Question

Electromagnetic radiation of wavelength \(242 \mathrm{~nm}\) is just sufficient to ionise the sodium atom. What is the ionisation energy of sodium per atom? (a) \(494.5 \times 10^{-6} \mathrm{~J} /\) atom (b) \(8169.5 \times 10^{-10} \mathrm{~J} /\) atom (c) \(5,85 \times 10^{-15} \mathrm{~J} /\) atom (d) \(8.214 \times 10^{-19} \mathrm{~J} / \mathrm{atom}\)

Short answer

\(8.214 \times 10^{-19} \mathrm{J} / \mathrm{atom}\)

Step-by-step solution

Understanding the Problem

The exercise involves calculating the ionization energy of a sodium atom, which is the energy required to remove an electron from the atom, using the provided wavelength of electromagnetic radiation that is just sufficient to ionize the atom.

Calculate the Energy Using Photon Energy Formula

Energy of a photon can be calculated using the equation: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is the Planck's constant \( (6.626 \times 10^{-34} \mathrm{J \cdot s}) \), \( c \) is the speed of light in a vacuum \( (3.00 \times 10^8 \mathrm{m/s}) \), and \( \lambda \) is the wavelength. Substitute the given wavelength into the formula to calculate the energy.

Conversion of Wavelength from nm to m

The provided wavelength is in nanometers (nm). Convert it to meters (m) by multiplying with \( 10^{-9} \) as 1 nm = \( 10^{-9} \) m.

Calculation of Ionisation Energy

Combine the values of \( h \), \( c \), and the converted wavelength in meters into the equation to solve for the energy \( E \), which is the ionisation energy per sodium atom.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that is all around us and takes many forms such as X-rays, radio waves, and visible light. It travels through space in waves and is also described as a stream of photons, which are packets of energy with properties of both particles and waves.

Each photon contains a certain amount of energy that depends on its wavelength; shorter wavelengths have higher energy photons compared to longer wavelengths. For example, ultraviolet light has shorter wavelengths than visible light and thus each photon carries more energy. This concept is critical when we're trying to understand the ionization energy of elements, such as sodium, because ionization requires an exact or higher amount of energy than what is bound in the electron.

Planck's Constant

Planck's constant is a fundamental constant in quantum physics named after Max Planck, who discovered that energy is quantized. It is denoted by the symbol 'h' and has a value of approximately 6.626 x 10^-34 joule seconds (Js).

This constant is crucial in the field of quantum mechanics as it relates the energy carried by a photon to its frequency. It also appears in the formula that provides the relation between energy and wavelength of electromagnetic radiation which is at the core of estimating the ionization energy for any atom. A deep understanding of Planck's constant and its application is essential for calculating the energy levels of electrons in an atom, and thus, for understanding the basic principles of atomic structures and reactions.

Photon Energy Calculation

The energy of a photon can be calculated if the wavelength of the electromagnetic radiation is known. It's derived from the formula: \( E = \frac{hc}{\lambda} \)Here, \( E \) represents the energy of a photon, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon.

To find the ionisation energy of sodium per atom, we first convert the wavelength from nanometers to meters, as the speed of light is usually expressed in meters per second. The speed of light, one of the most fundamental constants in physics, is about 3.00 x 10^8 m/s. After substitution and calculation, we end up with the photon's energy, which is essentially the ionisation energy of sodium when one electron is removed. This value is vital in understanding the electronic structure and reactivity of elements.