A simple picture of an H₂ molecule sharing two electrons is shown in Fig. 40–56. We assume the electrons are symmetrically located between the two protons, which are separated by r₀ = 0.074 nm. Determine analytically the value of d that gives minimum U (greatest stability).

Verified step by step guidance

1. Step 1: Understand the problem setup. The H₂ molecule consists of two protons separated by a distance r₀ = 0.074 nm, and two electrons symmetrically located above and below the midpoint of the proton-proton axis at a distance d. The goal is to find the value of d that minimizes the potential energy U of the system.
2. Step 2: Write the expression for the total potential energy U of the system. The potential energy includes contributions from: (1) the electron-proton attraction, (2) the electron-electron repulsion, and (3) the proton-proton repulsion. Use Coulomb's law for each interaction: U = k_e * (q₁q₂ / r), where k_e is Coulomb's constant.
3. Step 3: Calculate the distances involved. The distance between each electron and its nearest proton is √(d² + (r₀/2)²). The distance between the two electrons is 2d. The distance between the two protons is r₀.
4. Step 4: Substitute these distances into the potential energy expression. The total potential energy U can be written as: U = -2 * k_e * (e² / √(d² + (r₀/2)²)) + k_e * (e² / 2d) + k_e * (e² / r₀), where the negative term represents attraction and the positive terms represent repulsion.
5. Step 5: Minimize U with respect to d. To find the value of d that minimizes U, take the derivative of U with respect to d, set it equal to zero, and solve for d. This involves calculus and careful algebraic manipulation.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged particles. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is crucial for understanding the interactions between the protons and electrons in the H₂ molecule.

Potential Energy in Electrostatics

The potential energy (U) in an electrostatic system is determined by the positions of charged particles. For two charges, it is given by U = k * (q₁ * q₂) / r, where k is Coulomb's constant, q₁ and q₂ are the charges, and r is the distance between them. Minimizing this potential energy leads to the most stable configuration of the system.

Equilibrium Position

The equilibrium position in a molecular system is where the net force acting on the particles is zero, leading to stability. In the context of the H₂ molecule, this occurs when the attractive forces between the electrons and protons balance the repulsive forces between the protons. Finding the distance 'd' that minimizes potential energy corresponds to identifying this equilibrium position.