Question

A chemist has one mole of \(X\) atoms. He finds that when half of the \(\mathrm{X}\) atoms transfer one electron to the other half, \(409 \mathrm{~kJ}\) must be added. If all of the resulting \(\mathrm{X}^{-}\) ions are subsequently converted to \(\mathrm{X}+\) ions, an additional \(733 \mathrm{~kJ}\) must be added. Find the ionization potential, and the electron affinity of \(\mathrm{X}\).

Short answer

The ionization potential of X is \(818 \mathrm{~kJ/mol}\), and the electron affinity of X is \(85 \mathrm{~kJ/mol}\).

Step-by-step solution

Understanding the information given

The first part of the problem states that when half of the X atoms transfer one electron to the other half, 409 kJ must be added. This means that half a mole of X atoms loses electrons and becomes X+ ions (ionization), and the other half gains electrons and becomes X- ions (electron affinity). In the second part of the problem, all X- ions are converted to X+ ions with an additional 733 kJ.

Write equations for ionization potential and electron affinity based on the given information

Let the ionization potential be I and the electron affinity be A. Both will have units kJ/mol. Then we can write two equations: 1. \(\frac{1}{2}I - \frac{1}{2}A = 409\) 2. \(I - A = 733\)

Solve the system of equations to get the ionization potential and electron affinity

We can rewrite equation 2 as \(A = I - 733\). Then replace A in equation 1 with the new expression: \(\frac{1}{2}I - \frac{1}{2}(I - 733) = 409\) Now we can solve for I: \(\frac{1}{2}I - \frac{1}{2}I + \frac{1}{2}(733) = 409\) \(\frac{733}{2} = 409\) Multiplying both sides by \(\frac{2}{733}\) we get: \(I=818\) Now use the expression for A to find the electron affinity: \(A = 818 - 733\) \(A = 85\)

Write the final answer

The ionization potential of X is 818 kJ/mol and the electron affinity of X is 85 kJ/mol.

Key concepts

Chemical Energetics

Chemical energetics, often referred to as thermodynamics, deals with the study of energy transfer in chemical processes. This field explores how different forms of energy, such as heat and work, are converted in chemical reactions. The key to understanding chemical energetics lies in the principles of energy conservation and how energy changes affect spontaneity of reactions.

In the given exercise, the energy required to remove an electron from an atom is associated with the ionization potential. Contrastingly, electron affinity measures the energy change when an atom gains an electron. To accurately capture these processes, the energetics of both ionization and electron addition must be measured in kilojoules per mole, enabling a clear quantification of the energy involved in these transformations.

Considering energy in terms of moles relates to the mole concept, which introduces a link between the microscopic scale of atoms and ions and the macroscopic world we measure in laboratories. Understanding the chemical energetics of electron transfer in ions assists in comparing and predicting the behavior of different substances under various conditions.

Electron Transfer

Electron transfer is a fundamental concept in chemistry that occurs in a myriad of processes, from the simple formation of ions to the complex reactions in electrochemistry. It involves the movement of electrons from one atom, molecule, or ion to another, altering their oxidation states and chemical properties.

In the exercise above, the transfer of electrons from half the X atoms to the other half delineates a crucial aspect of electron transfer: ionization and the gaining of electrons, known as reduction. The energy considerations for these transfers are crucial, as one electron exits an atom's influence, it requires energy, which is the ionization energy, and conversely, energy is released when an electron is gained, which is conceptualized by electron affinity.

This balance of energy during electron transfer provides insight into the reactivity and stability of chemical species, and it is essential for understanding the nature of bonds in both ionic compounds and in more complex molecular structures.

Mole Concept

The mole concept is a bridge between the atomic and the macroscopic worlds. It allows chemists to count particles like atoms and molecules by weighing them. One mole is defined as the amount of substance that contains as many elementary entities as there are atoms in 12 grams of carbon-12, which is Avogadro's number (approximately 6.022x10²³ particles).

In the context of the given exercise, calculating ionization potential and electron affinity involves using the mole concept to express the energies per mole. The mole concept simplifies these vast numbers of atoms and ions into manageable quantities, which can then be used to determine macroscopic energy changes when electrons are transferred, such as in ionization and in the acquisition of electrons.

Through this concept, the exercise demonstrates how large-scale energetic quantities relate to a fundamental chemical unit—the mole. It not only underscores the consistency in energy changes whether we are dealing with a single atom or a mole of atoms, but also aids in making these abstract concepts tangible.