Question 1

Consider two protons separated by a distance $$r. $$The electrostatic potential energy between them is given by $$U(r) = +\frac{k e^2$}{r}. $$What does the gradient (slope) of this potential energy curve represent physically?

Accepted Answer

The magnitude of the repulsive force between the protons.

Question 2

When a single electron is added to the two-proton system (forming $$\mathrm{H}_2^+$$), where is the electrostatic potential energy most negative for the electron?

Accepted Answer

When the electron is midway between the two protons.

Question 3

Calculate the total electrostatic potential energy of the $$\mathrm{H}_2^+$$ system when the electron is exactly midway between the two protons separated by distance $$r. $$Which expression is correct?

Accepted Answer

$$U = -3 \frac{k e^2$}{r}$$

Question 4

Using the Heisenberg Uncertainty Principle, estimate the minimum kinetic energy of an electron confined between two protons a distance $$r$$ apart (as in $$\mathrm{H}_2^+). $$Which formula is correct?

Accepted Answer

$$K_{min} = \frac{\$$hbar^2$$}{8 m_e r^2$}$$

Question 5

The total energy of the $$\mathrm{H}_2^+$$ system as a function of proton separation $$r is $$given by $$E_{tot}(r) = -3 \frac{k e^2$}{r} + \frac{\$$hbar^2$$}{8 m_e r^2$}. $$What does the minimum of this function represent physically?

Accepted Answer

The most stable (bound) configuration of the molecule.

Question 6

For the $$\mathrm{H}_2^+$$ ion, the equilibrium separation $$r^*$$ and minimum energy $$E_{min}$$ are given by $$r^* = \frac{1}{12} \frac{\$$hbar^2$$}{k e^2$ m_e}$$ and $$E_{min} = -18 \frac{(k e^2$)^2 m_e$$}{\$$hbar^2$$}$$. How do these compare to the Bohr radius a_0$ and the ground state energy of the hydrogen atom?

Accepted Answer

They are of the same order of magnitude as $$a_0$$ and the hydrogen ground state energy.

Question 7

When a second electron is added to $$\mathrm{H}_2^+ to $$form $$\mathrm{H}_2$$, why does the second electron enter the same quantum state as the first (except for spin)?

Accepted Answer

Because of the Pauli exclusion principle, which allows two electrons with opposite spins in the same state.

Question 8

Why is the $$\mathrm{H}_2$$ molecule more tightly bound (smaller $$r^*$$ and more negative $$E_{min}) $$than $$\mathrm{H}_2^+$$?

Accepted Answer

Because the second electron increases the attractive potential energy.

Question 9

Why can we treat the protons as classical particles but must treat the electrons quantum mechanically in the $$\mathrm{H}_2$$ molecule?

Accepted Answer

Because electrons have much smaller mass and move much more rapidly than protons.

Question 10

For small displacements from equilibrium, the total energy of the $$\mathrm{H}_2$$ molecule can be approximated as $$E_{tot}(r) \approx E_{min} + \frac{1}{2} k (r - r^*)^2$. What does the parameter k$ represent in this context?

Accepted Answer

The effective spring constant for the vibrational motion of the protons.

Question 11

Quantum mechanically, the allowed vibrational energy levels of the $$\mathrm{H}_2$$ molecule are given by $$E_{n, vib} = \left(n + \frac{1}{2}\right) \hbar \sqrt{\frac{k}{\mu}}$$, where $$\mu is $$the reduced mass. What is the physical significance of the $$\frac{1}{2}$$ term?

Accepted Answer

It represents the zero-point energy, the lowest possible energy even at $$n=0.$$

Question 12

The rotational energy levels of a diatomic molecule are given by $$E_{\ell, rot} = \ell(\ell+1) \frac{\$$hbar^2$$}{2I}$$, where $$I is $$the moment of inertia. For which values of $$\ell$$ are these energy levels allowed?

Accepted Answer

$$\ell = 0, 1, 2, ...$$

Question 13

Which of the following best explains why the vibrational energy levels of a diatomic molecule are evenly spaced only for small oscillations?

Accepted Answer

Because for small displacements, the potential energy can be approximated as quadratic (harmonic oscillator).

Question 14

In molecular spectra, why are rotational transitions typically observed in the far infrared, while vibrational transitions are in the near infrared?

Accepted Answer

Because rotational transitions involve much smaller energy differences than vibrational transitions.

Question 15

If the reduced mass $$\mu of a $$diatomic molecule increases, what happens to the vibrational energy level spacing $$\Delta E = \hbar \sqrt{\frac{k}{\mu}}$$?

Accepted Answer

The spacing decreases.

Question 16

Which of the following statements about the quantum mechanical treatment of the $$\mathrm{H}_2$$ molecule is correct?

Accepted Answer

The electron probability distribution is concentrated between the two protons.

Question 17

Which of the following best describes the analogy between the narrowing of vibrational energy level spacing at higher energies in molecules and the hydrogen atom?

Accepted Answer

In both cases, the energy levels become more closely spaced at higher energies.

Question 18

A photon is emitted when a molecule transitions from a higher to a lower vibrational energy level. What is the energy of the emitted photon?

Accepted Answer

The difference in total energy between the two vibrational levels.

Step-by-Step Guidance: Molecular Bonds and Molecular Spectra (H₂ Molecule)

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