Problem 14 - University Physics with Modern Physics

Question

When a hydrogen atom undergoes a transition from the \(n=2\) to the \(n=1\) level, a photon with \(\lambda=122 \mathrm{nm}\) is emitted. (a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the \(n=2\) to \(n=1\) transition to correspond to emission of a photon of this energy? (b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (Hint: Compare the spacing between adjacent energy levels as a function of \(n . )\)

Step-by-Step Solution

Answer

(a) The box width is 0.11 nm. (b) Ground-state energy is 9.37 eV, different from hydrogen's 13.6 eV. (c) A 1D box is not a good hydrogen model.

Step 1: Determine Energy of Emitted Photon

First, we need to determine the energy of the emitted photon. Use the formula for energy of a photon \(E = \frac{hc}{\lambda}\), where \(h = 6.626 \times 10^{-34}\ \text{J}\cdot\text{s}\) and \(c = 3 \times 10^8\ \text{m/s}\). With \(\lambda=122 \text{ nm} = 122 \times 10^{-9} \text{ m}\), calculate \(E\). \[E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{122 \times 10^{-9}}\]\ Calculate this to find \(E = 1.63 \times 10^{-18} \text{ J}\).

Step 2: Relate Energy to Particle in a Box Model

The energy levels \(E_n\) in a one-dimensional box are given by the formula \(E_n = \frac{n^2 h^2}{8mL^2}\), where \(m\) is the electron's mass \(9.109 \times 10^{-31}\ \text{kg}\), \(h\) is Planck's constant, and \(L\) is the width of the box. For the transition \(n=2\) to \(n=1\), the change in energy \(\Delta E\) is:\[\Delta E = E_2 - E_1 = \frac{3h^2}{8mL^2}\]\ Set \(\Delta E = 1.63 \times 10^{-18} \text{ J}\) to solve for \(L\).

Step 3: Solve for Width of the Box

Rearrange the equation from Step 2 to solve for \(L\): \[L^2 = \frac{3h^2}{8m\Delta E}\]Simplify and solve for \(L\):\[L = \sqrt{\frac{3 \times (6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times 1.63 \times 10^{-18}}}\]Calculate \(L\) to find that \(L \approx 1.10 \times 10^{-10} \text{ m}\) or 0.11 nm.

Step 4: Calculate Ground-State Energy

Now, calculate the ground-state energy \(E_1\) using \(E_1 = \frac{h^2}{8mL^2}\). Substitute \(L = 1.10 \times 10^{-10} \text{ m}\) into the equation:\[E_1 = \frac{(6.626 \times 10^{-34})^2}{8 \times 9.109 \times 10^{-31} \times (1.10 \times 10^{-10})^2}\]Perform the calculation to find \(E_1 \approx 1.50 \times 10^{-18} \text{ J}\). Convert to electron volts: \(E_1 \approx 9.37 \text{ eV}\).

Step 5: Compare with Hydrogen Atom Energy

In the Bohr model, the ground-state energy of a hydrogen atom is \(-13.6 \text{ eV}\). The calculated value \(9.37 \text{ eV}\) is different, demonstrating that the one-dimensional box model does not accurately represent the hydrogen atom.

Step 6: Evaluate the Model

As energy levels in the particle in a box model increase quadratically (\(n^2)\) and the hydrogen atom energy levels increase inversely with \(n^2\), the one-dimensional box is not a good model for hydrogen atoms. The difference in energy relationships and level spacings confirm this.

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom, consisting of a single electron orbiting a solitary proton. This seemingly simple structure has profound implications in quantum mechanics, and it provides a foundational model for atomic physics. One key feature of the hydrogen atom is its discrete energy levels. These levels are determined by quantum numbers and have implications for the movement of the electron around the nucleus. The energy levels are quantized, meaning the electron can only occupy certain allowed states.

Energy Levels

Energy levels are a critical concept in understanding the behavior of electrons in atoms. Every atom has specific, quantized energy states. For hydrogen, these energy levels were first described by the Bohr model, where the electron can only exist in orbits with specific energies. The energy difference between these levels can be calculated using a formula relating to the principal quantum number, denoted by \(n\).

• These levels follow the relationship \(E_n = - rac{13.6 ext{ eV}}{n^2}\) for a hydrogen atom.
• Electrons can transition between these levels by absorbing or emitting photons.

The difference in energy corresponds to the frequency of the emitted or absorbed photons. In the exercise, understanding these energy levels helps in comparing the one-dimensional box model to the realistic atom.
This comparison shows the shortcomings of simplified models.

Particle in a Box

The concept of a particle in a one-dimensional box is a fundamental model used in quantum mechanics. It provides insights into quantum states and wave functions.

• In this model, a particle, such as an electron, is assumed to move freely within a confined space, defined by a box.
• Within this space, the energy levels are determined by the particle's quantum state, quantified as \(E_n = \frac{n^2 h^2}{8mL^2}\).

The model assumes infinite potential barriers at the boundaries of the box, meaning the particle cannot exist outside it.
For a hydrogen atom, simplifying as a particle in a box allows for direct energy level calculations, but the assumptions often fall short of accurately modeling atomic interactions.

Photon Emission

Photon emission occurs when an electron transitions from a higher energy level to a lower one, releasing energy in the form of light (a photon). This principle is central to many physical phenomena, including spectroscopy and quantum mechanics applications.
When dealing with a hydrogen atom or any quantum system:

• A transition from a higher to a lower energy level releases a photon with an energy equivalent to the difference in energy levels.
• The wavelength of this emitted photon can be calculated using the formula \(E = \frac{hc}{\lambda}\), where \(E\) is the energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength.

Understanding photon emissions allows us to deduce information about an atom's energy structure and the nature of electron transitions. This concept is pivotal in interpreting experimental data in quantum mechanics.