Question

What is the wavelength of light emitted (in nm) when an electron transitions from the n=5 to the ground state of a particle in a box of length 168 pm? (do not include units in your answer. if you round during your calculation make sure to keep at least 3 decimal places. report your answer to 2 decimal places.)

Answer 1

To determine the wavelength of light emitted during an electron's transition from n=5 to the ground state in a 168 pm box, we apply the particle in a box model. The energy difference between levels is linked to the photon's energy, and the wavelength is calculated using Planck's constant, electron mass, speed of light, and box length.

Step-by-step explanation:

To calculate the wavelength of light emitted by an electron transitioning from n=5 to the ground state in a box of length 168 pm, we use the particle in a box model, which follows the principles of quantum mechanics.

The energy levels are quantized and given by the equation En = (n2h2)/(8mL2), 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.

For the electron transition from n=5 to n=1, the change in energy ∆E is E5 - E1. This energy difference corresponds to the photon's energy, Ephoton = ∆E = hν, where ν is the frequency of the emitted photon. To find the wavelength λ, we use the relationship λ = c/ν, where c is the speed of light. By inserting ∆E in terms of h, c, and λ, we can solve for the wavelength.

Using the values h = 6.626×10-34 J·s, m = 9.109×10-31 kg, L = 168×10-12 m, and c = 3.00×108 m/s, and solving the equation, we find the wavelength of the emitted photon in meters, which can then be converted to nanometers (1 m = 1×109 nm) to provide the answer in the desired units.