Problem 9 - College Physics

Question

\(\bullet\) The photoelectric threshold wavelength of a tungsten surface is 272 \(\mathrm{nm}\) . (a) What are the threshold frequency and work function (in eV) of this tungsten? (b) Calculate the maximum kinetic energy (in eV) of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency \(1.45 \times 10^{15} \mathrm{Hz}\)

Step-by-Step Solution

Answer

(a) Threshold frequency: \(1.10 \times 10^{15}\) Hz, Work function: 4.55 eV. (b) Maximum kinetic energy: 1.45 eV.

Step 1: Determine the Threshold Frequency

To find the threshold frequency, use the relation between wavelength and frequency given by the equation \( c = \lambda \cdot f \), where \( c \) is the speed of light (\(3 \times 10^8 \ m/s\)), \( \lambda \) is the threshold wavelength (272 nm or 272 \( \times 10^{-9} \ m\)), and \( f \) is the frequency. Rearranging the equation gives:\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8}{272 \times 10^{-9}} = 1.10 \times 10^{15} \ Hz.\]

Step 2: Calculate the Work Function

The work function \( \phi \) (in eV) can be calculated using the equation \( \phi = h \cdot f_0 \), where \( h \) is Planck's constant (\(4.135667696 \times 10^{-15} \ eV \cdot s\)), and \( f_0 \) is the threshold frequency \(1.10 \times 10^{15} \ Hz\). Thus:\[ \phi = 4.135667696 \times 10^{-15} \ eV \cdot s \times 1.10 \times 10^{15} \ Hz \approx 4.55 \ eV.\]

Step 3: Calculate Maximum Kinetic Energy of Ejected Electrons

Use the photoelectric equation \( KE = h \cdot f - \phi \), where \( KE \) is the kinetic energy, \( f \) is the given frequency \(1.45 \times 10^{15} \ Hz\), and \( \phi \) is the work function calculated in the previous step. Plug in the values:\[ KE = 4.135667696 \times 10^{-15} \ eV \cdot s \times 1.45 \times 10^{15} \ Hz - 4.55 \ eV \ KE \approx 6.00 \ eV - 4.55 \ eV = 1.45 \ eV.\]

Key Concepts

Work Function

The work function is the minimum energy needed to remove an electron from the surface of a metal. In the photoelectric effect, if the energy of incoming light is less than the work function, no electrons will be ejected, because the light doesn't give the electrons enough energy to "jump out" of the material. This concept is important in understanding how light interacts with different materials.

• The work function is usually measured in electronvolts (eV), providing a convenient unit for dealing with the small amounts of energy involved in atomic-level processes.
• It varies across different materials; some materials require more energy to release electrons than others.

For the tungsten example from the exercise, the work function was calculated using the formula \( \phi = h \cdot f_0 \), which signifies the energy required to start the photoelectric process. This amounted to 4.55 eV, indicating the threshold energy needed for electrons to escape from the tungsten surface.

Threshold Frequency

Threshold frequency is the minimum frequency of light required to emit electrons from a material's surface in the photoelectric effect. If light hits the material with a frequency below this threshold, no photoelectrons are emitted regardless of the light's intensity.

• The threshold frequency is determined by the work function of the material. It sets the "starting point" for the photoelectric effect.
• To calculate it, you can use the equation: \( f = \frac{c}{\lambda} \), where \( c \) is the speed of light, and \( \lambda \) is the wavelength.

In our exercise, the threshold frequency for the tungsten surface came out to be 1.10 \( \times 10^{15} \) Hz. This means any light below this frequency won't have enough energy to free electrons from the tungsten, regardless of its intensity.

Kinetic Energy

Kinetic energy in the photoelectric effect refers to the energy that ejected electrons possess after they've absorbed light energy sufficient to overcome the work function. It's the "leftover" energy from the light, transformed into the motion of electrons.

• The kinetic energy of these electrons depends on the frequency of the incident light relative to the threshold frequency.
• According to the formula \( KE = h \cdot f - \phi \), the kinetic energy is the difference between the energy of the incident photons \( h \cdot f \) and the work function \( \phi \).

From the original exam problem, once you calculated the work function, you could find the maximum kinetic energy of ejected electrons with the given ultraviolet light frequency. The resulting 1.45 eV shows how much energy the electrons can carry as they depart the tungsten surface.