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.

검증된 단계별 안내

Step 1: Understand the photoelectric effect equation, which is given by $ K_{\text{max}} = h \nu - \phi $, where $ K_{\text{max}} $ is the maximum kinetic energy of the ejected electrons, $ h $ is Planck's constant, $ \nu $ is the frequency of the incident radiation, and $ \phi $ is the work function of the material.

Step 2: Calculate the work function $ \phi $ using the threshold wavelength $ \lambda_{\text{threshold}} $. The relationship is $ \phi = \frac{hc}{\lambda_{\text{threshold}}} $, where $ c $ is the speed of light. Substitute $ h = 6.626 \times 10^{-34} \, \text{J·s} $, $ c = 3.00 \times 10^8 \, \text{m/s} $, and $ \lambda_{\text{threshold}} = 272 \, \text{nm} $ (convert to meters: $ 272 \, \text{nm} = 272 \times 10^{-9} \, \text{m} $).

Step 3: Calculate the energy of the incident ultraviolet radiation using $ E = h \nu $, where $ \nu = 1.45 \times 10^{15} \, \text{Hz} $. Substitute $ h = 6.626 \times 10^{-34} \, \text{J·s} $ and $ \nu $ into the equation.

Step 4: Use the photoelectric effect equation $ K_{\text{max}} = h \nu - \phi $ to find the maximum kinetic energy of the ejected electrons. Subtract the work function $ \phi $ (calculated in Step 2) from the energy of the incident radiation $ h \nu $ (calculated in Step 3).

Step 5: Convert the maximum kinetic energy $ K_{\text{max}} $ from joules to electron volts (eV). Use the conversion factor $ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} $. Divide $ K_{\text{max}} $ in joules by $ 1.602 \times 10^{-19} $ to express the result in electron volts.

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주요 개념

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Photoelectric Effect

The photoelectric effect is the phenomenon where electrons are emitted from a material when it absorbs light or electromagnetic radiation. This effect demonstrates the particle nature of light, where photons transfer energy to electrons. The energy of the incoming photons must exceed a certain threshold to liberate electrons, which is determined by the material's work function.

Threshold Wavelength

The threshold wavelength is the maximum wavelength of light that can cause the photoelectric effect in a given material. It is inversely related to the energy of the photons; shorter wavelengths correspond to higher energy. For tungsten, a threshold wavelength of 272 nm indicates that photons with wavelengths longer than this will not have enough energy to eject electrons.

Kinetic Energy of Ejected Electrons

The kinetic energy of ejected electrons in the photoelectric effect can be calculated using the equation KE = E_photon - φ, where E_photon is the energy of the incoming photon and φ is the work function of the material. The energy of the photon can be determined from its frequency using E_photon = h * f, where h is Planck's constant. The result is typically expressed in electron volts (eV), a common unit for measuring energy at the atomic scale.