Question

The threshold wavelength for the photoelectric effect in a specific alloy is \(400 . \mathrm{nm}\). What is the work function in \(\mathrm{eV} ?\)

Short answer

Answer: The work function of the alloy is 3.1 eV.

Step-by-step solution

Identify given information and constants

The information provided by the exercise is: - Threshold wavelength (\(λ_{threshold}\)) = 400 nm - Planck's constant (\(h\)) = 6.626 x 10^-34 Js - Speed of light (\(c\)) = 3 x 10^8 m/s - Electron charge (\(e\)) = 1.602 x 10^-19 C

Convert the threshold wavelength to meters

To work with the threshold wavelength in the equations, we need to have it in meters. So, we will convert the wavelength from nm to meters: \(λ_{threshold} = 400\,\text{nm} \times \frac{1\,\text{m}}{10^9\,\text{nm}} = 4 \times 10^{-7}\,\text{m}\)

Calculate the energy when electrons are ejected

Now, we can use Planck's equation to find the energy (\(E\)) needed to eject electrons as: \(E = \frac{h \times c}{λ_{threshold}}\) Substitute the known values into the equation: \(E = \frac{(6.626 \times 10^{-34}\,\text{Js}) \times (3 \times 10^8\,\text{m/s})}{(4 \times 10^{-7}\,\text{m})} = 4.9695 \times 10^{-19}\,\text{J}\)

Convert the energy into electron volts

To find the energy in electron volts (eV), we will divide the energy in joules by the electron charge: \(E_{eV} = \frac{E}{e} = \frac{4.9695 \times 10^{-19}\,\text{J}}{1.602 \times 10^{-19}\,\text{C}} = 3.1\,\text{eV}\)

Find the work function

The work function is the minimum energy required to eject electrons, and in this case, it is equal to the calculated energy in electron volts: Work function = \(E_{eV} = 3.1\,\text{eV}\) Hence, the work function of the specific alloy is \(3.1\,\text{eV}\).

Key concepts

Threshold Wavelength

The photoelectric effect involves understanding the concept of a **threshold wavelength**. This is the specific wavelength of light at which electrons begin to be ejected from a material's surface due to the absorption of photons. If the wavelength of the incident light is longer than the threshold wavelength, no electrons will be emitted. Conversely, if the wavelength is shorter, electrons are emitted.

The relationship between the wavelength of the light and its energy is fundamental: shorter wavelengths correspond to higher energy photons. This can be calculated using the formula:

• \( E = \frac{h \times c}{\lambda_{threshold}} \)

where \( E \) is energy in joules, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda_{threshold} \) is the threshold wavelength.
In our example, the threshold wavelength was given as 400 nm, which when converted to meters becomes \( 4 \times 10^{-7} \) m. This is a typical calculation to determine the start of electron emission in the photoelectric effect.

Work Function

The **work function** emerges as a key term when discussing the photoelectric effect. It represents the minimum energy needed to remove an electron from a solid's surface. This energy is unique to each material and depends on its electronic structure.

When we say the work function, we often express it in electron volts (eV). One electron volt is the energy gained by an electron when it is accelerated through one volt of electric potential. The work function can be calculated using the energy determined by threshold wavelength:

• First, find the energy (in joules) using Planck's equation: \( E = \frac{h \times c}{\lambda} \).
• Then convert the result from joules to electron volts by dividing by the electron charge: \( E_{eV} = \frac{E}{e} \).

In the given problem, this energy was found to be \( 3.1 \, \text{eV} \), indicating the minimum energy required to eject an electron from the alloy's surface with the given threshold wavelength of light.

Planck's Constant

Understanding **Planck's constant** is crucial when analyzing the photoelectric effect. This fundamental constant plays a central role in quantum mechanics, as it relates the energy of photons to their frequency. Planck's equation, \( E = h \times f \), shows how Planck's constant \( h \) bridges the gap between wave and particle nature of light.

Planck's constant has a value of \( 6.626 \times 10^{-34} \, \text{Js} \), where \( J \) stands for joules and \( s \) for seconds. It appears in equations related to both the threshold wavelength and the energy calculations in electron ejection scenarios:

• Through the formula \( E = \frac{h \times c}{\lambda_{threshold}} \), it helps calculate the energy of the photon at the threshold wavelength.

Its small magnitude points to the incredibly minuscule scale of quantum effects, where even tiny energies correspond to sizable effects at the atomic or subatomic levels. This reinforces how Planck's constant is a pivot in understanding and predicting the behavior of electrons in materials subjected to electromagnetic radiation.