The figure below is the plot of potential energy versus internuclear distance (d) of H2 molecule in

Q: Why do we ignore the cross-attraction between Nucleus 1 and Electron 2?

The problem's phrasing 'at which repulsions are absent' is a classic simplification trick. By removing the complex multi-body interactions (both repulsions and cross-attractions that balance them out in this simplified model), we are left to calculate the energy of two isolated hydrogen atoms.

Q: Why is the potential energy of an H atom calculated using just Coulomb's law?

In the Bohr model, the potential energy of the electron is purely electrostatic. While the total energy is half of the potential energy (due to kinetic energy), the question specifically asks for the *potential energy* component alone.

Q: Why do we multiply by 2 at the very end?

The question asks for the potential energy of the $H_2$ molecule. A single molecule consists of two hydrogen atoms. Since we calculated the energy for one mole of individual H atoms, we must double it to represent one mole of $H_2$ molecules.

Q: What is the physical significance of $d_0$ in this context?

$d_0$ is the equilibrium bond length where the actual molecule is most stable. However, in this hypothetical scenario, it merely serves as the reference distance where we evaluate the isolated attractive terms, assuming all other forces magically vanish.

Q: Why is the final answer so much larger than typical bond energies?

Real bond energies (like -436 kJ/mol for $H_2$) include the massive positive (destabilizing) contributions from nucleus-nucleus and electron-electron repulsions, as well as kinetic energy changes. By ignoring repulsions, we get an artificially huge negative potential energy.

Question

The figure below is the plot of potential energy versus internuclear distance ($d$) of H$_2$ molecule in the electronic ground state. What is the value of the net potential energy $E_0$ (as indicated in the figure) in kJ mol$^{-1}$, for $d=d_0$ at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of H atom is taken as zero when its electron and the nucleus are infinitely far apart.
[Use Avogadro constant as $6.023 	imes 10^{23} 	ext{ mol}^{-1}$.]

Sigma Percentile · Accepted Answer

### Decoding the Potential Energy Curve

When two hydrogen atoms approach each other to form a molecule, their potential energy changes dramatically. The graph provided in the question is a classic representation of this phenomenon. At large distances, the atoms barely interact, and the potential energy is effectively zero. As they move closer, the attractive forces between the electrons and the nuclei begin to dominate, causing the potential energy to drop and the system to become more stable.

Eventually, the atoms reach an optimal distance known as the equilibrium bond length, denoted as $d_0$. At this exact point, the attractive forces are perfectly balanced by the repulsive forces between the two positively charged nuclei and the two negatively charged electrons. The energy at this minimum point is $E_0$, which corresponds to the bond energy of the molecule.

### The "Absent Repulsion" Trick

Here is where the problem introduces a fascinating, hypothetical twist. It asks us to calculate the net potential energy $E_0$ at the bond length $d_0$, but under the strict condition that **both the electron-electron repulsion and the nucleus-nucleus repulsion energies are completely absent**.

Imagine what happens when you magically turn off all repulsive forces in a molecule. The complex multi-body interactions vanish. What remains is purely the electrostatic attraction between each proton and its respective electron. In this simplified, hypothetical scenario, the hydrogen molecule essentially behaves as two independent, non-interacting hydrogen atoms sitting next to each other.

### Calculating the Electrostatic Attraction

Since the system now acts like two isolated atoms, our strategy is straightforward: calculate the potential energy of a single hydrogen atom and multiply it by two. According to the Bohr model, the potential energy of an electron in the first orbit is purely electrostatic and is given by Coulomb's law:

$$U_{	ext{atom}} = \frac{-K \cdot e^2}{r}$$

Let's plug in the standard physical constants. The Coulomb constant $K$ is $9 	imes 10^9 	ext{ N m}^2/	ext{C}^2$, the elementary charge $e$ is $1.6 	imes 10^{-19} 	ext{ C}$, and the Bohr radius $r$ is $0.529 	imes 10^{-10} 	ext{ m}$.

$$U_{	ext{atom}} = \frac{-(9 	imes 10^9)(1.6 	imes 10^{-19})^2}{0.529 	imes 10^{-10}} 	ext{ J}$$

Executing this calculation yields the potential energy for one single atom:

$$U_{	ext{atom}} = -4.355 	imes 10^{-18} 	ext{ J}$$

To make the numbers more manageable and align with standard chemical units, we convert Joules to kilo-Joules:

$$U_{	ext{atom}} = -4.355 	imes 10^{-21} 	ext{ kJ}$$

### Scaling up to a Mole

The question specifically requests the energy in $	ext{kJ mol}^{-1}$. Therefore, we must scale our single-atom energy up to a macroscopic level by multiplying it by Avogadro's constant ($N_A = 6.023 	imes 10^{23} 	ext{ mol}^{-1}$).

$$U_{	ext{per mole}} = (-4.355 	imes 10^{-21} 	ext{ kJ}) 	imes (6.023 	imes 10^{23} 	ext{ mol}^{-1})$$

$$U_{	ext{per mole}} = -2623.25 	ext{ kJ/mol}$$

Finally, we must remember that a hydrogen molecule ($H_2$) consists of exactly two hydrogen atoms. To find the total hypothetical potential energy of the molecule, we simply double the molar energy we just calculated:

$$E_0 = 2 	imes (-2623.25 	ext{ kJ/mol}) = -5246.50 	ext{ kJ/mol}$$

This massive negative value is our final answer. It serves as a powerful reminder of why real molecules have much lower bond energies (around $-436 	ext{ kJ/mol}$ for $H_2$). In reality, the repulsive forces we ignored are incredibly strong and significantly destabilize the system, counteracting much of the attractive potential energy.

Enter Numerical Value:

Visualized Solution

Visualizing the Molecule

The Hypothetical Condition

Net Potential Energy

Electrostatic Potential Energy

Substituting Values

Energy per Atom

Energy per Mole

Total Energy of Molecule

Reality Check

The Sigma Insight: Bohr's Model

Decoding the Potential Energy Curve

The "Absent Repulsion" Trick

Calculating the Electrostatic Attraction

Scaling up to a Mole

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Enter Numerical Value:

Visualized Solution

Visualizing the Molecule

The Hypothetical Condition

Net Potential Energy

Electrostatic Potential Energy

Substituting Values

Energy per Atom

Energy per Mole

Total Energy of Molecule

Reality Check

The Sigma Insight: Bohr's Model

Decoding the Potential Energy Curve

The "Absent Repulsion" Trick

Calculating the Electrostatic Attraction

Scaling up to a Mole

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