Chapter 38: Problem 6
The photoelectric threshold wavelength of a tungsten surface is 272 nm. Calculate the maximum kinetic energy of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency 1.45 \(\times\) 10\(^{15}\) Hz. Express the answer in electron volts.
Short Answer
Expert verified
The maximum kinetic energy is 1.45 eV.
Step by step solution
01
Understand the Problem
We're given the threshold wavelength of tungsten and asked to calculate the maximum kinetic energy of electrons ejected when the surface is illuminated by UV light with a specified frequency. We'll use the photoelectric effect equations to find this energy.
02
Convert Threshold Wavelength to Threshold Frequency
We know that frequency \( u \) and wavelength \( \lambda \) are related by the speed of light \( c \): \( c = u \lambda \). Using this, the threshold frequency (\( u_0 \)) can be calculated: \[ u_0 = \frac{c}{\lambda_0} \] where \( \lambda_0 = 272 \text{ nm} = 272 \times 10^{-9} \text{ m} \) and \( c = 3 \times 10^8 \text{ m/s} \).
03
Calculate the Threshold Frequency
Using the formula from Step 2, calculate \( u_0 \): \[ u_0 = \frac{3 \times 10^8}{272 \times 10^{-9}} = 1.10 \times 10^{15} \text{ Hz} \]
04
Find the Energy of Incident Photons
The energy of the incident photons \( E \) is given by Planck's equation: \( E = h u \). Here, \( u = 1.45 \times 10^{15} \text{ Hz} \), and \( h = 6.63 \times 10^{-34} \text{ J s} \).
05
Calculate the Energy of Incident Photons
Substitute the values into Planck's equation: \[ E = 6.63 \times 10^{-34} \times 1.45 \times 10^{15} = 9.61 \times 10^{-19} \text{ J} \]
06
Calculate the Work Function
The work function \( \phi \) is the minimum energy required to eject an electron, calculated using \( \phi = h u_0 \), where \( u_0 = 1.10 \times 10^{15} \text{ Hz} \).
07
Determine the Work Function
Substitute into the equation: \[ \phi = 6.63 \times 10^{-34} \times 1.10 \times 10^{15} = 7.29 \times 10^{-19} \text{ J} \]
08
Calculate the Maximum Kinetic Energy
The maximum kinetic energy (KE) of the ejected electrons is given by \( KE = E - \phi \). Substitute the calculated values: \[ KE = 9.61 \times 10^{-19} - 7.29 \times 10^{-19} = 2.32 \times 10^{-19} \text{ J} \]
09
Convert Joules to Electron Volts
To convert the kinetic energy from joules to electron volts, use the conversion \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \). Thus, \[ KE = \frac{2.32 \times 10^{-19}}{1.602 \times 10^{-19}} \text{ eV} = 1.45 \text{ eV} \]
10
Final Answer
The maximum kinetic energy of the electrons ejected from the tungsten surface is \( 1.45 \text{ eV} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Threshold Wavelength
In the realm of photoelectric effect, the threshold wavelength plays a pivotal role. It refers to the longest wavelength of light that is capable of ejecting electrons from a material's surface.
When a light of wavelength longer than this threshold strikes the substance, no electrons are ejected, no matter how intense the light may be. This happens because the energy of the photons is simply not enough to liberate an electron from the material's atomic structure.
To calculate the threshold frequency, we use the equation relating the speed of light (\( c \)) to frequency (\( u \)) and wavelength (\( \lambda \)): = u \lambda .
When a light of wavelength longer than this threshold strikes the substance, no electrons are ejected, no matter how intense the light may be. This happens because the energy of the photons is simply not enough to liberate an electron from the material's atomic structure.
To calculate the threshold frequency, we use the equation relating the speed of light (\( c \)) to frequency (\( u \)) and wavelength (\( \lambda \)):
- Convert the given threshold wavelength to meters.
- Use the speed of light in meters per second (\(3 \times 10^8 \) m/s).
- Determine the threshold frequency (\( u_0 \)), using the formula: \( u_0 = \frac{c}{\lambda_0}\).
Kinetic Energy Calculation
The calculation of kinetic energy of electrons plays a central role in understanding the photoelectric effect. When light hits the surface of a material, it ejects electrons with specific kinetic energy.
The energy of each photon is given by Planck's equation \(E = h u \), where \(E\) is the energy, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \text{ J s}\)), and \(u\) is the frequency of the light.
Once the incident energy is known, the next step is to calculate the work function, which we will discuss later. Subtract the work function from the photon energy to find the kinetic energy of the ejected electrons:
The energy of each photon is given by Planck's equation \(E = h u \), where \(E\) is the energy, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \text{ J s}\)), and \(u\) is the frequency of the light.
Once the incident energy is known, the next step is to calculate the work function, which we will discuss later. Subtract the work function from the photon energy to find the kinetic energy of the ejected electrons:
- Find the incident photon energy.
- Use the work function to get the effective energy of electrons.
- Calculate maximum kinetic energy using \(KE = E - \phi \).
Work Function
The work function (\(\phi\)) is a key concept when examining the photoelectric effect. It is defined as the minimum energy required to dislodge an electron from the surface of a material.
In most cases, the work function is tied to the intrinsic properties of the material itself. To calculate it, we multiply Planck’s constant (\(h\)) by the threshold frequency (\(u_0\)), which was derived from the threshold wavelength: \(\phi = h u_0\).
This value serves as a threshold, representing the barrier that photon energy must overcome to emit electrons.
In most cases, the work function is tied to the intrinsic properties of the material itself. To calculate it, we multiply Planck’s constant (\(h\)) by the threshold frequency (\(u_0\)), which was derived from the threshold wavelength: \(\phi = h u_0\).
This value serves as a threshold, representing the barrier that photon energy must overcome to emit electrons.
- Calculate using the threshold frequency and Planck's constant.
- Represents the energy barrier for electron ejection.
- A crucial parameter in predicting electron emissions.