Problem 8 - University Physics with Modern Physics

Question

The photoelectric threshold wavelength of a tungsten surface is 272 \(\mathrm{nm}\) . Calculate the maximum kinetic energy of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency \(1.45 \times 10^{15} \mathrm{Hz}\) . Express the answer in electron volts.

Step-by-Step Solution

Answer

The maximum kinetic energy is approximately 1.45 eV.

Step 1: Convert Threshold Wavelength to Frequency

First, we need to find the frequency corresponding to the threshold wavelength for the photoelectric effect. Use the equation:\[ u_0 = \frac{c}{\lambda_0} \]where \( c = 3.00 \times 10^8 \mathrm{m/s} \) is the speed of light and \( \lambda_0 = 272 \mathrm{nm} = 272 \times 10^{-9} \mathrm{m} \) is the threshold wavelength.Substitute the values to find the threshold frequency:\[ u_0 = \frac{3.00 \times 10^8 \mathrm{m/s}}{272 \times 10^{-9} \mathrm{m}} \approx 1.10 \times 10^{15} \mathrm{Hz} \]

Step 2: Find the Work Function

The work function (\( \phi \)) of the material can be determined using Planck's equation in terms of threshold frequency:\[ \phi = h u_0 \]where \( h = 6.626 \times 10^{-34} \mathrm{J \cdot s} \) is Planck's constant.Substitute the values to find the work function:\[ \phi = 6.626 \times 10^{-34} \mathrm{J \cdot s} \times 1.10 \times 10^{15} \mathrm{Hz} \approx 7.29 \times 10^{-19} \mathrm{J} \]

Step 3: Calculate the Photon Energy

Next, calculate the energy of the incident ultraviolet photons using their frequency:\[ E = h u \]For the given frequency \( u = 1.45 \times 10^{15} \mathrm{Hz} \), substitute into the equation:\[ E = 6.626 \times 10^{-34} \mathrm{J \cdot s} \times 1.45 \times 10^{15} \mathrm{Hz} \approx 9.61 \times 10^{-19} \mathrm{J} \]

Step 4: Determine Maximum Kinetic Energy

The maximum kinetic energy of the ejected electrons can be found by subtracting the work function from the photon energy:\[ K_{\text{max}} = E - \phi \]Substitute the calculated values:\[ K_{\text{max}} = 9.61 \times 10^{-19} \mathrm{J} - 7.29 \times 10^{-19} \mathrm{J} \approx 2.32 \times 10^{-19} \mathrm{J} \]

Step 5: Convert Kinetic Energy to Electron Volts

Finally, convert the kinetic energy from joules to electron volts using the conversion factor \(1 \, \mathrm{eV} = 1.602 \times 10^{-19} \, \mathrm{J}\):\[ K_{\text{max}} = \frac{2.32 \times 10^{-19} \, \mathrm{J}}{1.602 \times 10^{-19} \, \mathrm{J/eV}} \approx 1.45 \, \mathrm{eV} \]

Key Concepts

Threshold Wavelength

In the fascinating realm of quantum physics, the concept of threshold wavelength is crucial when discussing the photoelectric effect. The threshold wavelength is the maximum wavelength of light that can still eject electrons from a material's surface in a photoelectric effect experiment. This phenomenon occurs because only photons with enough energy can overcome the work function of the material.
The relationship between wavelength and photon energy is inversely proportional:

• Shorter wavelengths correspond to higher energy photons.
• If the wavelength is below the threshold, photons can knock electrons out.

For tungsten, the threshold wavelength is 272 nm, meaning that any light with a wavelength longer than this will not eject electrons. To determine the threshold frequency, which is another way to express this limit, we use the formula \( u_0 = \frac{c}{\lambda_0} \), where \( c \) is the speed of light. Threshold frequency tells us the lowest frequency needed to emit electrons. Understanding threshold wavelength helps illuminate why certain materials require specific light to exhibit the photoelectric effect.

Kinetic Energy

When discussing the photoelectric effect, kinetic energy is vital. It's the energy that the ejected electrons gain after absorbing energy from photons. This energy manifests as the motion of electrons as they speed away from the surface. As part of Einstein’s explanation of the photoelectric effect, he proposed the equation: \[ K_{\text{max}} = E - \phi \] Where:

• \( K_{\text{max}} \) is the maximum kinetic energy of ejected electrons.
• \( E \) is the energy of the incoming photon.
• \( \phi \) is the work function, the minimum energy required to release electrons.

The maximum kinetic energy of electrons depends on the difference between the energy of incident photons and the work function of the material. If the photon's energy is barely enough to overcome the work function, the kinetic energy will be very low. Conversely, higher energy photons result in electrons with more kinetic energy. For practical applications, such as this problem, kinetic energy is often measured in electron volts (eV) to conveniently express small amounts of energy.

Planck's Constant

At the very heart of quantum mechanics lies Planck's constant, a fundamental quantity symbolized by \( h \). Max Planck introduced this constant to his theory about blackbody radiation, a concept that's now pivotal in explaining many quantum phenomena.Planck's constant is valued at \( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \) and connects the energy of a photon to its frequency by the formula:\[ E = hu \]Where:

• \( E \) is the energy of a photon.
• \( h \) is Planck's constant.
• \( u \) is the frequency of the radiation.

This relationship tells us that the energy of a photon is directly proportional to its frequency. Higher frequency radiation (like ultraviolet light) carries more energy, thanks to Planck's constant. This insight allows us to predict and calculate the behavior of light and matter at microscopic levels, providing solutions to phenomena like the photoelectric effect.

Ultraviolet Radiation

Ultraviolet radiation is a type of electromagnetic radiation with a frequency higher than visible light but lower than X-rays. Ultraviolet light is invisible to the naked eye but plays a significant role in activities like the photoelectric effect. In the specific context of the photoelectric effect, ultraviolet radiation provides enough energy to release electrons from a surface. Each photon in ultraviolet light possesses more energy due to its higher frequency. The typical range of ultraviolet radiation is:

• Wavelengths from about 10 nm to 400 nm.
• Higher frequencies beyond visible light.

Because of this significant amount of energy, ultraviolet radiation is often used in experiments where lower energy, visible light is insufficient. It acts as an excellent source to demonstrate the principles of the photoelectric effect, releasing electrons from metal surfaces such as tungsten when other types of light fail. This is because photons in ultraviolet light have the right mix of frequency and energy to overcome the material's work function, leading to the emission of electrons.