Question

The absorption spectrum of the hydrogen atom shows lines at \(5334,7804,9145,9953,\) and \(10,478 \mathrm{cm}^{-1} .\) There are no lower frequency lines in the spectrum. Use the graphical methods discussed in Example Problem 22.6 to determine \(n_{\text {initial }}\) and the ionization energy of the hydrogen atom in this state. Assume values for \(n_{\text {initial}}\) of \(1,2,\) and 3.

Short answer

#tag_title# Step 2: Plot the data and draw a curve #tag_content# Make a plot using \(1/\lambda\) on the x-axis and \(1/\Delta E\) on the y-axis, where \(\Delta E = h\nu\), with \(h = 6.626 \times 10^{-27}\, \text{erg s}\) being the Planck's constant. Fit a smooth curve through the data points. #tag_title# Step 3: Determine \(n_{\text{initial}}\) and ionization energy #tag_content# Examine the plot and find of which value of \(n_{\text{initial}}\) it is a straight line. Determine the corresponding ionization energy (where \(n_f \to \infty\)) from the plot as the zero crossing point on the y-axis. Answer: Based on the graphical method using the Rydberg formula and the given absorption lines, we can determine that \(n_{\text{initial}} = 2\) and the ionization energy is approximately \(1312\, \mathrm{cm}^{-1}\) in this state.

Step-by-step solution

Convert the given frequencies to wavelengths

Using the \(\nu = \frac{c}{\lambda}\) formula mentioned above, find the wavelengths corresponding to the given frequencies.

Key concepts

Spectral Lines

Imagine the hydrogen atom as an incredibly small system with its electron orbiting a nucleus. When this electron moves between different orbits or energy levels, it either absorbs or emits energy. This energy difference corresponds to particular wavelengths of light, which, when absorbed or emitted, produce what we call the spectral lines.

These lines are unique to each element, much like a fingerprint, because they reflect the specific energy level transitions that are possible within an atom. In the case of the hydrogen atom, these spectral lines are due to electrons jumping to and from its first few orbits. The spectral lines in an absorption spectrum, such as the one in our exercise, correspond to the wavelengths of light that are absorbed when an electron transitions to a higher energy level.

Ionization Energy

The term ionization energy refers to the amount of energy required to completely remove an electron from an atom. For a hydrogen atom, this means supplying enough energy to free the electron from its attraction to the proton in the nucleus, effectively creating a hydrogen ion.

Determining this energy is crucial because it gives us fundamental information about the stability of an atom and helps to explain the chemical properties of the element. The last spectral line observed in the absorption spectrum we are looking at, which represents the highest energy transition, can be used to approximate the ionization energy of a hydrogen atom in an excited state.

Quantum Mechanics

To really understand what's happening when we talk about spectral lines and absorption, we must dive into the mysterious world of quantum mechanics. This is the branch of physics that deals with the behavior of atoms and subatomic particles, which don't always follow the rules of the larger, visible world we're used to.

Quantum mechanics tells us that electrons in an atom can only exist in certain discrete energy levels. When an electron jumps from one level to another, it must absorb or release a specific amount of energy, equal to the difference between these levels. This concept is fundamental to deciphering why atoms only absorb or emit light at specific wavelengths, leading to the creation of a unique absorption spectrum for each element.

Rydberg Formula

One of the milestones of quantum mechanics in explaining the spectral lines of hydrogen is the Rydberg formula. It is an equation used to predict the wavelengths of spectral lines in various chemical elements, particularly for hydrogen. The formula is usually expressed as \[ 1/\lambda = R(1/n_{1}^{2} - 1/n_{2}^{2}) \] where \(\lambda\) is the wavelength of the light, \(R\) is the Rydberg constant (approximately \(1.097 \times 10^{7} m^{-1}\)), and \(n_{1}\) and \(n_{2}\) represent the principal quantum numbers of the lower and higher energy levels, respectively.

For hydrogen, when an electron transitions from a higher to a lower energy level, the frequency of light emitted fits neatly into the Balmer series, Lyman series, and other series that are subsets of what the Rydberg formula describes. By using the known wavelengths and the Rydberg formula, we can calculate the initial energy level \(n_{\text{initial}}\) for an electron before it absorbs energy and transitions to a higher level or the energy required to ionize the atom.