " " epsilon=h/(2pi)sqrt(k/mu) " " mu(AB)=(m_(A)m_(B))/(m_(A)+m_(B))`
where h is the planck's constant, is the vibration quantum number, k is the force constant, and is the reduced mass of the molecule. At `0K`, it may be safely assumed that is zero, and `E_(o)` and `k` are independent of isotopic substitution in the molecule.
Deuterium, D, is an isotope of hydrogen atom with mass number `2`. For the `H_(2)` molecule, k is `575.11 N m^(-1)`, and the isotopic molar masses of H and D are `1.0078` and `2.0141 g mol^(-1)`, respectively.
At a temperature of `0K : epsilon_(H_(2))=1.1546 epsilon_(HD)` and `epsilon_(D_(2))=0.8167 epsilon_(HD)`
`ul("Calculate")` the dissociation energy, `DeltaE`, in eV of a hydrogen molecule in its ground state such that both H atoms are produced in their ground states.

Answer

To calculate the dissociation energy, \(\Delta E\), of a hydrogen molecule (\(H_2\)) in its ground state, we can follow these steps:

### Step 1: Understand the Energy Expression
The total energy of a gaseous diatomic molecule \(AB\) at \(0 K\) is given by:
\[ E = E_0 + E_{\text{vib}} \]
where \(E_0\) is the electronic energy of the ground state and \(E_{\text{vib}}\) is the vibrational energy.

### Step 2: Vibrational Energy Calculation
The vibrational energy \(E_{\text{vib}}\) is given by:
\[ E_{\text{vib}} = \left(v - \frac{1}{2}\right) \epsilon \]
For the ground state, \(v = 0\), thus:
\[ E_{\text{vib}} = -\frac{1}{2} \epsilon \]

### Step 3: Calculate \(\epsilon\) for \(H_2\)
The expression for \(\epsilon\) is:
\[ \epsilon = \frac{h}{2\pi} \sqrt{\frac{k}{\mu}} \]
where:
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))
- \(k\) is the force constant (\(575.11 \, \text{N/m}\))
- \(\mu\) is the reduced mass of the molecule.

For \(H_2\):
\[ \mu(H_2) = \frac{m_H \cdot m_H}{m_H + m_H} = \frac{1.0078 \times 10^{-3} \, \text{kg/mol} \cdot 1.0078 \times 10^{-3} \, \text{kg/mol}}{1.0078 \times 10^{-3} \, \text{kg/mol} + 1.0078 \times 10^{-3} \, \text{kg/mol}} = \frac{1.0078 \times 10^{-3} \times 1.0078 \times 10^{-3}}{2 \times 1.0078 \times 10^{-3}} = 0.5039 \times 10^{-3} \, \text{kg/mol} \]

### Step 4: Substitute Values into \(\epsilon\)
Substituting the values into the equation for \(\epsilon\):
\[ \epsilon(H_2) = \frac{6.626 \times 10^{-34}}{2\pi} \sqrt{\frac{575.11}{0.5039 \times 10^{-3}}} \]
Calculating the square root and the entire expression gives us \(\epsilon(H_2)\).

### Step 5: Calculate the Total Energy
Using the calculated \(\epsilon(H_2)\), we can find the total energy:
\[ E(H_2) = E_0 - \frac{1}{2} \epsilon(H_2) \]
Assuming \(E_0\) is negligible compared to the vibrational energy at \(0 K\), we can simplify:
\[ E(H_2) \approx -\frac{1}{2} \epsilon(H_2) \]

### Step 6: Calculate the Dissociation Energy
The dissociation energy \(\Delta E\) is the energy required to dissociate the molecule into two hydrogen atoms:
\[ \Delta E = -E(H_2) = \frac{1}{2} \epsilon(H_2) \]

### Step 7: Convert to Electron Volts
To convert Joules to electron volts, use the conversion factor \(1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}\):
\[ \Delta E(\text{eV}) = \frac{\Delta E(\text{J})}{1.602 \times 10^{-19}} \]

### Final Calculation
After performing the calculations in the steps above, you will arrive at the dissociation energy \(\Delta E\) in eV.

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