Question

Model a hydrogen atom as an electron in a cubical box with side length \(L\). Set the value of \(L\) so that the volume of the box equals the volume of a sphere of radius \(a\) = 5.29 \(\times\) 10\(^{-11}\) m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

Short answer

Set \(L = 3.94 \times 10^{-11}\) m. The energy separation \(\Delta E\) is approximately smaller than the Bohr model's 10.2 eV, indicating the box model's less quantized levels.

Step-by-step solution

Find the Volume of the Sphere

To start, calculate the volume of the sphere using the Bohr radius. The formula for the volume of a sphere is \[V = \frac{4}{3} \pi a^3\]where \(a = 5.29 \times 10^{-11}\) m. Substituting in this value, we get\[V = \frac{4}{3} \pi (5.29 \times 10^{-11})^3 \approx 6.22 \times 10^{-31} \text{ m}^3.\]

Determine Side Length of Cube

Next, set the volume of the cube equal to the volume of the sphere. The volume of the cube is given by \(L^3\), so\[L^3 = 6.22 \times 10^{-31} \text{ m}^3.\]Solving for \(L\), we get\[L = (6.22 \times 10^{-31})^{1/3} \approx 3.94 \times 10^{-11} \text{ m}.\]

Calculate Energy Levels in the Box

For a particle in a 3D cubical box, the energy levels are given by\[E_{n_x,n_y,n_z} = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)\]where \(n_x, n_y, n_z\) are quantum numbers, \(h\) is Planck's constant, and \(m\) is the mass of the electron. For the ground state, \(n_x = n_y = n_z = 1\). Thus,\[E_{1,1,1} = \frac{h^2}{8mL^2}(1^2 + 1^2 + 1^2) = \frac{3h^2}{8mL^2}.\]

Calculate First Excited State

The first excited state will have the quantum numbers such as \((n_x, n_y, n_z) = (1,1,2), (1,2,1), (2,1,1)\). Calculate for one case: \[E_{1,1,2} = \frac{h^2}{8mL^2}(1^2 + 1^2 + 2^2) = \frac{6h^2}{8mL^2}.\]

Determine Energy Separation

The energy separation between the ground and first excited states is\[\Delta E = E_{1,1,2} - E_{1,1,1} = \frac{6h^2}{8mL^2} - \frac{3h^2}{8mL^2} = \frac{3h^2}{8mL^2}.\]

Compare with Bohr Model

The energy separation in the Bohr model for a hydrogen atom is given by\[\Delta E_{Bohr} = 13.6 \text{ eV} \times \left(\frac{1}{1^2} - \frac{1}{2^2}\right) = 10.2 \text{ eV}.\] Compare this with the previously calculated \(\Delta E\).

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest atom in quantum mechanics, consisting of only one electron orbiting a nucleus with a single proton. In models of atomic structure, the hydrogen atom serves as a foundational building block, helping us to understand more complex systems. In this exercise, a hydrogen atom is modeled as an electron trapped in a tiny cubical box, mimicking the confinement of an electron around a nucleus. The side length of this cubical box is set to equal the size of the Bohr radius, a fundamental constant in quantum mechanics representing the most probable distance between the electron and the nucleus in a hydrogen atom when the system is in its ground state. This model helps illustrate concepts such as quantized energy levels and quantum confinement, which are pivotal in understanding how electrons behave at atomic scales.

Particle in a Box

The 'particle in a box' model is a key concept in quantum mechanics, explaining how particles like electrons behave when confined to a limited space. It's akin to imagining an electron trapped inside an impenetrable, 3D cube. Despite its simplicity, this concept offers significant insights into quantum systems. For a 3D cubical box, the quantum state of an electron is described by three quantum numbers, corresponding to its movement in each spatial dimension. These states are not continuous, but rather quantized, meaning electrons can only occupy specific energy levels. This model is useful in this exercise as it helps calculate the energy levels of an electron in a cubical box which mimics the electron in a hydrogen atom. By understanding these energy levels, we can compare them to more complex models like the Bohr model.

Bohr Model

The Bohr model provides a historical perspective on how scientists first understood atomic structure. Proposed by Niels Bohr in 1913, it depicts electrons orbiting the nucleus much like planets around the sun, but only in fixed paths or orbits with specific energies. In the hydrogen atom, these orbits correspond to quantized energy levels. Electrons can "jump" from one orbit to another when they gain or lose energy. The Bohr model efficiently describes the distribution of energy levels in hydrogen, predicting the spectral lines emitted when electrons transition between these levels. While more advanced quantum models have surpassed it, the Bohr model is still crucial for its simplicity and its vast correctness in predicting energy differences, like the energy separation between the ground and excited states, as seen in this exercise.

Energy Levels

Energy levels in quantum mechanics reflect the specific energies that an electron can possess when confined in a given space. In scenarios like a hydrogen atom or a particle in a box, these energy levels are discrete, not continuous.For a particle in a box, energy levels are determined by the formula: \[E_{n_x,n_y,n_z} = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)\]where \(n_x, n_y, n_z\) are quantum numbers, \(h\) is Planck's constant, and \(L\) is the length of the box's side.Understanding these levels helps in calculating energy separations, which are differences in energy between states. Comparing these with the Bohr model provides insights into the accuracy and limits of different quantum mechanical models.

Quantum Numbers

Quantum numbers are essential in describing the state of an electron in quantum mechanics. They define its position and energy within an atom or a confined region, like a particle in a box.In a hydrogen atom or a cubic box, the three main quantum numbers, \(n_x, n_y,\) and \(n_z\), indicate the electron's respective state in each spatial dimension within the box. These numbers are integral values representing the quantized nature of energy at the microscopic scale.In practical terms, quantum numbers enable the calculation of possible energy levels and prediction of electron behavior, underpinning much of quantum theory and its applications. Understanding how these numbers influence energy levels is critical for solving quantum systems and for exercises such as calculating energy separations in both the box model and the Bohr model.