What wavelength photon does a hydrogen atom emit in a 200→199 transition?

Verified step by step guidance

1. Step 1: Understand the problem. The hydrogen atom emits a photon when transitioning from a higher energy level (n=200) to a lower energy level (n=199). The energy difference between these levels determines the wavelength of the emitted photon.
2. Step 2: Use the Rydberg formula to calculate the energy difference between the two levels. The formula for the energy of a hydrogen atom's electron in a given energy level is: $E=-\frac{R_H}{h^2}/n^2$, where RH is the Rydberg constant, h2 is Planck's constant squared, and n is the principal quantum number.
3. Step 3: Calculate the energy difference between the two levels using the formula: $\mathrm{\Delta E}=-E_{200}-E_{199}$. Substitute the values of n = 200 and n = 199 into the energy formula to find the energy difference.
4. Step 4: Relate the energy difference to the wavelength of the emitted photon using the equation: $\lambda =\frac{\mathrm{hc}}{\mathrm{\Delta E}}$, where h is Planck's constant, c is the speed of light, and ΔE is the energy difference calculated in Step 3.
5. Step 5: Substitute the known constants (h, c, and RH) and the calculated energy difference into the wavelength formula to determine the wavelength of the emitted photon. Ensure units are consistent throughout the calculation.

Key Concepts

Photon Emission

Photon emission occurs when an electron in an atom transitions from a higher energy level to a lower one, releasing energy in the form of a photon. The energy of the emitted photon corresponds to the difference in energy between the two levels, which can be calculated using the formula E = hf, where E is energy, h is Planck's constant, and f is the frequency of the photon.

Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, which are quantized. The energy levels can be described by the formula E_n = -13.6 eV/n², where n is the principal quantum number. The transition from one level to another, such as from n=200 to n=199, results in the emission of a photon with a specific wavelength determined by the energy difference between these levels.

Wavelength Calculation

The wavelength of a photon can be calculated using the relationship between energy and wavelength given by the equation λ = hc/E, where λ is the wavelength, h is Planck's constant, c is the speed of light, and E is the energy of the photon. By determining the energy difference for the transition and applying this formula, one can find the wavelength of the emitted photon.