Question

Suppose an electron confined to a box emits photons. The longest wavelength that is registered is 500.0 nm. What is the width of the box?

Answer 1

The width of the box is approximately
`\(1.98 * 10^(-6)\)` meters.

Step-by-step explanation:

The longest wavelength
`(\(\lambda\))` emitted by an electron confined to a box corresponds to the fundamental frequency of the box. For a one-dimensional box, the relationship between the wavelength and the length of the box L is given by the formula
`\(\lambda = (2L)/(n)\)`, where n is the mode number.

In this case, the longest wavelength is given as 500.0 nm. To find the width of the box, we rearrange the formula to solve for L :
`\(L = (\lambda n)/(2)\)`. Since the box is likely in its fundamental mode n = 1, the width is
`\(L = \frac{500.0 \, \text{nm} * 1}{2} = 250.0 \, \text{nm} = 1.98 * 10^(-6)\)` meters.

Understanding the relationship between the confinement of particles in a box and the emitted wavelengths provides insights into quantum mechanics and the quantization of energy levels in such systems.

Quantum mechanics principles related to the quantization of energy levels in confined systems and the relationship between confinement dimensions and emitted wavelengths.

Answer 2

The width of the box (L) in which the electron is confined is 250.0 nanometers, or 250.0 nm.

To find the width of the box in which the electron is confined, you can use the concept of the particle-in-a-box model and relate the longest wavelength of emitted photons to the length of the box. The wavelength of emitted light is inversely proportional to the size of the box. The relationship is given by the following equation:

λ = 2L / n

Where:

- λ is the wavelength of the emitted light (500.0 nm in this case).

- L is the length (width) of the box.

- n is the mode number (the number of the standing waves inside the box).

In this case, we want to find L, so rearrange the equation:

L = (n * λ) / 2

Now, you need to determine the mode number (n). For the longest wavelength, n will be 1. Thus:

L = (1 * 500.0 nm) / 2

Convert nanometers (nm) to meters (m) for consistent units:

L = (1 * 500.0 × 10^(-9) m) / 2

L = 250.0 × 10^(-9) m

So, the width of the box (L) in which the electron is confined is 250.0 nanometers, or 250.0 nm.