With sufficient energy, it's possible to eject an electron from an inner atomic orbital. A higher-energy electron will then drop into the unoccupied state, emitting a photon with energy equal to the difference between the two levels. For inner-shell electrons, photon energies are in the keV range, putting them in the X-ray region of the spectrum. These characteristic X rays are labeled with the letter indicating the shell to which the electron drops, followed by a Greek letter indicating the higher level from which it drops; thus \(K \alpha\) designates a transition from the \(L\) shell to the \(K\) shell. Characteristic X rays provide scientists and physicians with an important diagnostic tool. Environmental scientists bombard pollution samples with high- energy electrons, knocking out inner-shell electrons and thus producing X-ray spectra that help identify contaminants (Fig. \(36.20 a\) ). Geologists do the same with rocks. Medical radiologists reverse the process, exploiting the fact that X rays cause inner-shell transitions as well as complete ejection of inner-shell electrons. In particular, radiologists use the element barium in this way to produce high-contrast X-ray images of the intestinal tract \((\text { Fig. } 36.20 b)\)(GRAPH CANNOT COPY) (a) An \(\mathrm{X}\) -ray spectrum from air pollutants trapped on a filter. The labeled peaks show the presence of lead (Pb) and arsenic (As), as evidenced by \(K \alpha, K \beta, L \alpha,\) and \(L \beta\) characteristic X rays. (b) \(\mathrm{X}\) -ray of an intestinal tract, made by coating the intestinal wall with X-ray-opaque barium Molybdenum's X-ray spectrum has its \(K \alpha\) peak at 17.4 keV. The corresponding X-ray wavelength is closest to a. \(1 \mathrm{pm}\) b. \(100 \mathrm{pm}\) c. \(1 \mathrm{nm}\) d. \(100 \mathrm{nm}\)

Step by step solution

01

Convert the Energy from keV to Joules

One must first convert the energy from keV to Joules before being able to apply said energy in Planck's formula. This is because Planck's constant (\(h\)) is measured in Joules, and it's crucial for the units to match to produce an accurate result. The conversion factor between keV and Joules is \(1.602 \times 10^{-16} J/keV\). So, in this case, 17.4 keV would equal \(17.4 keV \times 1.602 \times 10^{-16} J/keV = 2.79 \times 10^{-15}\) Joules

02

Substitute values into the Formula

The energy value calculated in the previous step can be substituted into Planck's formula, which has been rearranged for wavelength. Known values of Planck's constant (\(h = 6.626 \times 10^{-34} Js\)) and the speed of light (\(c = 3.00 \times 10^8 m/s\)) must also be substituted into the formula. This results in the formula \(2.79 \times 10^{-15} J = \frac{6.626 \times 10^{-34} Js \times 3.00 \times 10^8 m/s}{\lambda}\)

03

Solve for Wavelength

By solving the equation for \(\lambda\), the wavelength can be calculated. This results in \(\lambda = \frac{6.626 \times 10^{-34} Js \times 3.00 \times 10^8 m/s}{2.79 \times 10^{-15} J}\). This completes the calculation of the wavelength.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inner-Shell Electrons

The concept of inner-shell electrons is pivotal in understanding characteristic X-rays. Inner-shell electrons reside in the deepest layers of an atom's electron structure, close to the nucleus. These electrons are tightly bound, and it takes significant energy to eject them from their orbits. When an inner-shell electron is ejected due to high-energy interactions, such as those occurring during bombardment by high-energy electrons or X-rays, an electron from a higher shell can transition down to fill the vacancy, releasing energy in the form of a photon.

This energy release presents as an X-ray photon, a process fundamental for techniques utilized in medicine, geology, and environmental science to identify substances or to create diagnostic images. By analyzing the energies of these photons, scientists can deduce the identities of elements within a given sample.

Photon Energy

The concept of photon energy refers to the amount of energy that is carried by a single photon. According to quantum mechanics, photons are the basic unit or quantum of light and other forms of electromagnetic radiation, and the energy of a photon is directly proportional to its frequency. The energy of the photons emitted during electron transitions, specifically from higher-energy shells to lower-energy shells in an atom, falls within the X-ray region of the electromagnetic spectrum with energies typically in the keV range.

Understanding photon energy is crucial for interpreting X-ray spectra and for applications like medical imaging, where different tissues absorb X-rays to varying degrees. This absorption difference, attributable to the energy of the photons, allows radiologists to produce contrasting images that can aid in diagnosis.

Planck's Formula

Planck's formula, a principle cornerstone of quantum mechanics, describes the relationship between the energy of a photon and its wavelength. The energy (E) of a photon is equal to the product of Planck's constant (h) and its frequency (). Planck's constant is a fundamental constant of nature that's used in the quantification of energy packets within quantum systems. The formula is often represented as ), where h is Planck's constant (approximately 6.626 x ) Js, c is the speed of light in a vacuum (roughly 3.00 x ) m/s, and ) is the wavelength of the photon.

In the context of X-rays, Planck's formula is rearranged to solve for wavelength when the energy of an emitted photon is known. This calculation allows for the analysis of substances based on their characteristic X-rays. By employing the principles behind Planck's formula, scientists can accurately determine attributes such as atomic structure and the presence of specific elements.

X-ray Spectroscopy

X-ray spectroscopy is an analytical technique that leverages the principle of characteristic X-rays to identify and characterize materials. When a substance is bombarded with high-energy electrons or X-rays, the resulting inner-shell transitions produce X-rays with specific energies unique to each element. By measuring the spectrum of X-rays emitted—essentially the energies and intensities—scientists can determine which elements are present in a sample and gain insights into its composition.

This technique finds widespread use across various fields. For instance, environmental scientists use it to detect pollutants by analyzing their X-ray spectra for characteristic energies that indicate the presence of specific elements. Similarly, geologists apply it to study the elemental composition of rocks.

Electron Transitions

Electron transitions in an atom occur when electrons move between different energy levels or shells. An excited electron may 'fall' to a lower energy level, and this process is accompanied by the release of energy in the form of a photon. If the transition involves the inner shell of an atom, and the energy released falls within the X-ray range of the electromagnetic spectrum, the emitted radiation is known as characteristic X-rays. These transitions follow specific selection rules that govern the possible changes in an electron's energy state.

Understanding these transitions is crucial for interpreting X-ray spectra and provides the foundation for various imaging and analytical techniques. The characteristic lines, like the K and L lines referred to in the exercise, are named for the shells involved in the transitions, and their precise energies allow for the identification of the elements within a sample.