Threshold Wavelength and Work Function of a Metal:

The threshold wavelength of a metal can be determined using the equation:

λ = hc / E

where λ is the wavelength, h is the Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.0 x 10^8 m/s), and E is the energy. In this case, E represents the work function of the metal.

Understanding Work Function:

The work function of a metal is the minimum energy required to remove an electron from the surface of the metal. It is typically measured in electron volts (eV). When light or other forms of electromagnetic radiation are incident on a metal, electrons can absorb energy and be emitted from the metal's surface. However, this process only occurs if the energy of the incident photons exceeds the work function of the metal.

Determining Threshold Wavelength:

To find the threshold wavelength, we need to convert the work function from electron volts to joules, as the equation requires energy in SI units:

1 eV = 1.6 x 10^-19 J

Given that the work function of the metal is 2.8 eV, we can calculate its energy in joules:

Energy (E) = 2.8 eV * 1.6 x 10^-19 J/eV = 4.48 x 10^-19 J

Now, we can substitute the values into the equation to find the threshold wavelength:

λ = (6.626 x 10^-34 J·s * 3.0 x 10^8 m/s) / (4.48 x 10^-19 J)

Simplifying the equation yields:

λ = 4.68 x 10^-7 m

Explanation:

The threshold wavelength of the metal is approximately 4.68 x 10^-7 meters or 468 nm. This means that any incident light with a wavelength shorter than 468 nm will have sufficient energy to remove electrons from the metal's surface. As the wavelength increases beyond this threshold, the energy of the photons decreases, and the probability of electron emission decreases as well.

It is important to note that the determination of threshold wavelength assumes that the metal is in a vacuum and that there are no other factors influencing the emission of electrons, such as impurities or surface conditions. Additionally, this calculation assumes that the metal follows the photoelectric effect, where the energy of the incident photons is converted into kinetic energy of the emitted electrons.

In conclusion:

The threshold wavelength of a metal with a work function of 2.8 eV is approximately 468 nm. This threshold represents the minimum wavelength required for photons to possess sufficient energy to eject electrons from the metal's surface.