Problem 114 - Chemical Principles

Question

In the spectroscopic technique known as photoelectron spectroscopy (PES), ultraviolet radiation is directed at an atom or a molecule. Electrons are ejected from the valence shell, and their kinetic energies are measured. Because the energy of the incoming ultraviolet photon is known and the kinetic energy of the outgoing electron is measured, the ionization energy, \(I\), can be deduced from the fact that the total energy is conserved. (a) Show that the speed \(v\) of the ejected electron and the frequency \(\nu\) of the incoming radiation are related by \(h v=I+\frac{1}{2} m_{\mathrm{e}} v^{2}\) (b) Use this relation to calculate the ionization energy of a rubidium atom, given that radiation of wavelength \(58.4 \mathrm{~nm}\) produces electrons with a speed of \(2450 \mathrm{~km} \cdot \mathrm{s}^{-1}\); recall that \(1 \mathrm{~J}=1 \mathrm{~kg} \cdot \mathrm{m}^{2} \cdot \mathrm{s}^{-2}\).

Step-by-Step Solution

Answer

The ionization energy of the rubidium atom is calculated using the relation h = I + (1/2) me v^2 by first finding the frequency from the given wavelength, converting the electron's speed into meters per second, and using Planck's equation to determine the energy of the incoming photon.

Step 1: Identify Given Information

The wavelength of the UV radiation is given as 58.4 nm, which can be converted to meters, and the speed of the ejected electrons is 2450 km/s, which must also be converted to meters per second. Additional constants that will be used are Planck's constant, h (which is 6.626 × 10^-34 J·s), and the mass of an electron, me (which is 9.109 × 10^-31 kg).

Step 2: Convert Given Wavelength to Frequency

Using the relationship between frequency (), wavelength (), and the speed of light (c), find the frequency of the incoming radiation (). The speed of light is approximately 3.00 × 10^8 m/s. The equation to be used is = c / .

Step 3: Convert Electron Speed to Meters per Second

The given speed of the electrons is 2450 km/s, which is equivalent to 2.450 × 10^6 m/s in SI units.

Step 4: Use Planck's Equation to Find Ionization Energy

Planck's equation relates the energy of a photon to its frequency: E = h . Substitute the calculated frequency to find the energy of the incoming photon.

Step 5: Use Energy Conservation to Find Ionization Energy

The conservation of energy can be expressed as the energy of the photon equals the ionization energy plus the kinetic energy of the ejected electron. Mathematically, this can be written as h = I + (1/2) me v^2. We know h (from the previous step), me, and v. Solve this equation for the ionization energy, I.

Key Concepts

Spectroscopy

Spectroscopy is a scientific technique used to analyze the interaction between matter and electromagnetic radiation. In the context of photoelectron spectroscopy (PES), we're particularly concerned with how ultraviolet (UV) radiation affects atoms or molecules. When UV light is directed at an atom, it can impart enough energy to an electron in the valence shell—the outermost electron layer—to eject it from the atom.This ejection of an electron by a photon is a core aspect of PES, which allows us to measure the kinetic energy of the ejected electron. By knowing the amount of energy the UV light carried and measuring the energy of the freed electron, we can deduce various properties of the atom, including its ionization energy. PES is incredibly useful in chemistry for understanding the electronic structure and bonding in atoms and molecules.

By analyzing the spectrum of the ejected electrons, researchers can obtain information about the electronic configurations and energy levels within an atom, providing insight into chemical bonding and reaction dynamics.

Ionization Energy

Ionization energy (I) is a critical concept in chemistry, especially as it pertains to PES. It's defined as the minimum amount of energy required to remove an electron from an atom or molecule in its gaseous state. This energy reflects the strength of the attraction between an electron and the nucleus; the stronger the attraction, the higher the ionization energy required to eject the electron.In a PES experiment, when an electron is knocked free from the atom, it carries away kinetic energy, which we can measure. Since the energy provided by the UV photon is known, the ionization energy can be calculated as the difference between the photon's energy and the kinetic energy of the ejected electron. This allows us to understand how tightly an electron is bound to an atom, which has implications for reactivity and stability of elements and compounds.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics with a value of approximately 6.626 x 10^-34 Joule seconds (J·s). It plays a crucial role in the quantization of energy levels and is used to calculate the energy (E) of photons from their frequency () using the formula: E = h. This relationship, known as Planck's equation, was pivotal to the development of quantum theory.In photoelectron spectroscopy, Planck's constant helps relate the energy of the incoming UV light to the frequency of that light. Since the UV light is composed of photons, each with a discrete amount of energy, the constant ensures that energy transfer in the form of emitted electrons can be accurately measured and quantified. This allows for precise calculations when determining the ionization energies of different elements.

Energy Conservation in Photoelectron Spectroscopy

Energy conservation is a foundational principle in physics and is key to understanding how photoelectron spectroscopy works. According to this principle, energy cannot be created or destroyed in an isolated system; it can only change forms. In the context of PES, the principle tells us that the total energy before and after an electron is ejected remains constant.Thus, the initial energy of the incoming UV photon is used to do two things: first, to overcome the ionization energy and free the electron, and second, to provide the ejected electron with kinetic energy as it leaves the atom. Mathematically, this can be described with the equation: h = I + (1/2) m_e v^2, where h is Planck's constant, is the frequency of the photon, I is the ionization energy, m_e is the mass of the electron, and v is the velocity of the ejected electron. This equation is instrumental in calculating the ionization energy of an element during a PES experiment, using the measured kinetic energy of the emitted electrons and the known energy of the incoming UV light.