To solve the problem regarding the photoelectric effect for metallic silver, we need to follow a series of logical steps. The photoelectric effect describes how light can eject electrons from a material, and the key parameters involved are the work function, the wavelength of the incident light, and the kinetic energy of the emitted electrons. Let's break down the calculations step by step.

Determining the Work Function

The work function (φ) is the minimum energy required to remove an electron from the surface of a metal. It can be calculated using the threshold wavelength (λ₀) and the energy of the incident light.

• Given: Threshold wavelength of silver, λ₀ = 3800 Å (1 Å = 10^-10 m)
• Energy of the incident light, λ = 2600 Å

First, we need to convert the wavelengths from Angstroms to meters:

• λ₀ = 3800 Å = 3800 x 10^-10 m = 3.8 x 10^-7 m
• λ = 2600 Å = 2600 x 10^-10 m = 2.6 x 10^-7 m

Next, we can calculate the work function using the formula:

φ = (hc) / λ₀

Where:

• h = Planck's constant = 6.626 x 10^-34 J·s
• c = speed of light = 3 x 10⁸ m/s

Substituting the values:

φ = (6.626 x 10^-34 J·s * 3 x 10⁸ m/s) / (3.8 x 10^-7 m)

Calculating this gives:

φ ≈ 5.23 x 10^-19 J

To convert this energy into electron volts (eV), we use the conversion factor (1 eV = 1.6 x 10^-19 J):

φ (in eV) = φ (in J) / (1.6 x 10^-19 J/eV)

φ ≈ 5.23 x 10^-19 J / (1.6 x 10^-19 J/eV) ≈ 3.27 eV

Calculating Maximum Kinetic Energy of Emitted Electrons

The maximum kinetic energy (K.E.) of the emitted photoelectrons can be calculated using the formula:

K.E. = E_incident - φ

Where E_incident is the energy of the incident light:

E_incident = (hc) / λ

Substituting the values:

E_incident = (6.626 x 10^-34 J·s * 3 x 10⁸ m/s) / (2.6 x 10^-7 m)

Calculating this gives:

E_incident ≈ 7.65 x 10^-19 J

Now, we can find the maximum kinetic energy:

K.E. = 7.65 x 10^-19 J - 5.23 x 10^-19 J ≈ 2.42 x 10^-19 J

Converting this to eV:

K.E. ≈ 2.42 x 10^-19 J / (1.6 x 10^-19 J/eV) ≈ 1.51 eV

Finding the Maximum Velocity of Photoelectrons

The maximum velocity (v) of the emitted photoelectrons can be determined using the kinetic energy formula:

K.E. = (1/2)mv²

Rearranging gives:

v = sqrt((2 * K.E.) / m)

Substituting the values:

• K.E. = 2.42 x 10^-19 J
• m (mass of electron) = 9.11 x 10^-31 kg

Calculating this gives:

v = sqrt((2 * 2.42 x 10^-19 J) / (9.11 x 10^-31 kg))

v ≈ 0.7829 x 10^-6 m/s

In summary, we have calculated:

• Work function: φ ≈ 3.27 eV or 5.23 x 10^-19 J
• Maximum kinetic energy of emitted electrons: K.E. ≈ 1.5 eV
• Maximum velocity of photoelectrons: v ≈ 0.7829 x 10^-6 m/s

This process illustrates the relationship between light and electron emission, showcasing the principles of the photoelectric effect in action.