Question

The series of emission lines of the hydrogen atom for which \(n_{f}=3\) is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed. (b) Calculate the wavelengths of the first three lines in the Paschen series-those for which \(n_{i}=4,5,\) and 6

Short answer

(a) The Paschen series lines fall in the Infrared region of the electromagnetic spectrum with a range of wavelengths between \(\lambda_\text{min}\) and \(\lambda_\text{max}\). (b) The wavelengths of the first three lines in the Paschen series are: - For \(n_{i} = 4\): \(\lambda_1 \approx 1.87 \times 10^{-6} m\) - For \(n_{i} = 5\): \(\lambda_2 \approx 1.28 \times 10^{-6} m\) - For \(n_{i} = 6\): \(\lambda_3 \approx 1.09 \times 10^{-6} m\)

Step-by-step solution

Apply the Rydberg Formula

The Rydberg formula for the hydrogen atom is: \[\frac{1}{\lambda} = R_H \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right)\] Where, - \(\lambda\) is the wavelength of the emission line, - \(R_H\) is the Rydberg constant for hydrogen, which is approximately \(1.097 \times 10^7 m^{-1}\), - \(n_{f}\) is the final energy level (3 for the Paschen series), and - \(n_{i}\) is the initial energy level.

Determine the Range of Wavelengths for the Paschen Series

To find the range of wavelengths, we need to find the smallest and largest possible wavelengths for the Paschen series. The smallest wavelength occurs when \(n_{i}\) is just above \(n_{f}\), so \(n_{i} = n_{f} + 1 = 3 + 1 = 4\); whereas the largest possible wavelength occurs when \(n_{i} \to \infty\). Minimum wavelength: \[\frac{1}{\lambda_\text{min}} = R_H \left( \frac{1}{3^2} - \frac{1}{4^2} \right)\] Maximum wavelength: \[\frac{1}{\lambda_\text{max}} = R_H \left( \frac{1}{3^2} - \frac{1}{\infty} \right) = R_H \frac{1}{3^2}\] Next, we will solve these equations to find the range of wavelengths.

Identify the Region of the Electromagnetic Spectrum

Now we will solve for the minimum and maximum wavelengths and identify which region of the electromagnetic spectrum they fall into. From the electromagnetic spectrum, we know the ranges for various types of radiation: - Ultraviolet: \(10^{-9} m < \lambda < 400 \times 10^{-9} m\) - Visible: \(400 \times 10^{-9} m < \lambda < 700 \times 10^{-9} m\) - Infrared: \(700 \times 10^{-9} m < \lambda\) Calculate \(\lambda_\text{min}\) and \(\lambda_\text{max}\): \[\lambda_\text{min} = \frac{1}{R_H \times (1/3^2 - 1/4^2)}\] \[\lambda_\text{max} = \frac{1}{R_H \times (1/3^2)}\] After solving, we will identify which region the wavelengths fall into.

Calculate the Wavelengths of the First Three Lines

Finally, we will calculate the wavelengths of the first three lines in the Paschen series, where \(n_{i} = 4, 5\), and 6 using the Rydberg formula: For \(n_{i} = 4\): \[\frac{1}{\lambda_1} = R_H \left( \frac{1}{3^2} - \frac{1}{4^2} \right)\] \[\lambda_1 = \frac{1}{R_H \times (1/3^2 - 1/4^2)}\] For \(n_{i} = 5\): \[\frac{1}{\lambda_2} = R_H \left( \frac{1}{3^2} - \frac{1}{5^2} \right)\] \[\lambda_2 = \frac{1}{R_H \times (1/3^2 - 1/5^2)}\] For \(n_{i} = 6\): \[\frac{1}{\lambda_3} = R_H \left( \frac{1}{3^2} - \frac{1}{6^2} \right)\] \[\lambda_3 = \frac{1}{R_H \times (1/3^2 - 1/6^2)}\] Calculate the values of \(\lambda_1, \lambda_2\), and \(\lambda_3\) to find the first three lines in the Paschen series.

Key concepts

Rydberg Formula

The Rydberg formula is a fundamental equation in atomic physics that allows us to calculate the wavelengths of spectral lines of hydrogen and other hydrogen-like elements. Named after the Swedish physicist Johannes Rydberg, this formula is given by:
\[\begin{equation}\frac{1}{\lambda} = R \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right)\end{equation}\]
where:

• \(\lambda\) is the wavelength of the emitted or absorbed light,
• \(R\) is the Rydberg constant, approximately \(1.097 \times 10^7 m^{-1}\) for hydrogen,
• \(n_{f}\) is the principal quantum number of the final energy level, and
• \(n_{i}\) is the principal quantum number of the initial energy level.

In the context of the Paschen series, the formula is used to calculate the wavelengths when an electron transitions from a higher energy level \(n_{i}\) to the lower energy level \(n_{f} = 3\) for hydrogen. Each series in the hydrogen emission spectrum is named after the scientist who discovered it, and the Paschen series specifically refers to the series of infrared emissions as the electron cascades down to the third energy level.

Hydrogen Emission Spectrum

The hydrogen emission spectrum is a set of lines that represent the spectrum of light emitted when electrons within a hydrogen atom descend from higher energy levels to lower ones, releasing energy in the form of photons. Each of these spectral lines corresponds to a different electron transition and can be visibly seen through a spectroscope as an array of colors. However, not all these lines are visible to the human eye.
For instance, the Lyman series comprises ultraviolet wavelengths, while the Balmer series can be observed in the visible range. The Paschen, Brackett, and Pfund series fall in the infrared region of the spectrum, beyond the sensitivity of the human eye but detectable with special instruments. These series are crucial for understanding atomic structure, as they provide empirical evidence for the energy quantization in atoms – meaning that electrons exist at discrete energy levels, and not just at any level.

Wavelength Calculation

Calculating the wavelength of light emitted by electrons transitioning between energy levels in an atom is an essential aspect of understanding atomic spectra. The Rydberg formula is instrumental for this purpose. It allows us to calculate the specific wavelengths of light associated with electrons moving from one specified energy state to another. When given specific values for \(n_{i}\) and \(n_{f}\), the calculation merely involves substituting these numbers into the Rydberg equation and solving for \(\lambda\), the resulting wavelength.

Practical Example

Using the given Rydberg constant \(R_H\) and the energy levels for the Paschen series \(n_{f}=3\), and by knowing the initial energy levels \(n_{i}=4, 5, 6\), we input these into the formula to obtain the corresponding wavelengths for the first three lines of the Paschen series. These calculated wavelengths fall within the infrared region, confirming that when electrons transition to the third energy level from higher levels within a hydrogen atom, they emit infrared radiation. This calculation is pivotal in many scientific and technological fields, including astronomy, where it helps in identifying elements present in distant celestial objects by studying their emitted light spectra.