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In the context of the photoelectric effect, the term 'work function' refers to the minimum energy required to eject an electron from the surface of a metal. For metallic sodium, this energy is provided in units of electron volts (eV). The work function is a property that varies from one metal to another and is represented by the symbol \( W_0 \). If the energy of the incoming photon is less than the work function, no photoelectrons are emitted regardless of its intensity.
This foundational concept is key to understanding why only photons with sufficient energy can cause the emission of electrons. In our example, metallic sodium has a work function of 2.3 eV. To use this in our calculations, it's important to convert the value into joules, the SI unit of energy. Remember,\(1 \, \text{eV} = 1.6 \times 10^{-19} \text{J}\), thus making the work function in joules:\( W_0 = 3.68 \times 10^{-19} \text{J}\).
Understanding the work function is crucial because it sets the threshold energy necessary for the photoelectric effect to occur.

In the photoelectric effect, the kinetic energy of the emitted photoelectrons measures how much energy they carry after being knocked free from the metal. When a photon strikes the surface, it imparts energy to the electron, and once the work function energy is covered, any remaining energy becomes the kinetic energy of the electron.
The kinetic energy can be calculated using the classical equation \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the electron and \( v \) is its velocity. For electrons, \( m \) is approximately \( 9.11 \times 10^{-31} \text{kg} \), and in our example, they have a velocity of \( 1.2 \times 10^6 \text{m/s} \). Plug these values into the equation to obtain the kinetic energy:\( KE_{max} = 6.55 \times 10^{-19} \text{J}\).
This kinetic energy is essential because it directly influences the characteristics of the emitted electrons, ultimately allowing us to derive the energy of the incident light.

Calculating the wavelength of light involves understanding the relationship between energy and wavelength. In physics, this relationship is given by \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)), \( c \) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)), and \( \lambda \) is the wavelength.
By rearranging this equation, we solve for the wavelength: \( \lambda = \frac{hc}{E} \). Using the total energy calculated from the work function and kinetic energy (\(1.023 \times 10^{-18} \, \text{J} \)), we find the wavelength to be \(1.94 \times 10^{-7} \, \text{m} \).
Finally, to make the wavelength more interpretable, we convert it to nanometers by recognizing that \(1 \, \text{m} = 10^9 \, \text{nm}\), yielding a final result of \(194 \, \text{nm}\). This step helps connect the properties of the electromagnetic spectrum to real-world observations.

Einstein's photoelectric equation is fundamental for explaining how light interacts with matter to produce the photoelectric effect. It is expressed as \( E = W_0 + KE_{max} \), where \( E \) is the energy of the incident photon, \( W_0 \) is the work function, and \( KE_{max} \) represents the maximum kinetic energy of the emitted electrons.
This equation highlights one of the pivotal breakthroughs in physics by showing that light must have both particle and wave qualities. Through rearrangement and solving for \( E \), we can derive the kinetic energy and work function, affirming that only specific light frequencies can cause electron emission.
The application of this equation enables us to determine the energy and hence the wavelength required for an electron to be liberated from a metal's surface. In our example, the equation corroborates the combined energy value needed to surpass the work function of sodium and provide kinetic energy to the electrons.