Problem 128 - Chemistry The Science in Context

Question

When an atom absorbs an X-ray of sufficient energy, one of its 2 s electrons may be emitted, creating a hole that can be spontaneously filled when an electron in a higher-energy orbital-a \(2 p,\) for example - falls into it. A photon of electromagnetic radiation with an energy that matches the energy lost in the \(2 p \rightarrow 2 s\) transition is emitted. Predict how the wavelengths of \(2 p \rightarrow 2 s\) photons would differ between (a) different elements in the fourth row of the periodic table and (b) different elements in the same column (for example,

Step-by-Step Solution

Answer

Answer: The wavelengths of 2p → 2s photons would vary more between different elements in the fourth row of the periodic table compared to different elements in the same column. This is because the energy differences for the 2p → 2s transition are more distinct for elements with differing atomic numbers in the fourth row, while elements in the same column experience more similar energy differences due to the electrons involved in the transition occupying the same shell (n=2).

Step 1: Understand the energy levels of atoms and transitions

Electrons in an atom occupy specific energy levels (orbitals) depending on their position in the atom. In this case, the electron transitions from the higher-energy \(2p\) orbital to the lower-energy \(2s\) orbital. When this happens, a photon with an energy equal to the energy loss in the transition is emitted. To understand how the wavelengths of these photons would differ between different elements, we need to determine the difference in energy between the \(2p \rightarrow 2s\) transitions for these elements.

Step 2: Calculate energy difference for \(2p \rightarrow 2s\) transition

The energy difference (\(\Delta E\)) between the initial (\(2p\)) and final (\(2s\)) energy levels can be expressed as: \(\Delta E = E_{2p} - E_{2s}\) The energy levels of an electron in an atom can be determined by using the general formula: \(E = -\dfrac{Z^2e^2}{8\pi \epsilon_{0} na_0}\) where \(E\) is the energy, \(Z\) is the atomic number, \(e\) is the charge of an electron, \(\epsilon_{0}\) is the vacuum permittivity, \(n\) is the principal quantum number, and \(a_0\) is the Bohr radius.

Step 3: Calculate the energy of the emitted photon

The energy of a photon is related to its wavelength (\(\lambda\)) through the formula: \(E_{photon} = \dfrac{hc}{\lambda}\) where \(h\) is Planck's constant and \(c\) is the speed of light. Since the energy of the emitted photon is equal to the energy difference between the \(2p\) and \(2s\) orbitals, we have: \(\dfrac{hc}{\lambda} = \Delta E\)

Step 4: Find the wavelengths for different elements in the fourth row and same column

(a) For different elements in the fourth row of the periodic table, their atomic numbers (Z) differ. As a result, the energy levels and energy differences for the \(2p \rightarrow 2s\) transitions will also differ. This leads to a different wavelength for each element in the fourth row. (b) For different elements in the same column, their atomic numbers (Z) will differ as well. However, the electrons involved in the \(2p \rightarrow 2s\) transition are in the same shell (n=2) which leads to similar energy differences for the transition. Consequently, the wavelengths of the emitted photons for different elements in the same column will be more similar compared to those in the fourth row. In conclusion, the wavelengths of \(2p \rightarrow 2s\) photons would vary more between different elements in the fourth row of the periodic table compared to different elements in the same column.

Key Concepts

Energy Levels in Atoms

Atoms are composed of a nucleus and electrons that are distributed across various energy levels or orbitals. These energy levels are well-defined and are determined by the principles of quantum mechanics. An important aspect of an atom's structure is that electrons cannot occupy energies between these levels; they must jump from one level to another in discrete steps, absorbing or releasing energy as they do so.

Imagine each energy level as a step in a staircase. Electrons can move up the staircase by absorbing energy, typically in the form of photons, which excites them to higher energy levels. Conversely, when they move down, they lose energy and release it, usually as light or other electromagnetic radiation. This process is central to understanding the electronic configurations and chemical behaviors of elements.

• Lower energy levels are closer to the nucleus and are filled first as they are more stable.
• Higher energy levels are further away and can hold more electrons, also influencing the atom's reactivity.
• When an electron transitions from a higher to a lower energy level, it emits a photon, which carries away the energy difference between these levels.

The intricacies of these levels are crucial for explaining why elements produce photons with specific wavelengths, and how these wavelengths change between different elements or even isotopes of the same element.

Photon Emission

Photon emission occurs when an electron in an atom transitions from a higher to a lower energy level, releasing energy in the form of a photon. This photon's energy is precisely equal to the energy difference between the two levels involved in the transition.

The energy of the emitted photon can be calculated using the famous equation from Planck's theory of quantum mechanics: \( E_{photon} = \frac{hc}{\lambda} \) where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the photon's wavelength. The shorter the wavelength, the more energy the photon carries; inversely, the longer the wavelength, the less energy it has.

Significance of Photon Emission in Science

• Photon emission is fundamental in spectroscopy, a tool used to identify elements and compounds.
• In medical imaging and treatments, controlled photon emission is crucial for techniques like X-rays and radiation therapy.
• In everyday technology, photon emission is the basis for devices like lasers and LEDs.

Wrapping our heads around photon emission is not only critical for understanding atomic behavior but also for harnessing the properties of light across various technologies and scientific fields.

Periodic Table Trends

The periodic table is more than just an arrangement of elements; it's a map that reveals the underlying trends in the properties of these elements. Understanding these trends helps in predicting an element's behavior, including its energy levels and the types of photons it may emit during electron transitions.

As we move across a period (row) in the periodic table, elements increase in atomic number and electrons fill higher energy sublevels. This progression influences the energy differences between orbitals, which impacts the wavelengths of photons emitted during electron transitions. Moving down a group (column), elements have additional electron shells, making the outer electrons less tightly bound to the nucleus. However, within a shell, sublevels and orbitals have similar energies across different elements in the same group.

Key Aspects of Periodic Trends

• Electronegativity increases across a period and decreases down a group.
• Atomic radius decreases across a period and increases down a group.
• Ionization energy tends to increase across a period and decrease down a group.

These periodic trends play a significant role in the transitions of electrons between energy levels and thus the energy of the photons emitted or absorbed. As atomic numbers change, so do the energy requirements for an electron to jump between levels, making the periodic table an essential tool for students to predict and explain how the characteristics of an element influence photon emissions and chemical reactions.