What is the difference in the wavelengths emitted in a 199→2 transition and a 200→2 transition?

Verified step by step guidance

1. Identify the formula for the wavelength of light emitted during an electron transition in a hydrogen atom. This is given by the Rydberg formula: $\frac{1}{\lambda }=R(\frac{1}{n^2}-\frac{1}{m^2})$, where $\lambda$ is the wavelength, $R$ is the Rydberg constant, $n$ is the lower energy level, and $m$ is the higher energy level.
2. Substitute the values for the first transition (199 → 2) into the Rydberg formula. Here, $n$ = 2 and $m$ = 199. Calculate the term $\frac{1}{n^2}$ and $\frac{1}{m^2}$ separately, then find the difference.
3. Substitute the values for the second transition (200 → 2) into the Rydberg formula. Here, $n$ = 2 and $m$ = 200. Again, calculate the term $\frac{1}{n^2}$ and $\frac{1}{m^2}$ separately, then find the difference.
4. Using the results from the two transitions, calculate the difference in the wavelengths. Since the wavelength is inversely proportional to the difference in the terms, you can find the difference in wavelengths by taking the reciprocal of the differences calculated in the previous steps.
5. Express the final result as the difference in wavelengths between the two transitions. Ensure the units are consistent (e.g., nanometers or meters) and clearly state the difference in terms of the wavelength values.

Key Concepts

Wavelength and Frequency Relationship

The wavelength of light is inversely related to its frequency, as described by the equation c = λν, where c is the speed of light, λ is the wavelength, and ν is the frequency. This means that as the frequency of emitted light increases, its wavelength decreases, and vice versa. Understanding this relationship is crucial for analyzing transitions in atomic energy levels.

Energy Levels in Atoms

Atoms have quantized energy levels, meaning electrons can only occupy specific energy states. When an electron transitions between these levels, it emits or absorbs a photon with energy equal to the difference between the two levels. The energy of the photon is related to its wavelength, making it essential to know the energy levels involved in the transitions to calculate the emitted wavelengths.

Transition Notation

Transition notation, such as 199→2 and 200→2, indicates the initial and final energy levels of an electron in an atom. The first number represents the higher energy level from which the electron is transitioning, while the second number indicates the lower energy level. Understanding this notation is vital for determining the energy difference and, consequently, the wavelength of the emitted light during these transitions.