Question

Find the wavelength of light required for the ionization of sodium atoms to occur. The ionization potential of sodium is \(8.17 \times 10^{-19} \mathrm{~J}\)

Short answer

The wavelength of light required for the ionization of sodium atoms is approximately \(242.3 \mathrm{~nm}\).

Step-by-step solution

Identify the relevant constants and variables

We are given the ionization potential of sodium, \(E = 8.17 \times 10^{-19} \mathrm{~J}\). We need to find the wavelength of light, \(\lambda\). We will also need Planck's constant, \(h = 6.626 \times 10^{-34} \mathrm{~Js}\), and the speed of light, \(c = 2.998 \times 10^{8} \mathrm{~m/s}\).

Write down the Planck's Equation

Planck's Equation relates the energy, \(E\), of a photon with its frequency, \(f\), and wavelength, \(\lambda\): \(E=hf=\dfrac{hc}{\lambda}\)

Solve for the wavelength

Using the data given and Planck's Equation, we can solve for \(\lambda\). We were given the ionization energy, so we have: \(8.17 \times 10^{-19} \mathrm{~J} = \dfrac{(6.626 \times 10^{-34} \mathrm{~Js})(2.998 \times 10^{8} \mathrm{~m/s})}{\lambda}\)

Rearrange the equation and calculate the wavelength

Rearranging the equation to solve for \(\lambda\), we get: \(\lambda = \dfrac{(6.626 \times 10^{-34} \mathrm{~Js})(2.998 \times 10^{8} \mathrm{~m/s})}{8.17 \times 10^{-19} \mathrm{~J}}\) Now, plug in the given values and calculate \(\lambda\): \(\lambda = \dfrac{(6.626 \times 10^{-34})(2.998 \times 10^{8})}{8.17 \times 10^{-19}}\) \(\lambda = 2.423 \times 10^{-7} \mathrm{~m}\)

Convert the wavelength to nanometers

Since wavelengths are usually expressed in nanometers (nm), we can convert \(\lambda\) from meters to nanometers: \(\lambda = (2.423 \times 10^{-7} \mathrm{~m}) \times \dfrac{10^9 \mathrm{~nm}}{1 \mathrm{~m}} = 242.3 \mathrm{~nm}\) Thus, the wavelength of light required for the ionization of sodium atoms is approximately \(242.3 \mathrm{~nm}\).

Key concepts

Wavelength of Light

Light is a form of energy that travels in waves, much like waves on the surface of water. The wavelength of light is the distance between successive peaks or troughs in these waves. It's a critical measurement in physics because it's directly related to the energy and frequency of the light. Wavelengths are commonly measured in meters (m), but they can also be expressed in other units such as nanometers (nm) or angstroms (Å), depending on the context or the scale of the wavelengths involved.

In the context of the exercise, understanding the wavelength of light helps us determine the type of light needed to ionize sodium atoms. The ionization potential of sodium requires a specific amount of energy, and this energy corresponds to a certain wavelength of light according to the relationship provided by Planck's equation. Shorter wavelengths correspond to higher energy photons, and vice versa. For sodium, the necessary energy level is achieved with light of approximately 242.3 nm, which falls within the ultraviolet range of the electromagnetic spectrum.

Planck's Equation

Closely associated with the quantum theory of light, Planck's equation is a fundamental principle in quantum mechanics. It describes the quantized nature of energy as discrete packets called 'quanta' or photons when considering electromagnetic waves. The formula \(E=hf\) outlines the direct relationship between the energy \(E\) of a photon and its frequency \(f\), with \(h\) being Planck's constant (\(6.626 \times 10^{-34} J\cdot s\)). By introducing the speed of light (\(c\)), Planck's equation can be transformed to relate energy and wavelength: \(E = \dfrac{hc}{\lambda}\).

This equation signifies that the energy of a photon is inversely proportional to its wavelength—the longer the wavelength, the lower the energy of the photon, and vice versa. In the exercise for ionizing sodium, we use this equation to find the wavelength of light that matches the ionization potential.

Energy of a Photon

The concept of a photon portrays light as a particle carrying energy. The energy of a photon is a crucial idea when discussing phenomena at the quantum level, like the ionization of atoms. Each photon has a specific amount of energy that depends on its frequency or wavelength. As elucidated by Planck's equation, the shorter the wavelength of a photon, the greater its energy. Conversely, photons with long wavelengths will carry less energy.

Thinking of light in terms of photons and their associated energies allows us to understand processes like photoelectric effect or, as in this exercise, the ionization of atoms. To ionize a sodium atom, a photon must have energy equal to or greater than the atom's ionization potential. Using the known values of Planck's constant and the speed of light, we can calculate the energy of these photons and determine the wavelength of light required to ionize sodium, which we found to be approximately 242.3 nm.