AP Physics 1 Exam Question - Quantum Theory

A student performs an experiment to determine the energy levels of electrons in a hydrogen atom. The student finds that the lowest energy level, n=1, has an energy of -13.6 eV. The student wants to calculate the energy of an electron in the n=3 energy level.

a) What is the formula for the energy levels of electrons in a hydrogen atom? b) Calculate the energy difference between the n=3 and n=1 energy levels. c) Calculate the energy of an electron in the n=3 energy level. d) Express your final answer in joules.

Answer with Step-by-Step Detailed Explanation:

a) The formula for the energy levels of electrons in a hydrogen atom can be given by the equation:

E = -13.6 eV/n^2

Where E is the energy of the electron, -13.6 eV is the constant value for the lowest energy level (n=1), and n^2 represents the energy level.

b) To calculate the energy difference between the n=3 and n=1 energy levels, we need to find the difference between the energies at these two levels.

For n=1, we have: E₁ = -13.6 eV / 1^2 = -13.6 eV

For n=3, we have: E₃ = -13.6 eV / 3^2 = -13.6 eV / 9 = -1.511 eV

The energy difference between these two levels is:

ΔE = E₃ - E₁ = (-1.511 eV) - (-13.6 eV) = 12.089 eV

c) To calculate the energy of an electron in the n=3 energy level, we use the formula mentioned earlier:

E₃ = -13.6 eV / 3^2 = -13.6 eV / 9 = -1.511 eV

d) To express the final answer in joules, we need to convert the energy from electron volts (eV) to joules (J).

1 eV = 1.6 × 10^-19 J

Converting the energy from eV to J:

E₃ (in joules) = -1.511 eV × (1.6 × 10^-19 J/eV) = -2.4184 × 10^-19 J

Therefore, the energy of an electron in the n=3 energy level is approximately -2.4184 × 10^-19 J.