INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. a. What are the four lowest energy levels of the electron?

Verified step by step guidance

1. Identify the problem as a quantum mechanics scenario involving a particle (electron) in a one-dimensional box (rigid box). The energy levels of a particle in a box are quantized and can be calculated using the formula for the energy levels of a particle in a one-dimensional box: $E_n = \frac{n^2 h^2}{8mL^2}$, where $n$ is the principal quantum number, $h$ is Planck's constant, $m$ is the mass of the electron, and $L$ is the length of the box.
2. Substitute the given values into the formula. Use the length of the box $L = 0.100 \, \text{nm}$, Planck's constant $h = 6.626 \times 10^{-34} \, \text{Js}$, and the mass of the electron $m = 9.109 \times 10^{-31} \, \text{kg}$.
3. Calculate the energy for the first four quantum states ($n=1, 2, 3, 4$) by substituting these values of $n$ into the energy formula.
4. Simplify the calculations by using appropriate units and constants. Convert the length from nanometers to meters by multiplying by $10^{-9}$.
5. Interpret the results to understand the spacing and relative sizes of the energy levels, noting how the energy increases with increasing values of $n$. This will give insight into the electron's behavior in the box and how the confinement affects its energy states.

Key Concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces the concept of quantization, where certain properties, such as energy, can only take on discrete values. This framework is essential for understanding how electrons occupy specific energy levels in an atom.

Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics problem that simplifies the behavior of a particle confined to a rigid, one-dimensional space. In this model, the particle can only occupy certain energy levels determined by the size of the box and the boundary conditions. This concept is crucial for calculating the energy levels of the electron in the given atom.

Energy Levels and Quantum Numbers

Energy levels in quantum mechanics are quantized states that an electron can occupy, characterized by quantum numbers. For a particle in a box, the energy levels are determined by the formula E_n = n^2 * (h^2 / (8mL^2)), where n is a positive integer (quantum number), h is Planck's constant, m is the mass of the electron, and L is the length of the box. Understanding this relationship allows us to calculate the specific energy levels for the electron in the atom.