Question

What is the shortest wavelength of light that a hydrogen atom will emit?

Short answer

Answer: The shortest wavelength of light that a hydrogen atom will emit is approximately 91.1 nm.

Step-by-step solution

Understand the Rydberg formula

The Rydberg formula is used to predict the wavelengths of spectral lines in the hydrogen spectrum. The formula is as follows: \begin{equation} \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \end{equation} where \(\lambda\) is the wavelength, \(R_H\) is the Rydberg constant for hydrogen (approximately 1.097373 x \(10^7\) m\(^{-1}\)), \(n_1\) and \(n_2\) are the principal quantum numbers of the two energy levels involved in the transition.

Recognize the meaning of shortest wavelength

In order to find the shortest wavelength emitted by a hydrogen atom, we need to know that shorter wavelength corresponds to higher energy. Therefore, we need to find the energy transition that results in the highest energy being emitted, or in other words, we need to find the maximum possible difference in energy levels.

Find the appropriate energy levels

When an electron in a hydrogen atom transitions from a higher energy level to a lower energy level, it emits a photon. The shortest wavelength corresponds to the highest energy transition, which occurs when an electron jumps from the highest possible energy level to the lowest possible energy level. In the hydrogen atom, the lowest energy level is \(n_1 = 1\) (ground state), and the highest possible energy level in principle is infinity (\(n_2 = \infty\)). However, in a practical case, we can consider a very large value for \(n_2\) to make the calculation.

Apply the Rydberg formula

Using the Rydberg formula and the energy levels found in Step 3, we can now calculate the shortest wavelength: \begin{equation} \frac{1}{\lambda} = R_H \left(\frac{1}{1^2} - \frac{1}{\infty^2} \right) \end{equation} Since any number divided by infinity tends to zero, the equation becomes: \begin{equation} \frac{1}{\lambda} = R_H \left(\frac{1}{1^2} - 0 \right) \end{equation} Now plug in the value of \(R_H\): \begin{equation} \frac{1}{\lambda} = (1.097373 \times 10^7\,\text{m}^{-1}) \left(1 - 0 \right) \end{equation}

Solve for the shortest wavelength

Finally, solve for the shortest wavelength: \begin{equation} \lambda = \frac{1}{1.097373 \times 10^7\,\text{m}^{-1}} \end{equation} \begin{equation} \lambda \approx 9.11 \times 10^{-8}\,\text{m} = 91.1\,\text{nm} \end{equation} Thus, the shortest wavelength of light that a hydrogen atom will emit is approximately 91.1 nm.

Key concepts

Rydberg Formula

The Rydberg formula plays a fundamental role in understanding the emission spectrum of a hydrogen atom. It is an empirical formula used to predict the wavelengths of the photons emitted or absorbed by electrons jumping between energy levels in a hydrogen atom.

The formula is expressed as follows:
\[\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]
In this equation, \(\lambda\) represents the wavelength, \(R_H\) is the Rydberg constant specifically for hydrogen, and \(n_1\) and \(n_2\) are the principal quantum numbers of the initial and final energy levels, respectively. It's interesting that this formula is elegantly connected to the energy difference between two levels and beautifully demonstrates the quantization inherent in atomic systems.

When looking for the shortest wavelength—and thus the highest energy transition according to Planck's relation \(E = hu\)—we consider the limit as \(n_2\) approaches infinity, denoting a transition from infinitely high energy level down to the first energy level (ground state).

Quantum Numbers

Quantum numbers are integral to the understanding of atomic structure and behavior. They essentially serve as coordinates, describing the position and energy level of an electron in an atom. There are four quantum numbers: principal \(n\), azimuthal or orbital angular momentum \(l\), magnetic \(m\), and spin \(s\).

In the context of hydrogen's emission spectrum, the principal quantum number, \(n\), is most directly involved. It describes the energy level or shell of an electron within the atom and can take any positive integer value. As \(n\) increases, the electron orbits farther from the nucleus, and the energy associated with that level is greater. The transition of an electron from a higher principal quantum number to a lower one results in the emission of a photon, the energy and wavelength of which can be predicted by the Rydberg formula.

Energy Level Transitions

Energy level transitions are events where electrons in an atom move between energy states or levels. When an electron drops to a lower energy level, it emits energy in the form of a photon. Conversely, when an electron absorbs energy, it can jump to a higher energy level.

The energy difference between these levels is discrete and quantized, leading to the term 'quantum leap'. The spectral lines we observe in an emission spectrum directly correspond to these quantum leaps. For the hydrogen atom, the highest energy (and thus shortest wavelength) photon is emitted when an electron makes a transition from the highest energy level it can occupy to the n=1 level, the ground state. This concept is central to solving problems that deal with the emission spectrum, such as determining the shortest wavelength a hydrogen atom can emit.

Spectral Lines

Spectral lines are the fingerprints of elements in the field of spectroscopy. Each element emits or absorbs specific wavelengths of light, leading to the appearance of distinct lines in their spectra. These lines represent the energy level transitions of electrons within the atoms of that element.

In the hydrogen spectrum, these spectral lines are categorized in series such as Lyman, Balmer, and Paschen, with each series corresponding to transitions ending at a particular energy level. The specific wavelengths of these lines can be calculated using the Rydberg formula, giving us a clear and quantitative way of understanding and predicting the behavior of electrons in hydrogen atoms.

The intensity and distribution of these spectral lines provide valuable information about the physical properties of stars and other celestial bodies, as well as helping to understand atomic and quantum mechanics in greater detail.