Question

Use Coulomb's law to calculate the ionization energy in \(\mathrm{kJ} / \mathrm{mol}\) of an atom composed of a proton and an electron separated by \(100.00 \mathrm{pm} .\) What wavelength of light has sufficient energy to ionize the atom?

Short answer

The ionization energy is approximately 1312.00 kJ/mol and the wavelength of light sufficient to ionize the atom is approximately 95.0 nm.

Step-by-step solution

Calculate ionization energy using Coulomb's law

Use Coulomb's law, which is given by the formula: \(E = k_e \frac{|q_1 q_2|}{r}\), where \(E\) is the potential energy, \(k_e\) is Coulomb's constant (\(8.988 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2\)), \(q_1\) and \(q_2\) are the charges of the electron and proton respectively (\(1.602 \times 10^{-19} \mathrm{C}\)), and \(r\) is the separation distance (converted to meters). First, convert 100 pm to meters by multiplying by \(10^{-12}\). Then, plug the values into Coulomb's law to get the ionization energy in joules.

Convert ionization energy to kJ/mol

After calculating the ionization energy per atom in joules, convert this energy to kilojoules by dividing by \(1000\), and then convert it to energy per mole by multiplying by Avogadro's number (\(6.022 \times 10^{23} \mathrm{mol}^{-1}\)).

Calculate the wavelength of light using the energy

Using the relation between energy and wavelength given by Planck's equation, \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{J s}\)), \(c\) is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)), and \(\lambda\) is the wavelength. Solve for \(\lambda\) by rearranging the equation: \(\lambda = \frac{hc}{E}\). Use the ionization energy per photon (in joules) calculated in step 1 (not the molar energy) to find the wavelength in meters.

Key concepts

Coulomb's Law

Imagine you're trying to pull two magnets apart or push them together; the force you feel is related to the distance between them and their magnetic strength. Coulomb's Law works in a similar way, but for electrical forces. It describes how strong the force is between two charged particles, like protons and electrons, based on their charges and distance apart. In simple terms, like charges repel and opposite charges attract each other with a force that decreases as the square of the distance between them increases.

In our calculation, we use this fundamental law to determine the potential energy between a proton and an electron, which is effectively the energy needed to separate them - or the ionization energy. The equation is:
\( E = k_e \frac{|q_1 q_2|}{r} \),
where \( E \) is the potential energy in joules, \( k_e \) is Coulomb's constant (\( 8.988 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2 \)), \( q_1 \) and \( q_2 \) are the charges of the charged particles, and \( r \) is the separation distance between them.

Planck's Equation

Now think about waves at the beach - you can measure their energy by their height and how fast they hit the shore. Light also behaves like a wave (and a particle), and Planck's equation tells us how much energy light waves carry based on their wavelength. The shorter the wavelength, the more energy it has, like those tall, fast-crashing waves.

The equation \( E = \frac{hc}{\lambda} \) shows the relationship between energy (\(E\)) and wavelength (\(\lambda\)). Here, \(h\) is a super important constant named after Max Planck (\(6.626 \times 10^{-34} \mathrm{J s}\)) and \(c\) is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)). By rearranging this equation and inserting the energy we want to use to ionize the atom, we can figure out what wavelength of light has the mojo to do it - which can help us understand the kind of light that can affect different atoms.

Avogadro's Number

When you're cooking, you might measure ingredients like flour or sugar in cups or tablespoons. In chemistry, we have a special 'cup' for atoms and molecules called a mole, which helps us count huge quantities of tiny particles. The 'number of tablespoons' in this special cup is Avogadro's number: \(6.022 \times 10^{23}\).

Understanding Avogadro's number is crucial because it bridges the gap between an individual atom and the amount we handle in the lab. For ionization energy, once we know the energy needed to remove an electron from one atom, we multiply by Avogadro’s number to scale it up to a practical quantity in moles - just like you'd multiply the amount of sugar in one cup by the number of cups needed to bake a batch of cookies. It's a key step to translate from the microscopic to the everyday scale, and that's essential when we communicate chemical concepts in digestible terms.