Question

In the Paschen series, \(\mathbf{n}_{\text {lo }}=3\). Calculate the longest wavelength possible for a transition in this series.

Short answer

Answer: The longest wavelength possible for a transition in the Paschen series of the Hydrogen Spectrum is approximately 1.875 x 10^-6 meters.

Step-by-step solution

Identify the given values

The given values in the question are: - The low energy level(n_lo) for the Paschen series is n_lo = 3. - The Rydberg constant for Hydrogen (R_z) is approximately 1.097373 x 10^7 m^-1.

Determine the energy level the electron falls from

When we are asked for the longest wavelength, this translates to the smallest energy difference, which will be from the nearest higher energy level. So this will be from n=4 to n=3. Hence, n_hi = 4.

Calculate the inverse of the wavelength

By rearranging the Rydberg formula, the inverse of the wavelength can be found. The formula we are using is: 1/λ = R_z * (1/n_lo^2 - 1/n_hi^2) By substituting the given values, we have: 1/λ = 1.097373 x 10^7 * (1/3^2 - 1/4^2)

Calculate the inverse of the wavelength

We now just need to compute the expression calculated in the last step. Doing the calculation for each part of the expression within the brackets and then multiplying by the Rydberg constant, we get: 1/λ = 1.097373 x 10^7 * (0.11111 - 0.0625) = 1.097373 x 10^7 * 0.048611 ≈ 533479 m^-1

Find the wavelength

The wavelength is the reciprocal of the value calculated above. By computing the reciprocal of the above computed value, we get: λ = 1 / 533479 ≈ 1.875 x 10^-6 meters, which is the longest wavelength possible for a transition in the Paschen series of the Hydrogen Spectrum.

Key concepts

Rydberg Formula

The Rydberg formula is a mathematical equation used to predict the wavelength of light resulting from an electron moving between energy levels in an atom. Specifically, for hydrogen, this formula is expressed as:

\begin{align*} \frac{1}{\lambda} = R_\text{H} \left( \frac{1}{n_\text{lo}^2} - \frac{1}{n_\text{hi}^2} \right)\thanks{align*}
Here, \( \lambda \) represents the wavelength of the emitted light, \( R_\text{H} \) is the Rydberg constant for hydrogen, \( n_\text{lo} \) is the lower energy level, and \( n_\text{hi} \) is the higher energy level from where the electron is transitioning. The Rydberg constant has a value of approximately 1.097373 x 10^7 m^-1 for hydrogen. When solving problems using the Rydberg formula, careful substitution of values and arithmetic is necessary to arrive at the correct result.

• The formula helps in calculating the wavelengths of not just the visible spectrum but also ultraviolet and infrared emissions.
• To find the longest wavelength in a series, we must consider the smallest energy transition, which is between adjacent energy levels.
• The concept of the wavelength being inversely proportional to the energy difference between levels is a key understanding when using the Rydberg formula.

Energy Level Transitions

In an atom, electrons occupy certain energy levels, or orbits. Energy level transitions occur when an electron jumps from one level to another, either absorbing or emitting energy in the form of photons. The energy levels are quantized, meaning they have specific values and the electron can only occupy these certain levels, not the space in between.

• Each energy level is denoted by a principal quantum number \( n \) such as \( n=1,2,3 \), and so on.
• Transitions to lower energy levels (\( n_\text{hi} \) to \( n_\text{lo} \)) result in photon emission, whereas transitions to higher levels (\( n_\text{lo} \) to \( n_\text{hi} \)) absorb energy.
• The energy difference between levels determines the photon's wavelength; the greater the difference, the shorter the wavelength.
• The smallest energy difference, and thus the longest wavelength, will be when the electron falls from the nearest higher level to the specified lower level, as demonstrated in the exercise.

When electrons make a transition, they follow specific selection rules which dictate the possible changes in their quantum states, ensuring the conservation of energy within the system.

Hydrogen Spectrum

The hydrogen spectrum is the set of electromagnetic emissions produced when the electrons in a hydrogen atom make transitions between energy levels. This spectrum exists due to the unique quantized energy levels of electrons in the hydrogen atom.

Series in the Hydrogen Spectrum

The spectrum consists of several series named after the scientists who discovered them:

• The Lyman series encompasses transitions to the ground state \( n=1 \), and falls in the ultraviolet range.
• The Balmer series includes transitions to the first excited state \( n=2 \), resulting in visible light emissions.
• The Paschen series involves transitions to the second excited state \( n=3 \), being mainly in the infrared range, which is the focus of the original exercise.

Transitions in higher series like Brackett, Pfund, and Humphreys encompass larger \( n \) values and are also in the infrared range.
The wavelengths within these series can be predicted using the Rydberg formula, where the longest wavelength corresponds to an electron falling from the level just above the base level of the series. For the Paschen series, this is a fall from \( n=4 \) to \( n=3 \), as correctly calculated in the step-by-step solution of the exercise provided.