It takes \(208.4 \mathrm{~kJ}\) of energy to remove 1 mole of electrons from an atom on the surface of rubidium metal. How much energy does it take to remove a single electron from an atom on the surface of solid rubidium? What is the maximum wavelength of light capable of doing this?

Understanding the energy required to remove an electron from an atom is critical in the study of atomic structure and photoelectron spectroscopy. The process of removing an electron from an atom is known as ionization. The energy required to remove one electron is called the ionization energy.

The ionization energy can vary depending on the element and the state of the electron within the atom. In the case of rubidium metal, the exercise provided us with the energy needed to ionize a mole of electrons, which is 208.4 kJ/mol. This amount of energy refers to the collective energy needed to remove all electrons in a mole of rubidium atoms.

To find out how much energy it takes to remove just a single electron, we use Avogadro's number (\(6.022 \times 10^{23}\) particles/mol), which represents the number of particles in a mole. By dividing the provided energy per mole by Avogadro's number, we can pinpoint the energy required for a single electron. This meticulous calculation unravels the sheer magnitude of Avogadro's number and the tiny amount of energy to detach a single electron.

Avogadro's number is a fundamental constant in chemistry often used when dealing with quantities in the microscopic world. It represents the number of units (atoms, molecules, ions, etc.) in one mole of a substance and is approximately equal to \(6.022 \times 10^{23}\) units/mol.

When working with atoms or molecules on the macroscopic scale, such as in chemical reactions or when measuring substances in a lab, dealing with individual particles is impractical. Avogadro's number allows scientists to translate between the microscopic world of atoms and the macroscopic world of grams and liters. It also plays a crucial role in calculations involving the energy per particle, as seen in our problem where it helps in converting the energy required for a mole of electrons to the energy required for a single electron.

The maximum wavelength of light capable of ionizing an atom pertains to the threshold wavelength beyond which a photon has insufficient energy to eject an electron. This concept is rooted in the photoelectric effect, where electrons are emitted from materials when exposed to light.

According to quantum theory, light behaves both as a wave and as a particle. When considering its particle nature, light consists of photons, and each photon carries energy based on its frequency (\(E=h \times f\) where h is Planck's constant and f is the frequency). The relationship between the wavelength (\textlambda) and frequency (given that \textlambda \times f = c, where c is the speed of light) implies that photons of long wavelengths carry less energy.

The maximum wavelength of light that can ionize an atom correlates inversely with the minimum photon energy needed to remove an electron. To calculate this wavelength, we would use the energy per electron that we previously calculated and apply Planck's equation to solve for the wavelength. This relationship is pivotal in understanding the interaction between light and matter, especially in spectroscopy where it helps in determining the energy levels within an atom.