Ch.5 - Periodicity & Electronic Structure of Atoms

McMurry8th EditionChemistryISBN: 9781292336145Non è quello che usi tu?Cambia libro di testo

McMurry 8th Edition

Ch.5 - Periodicity & Electronic Structure of Atoms

Problema 136b

Capitolo 5, Problema 136b

A minimum energy of 7.21⨉10^-19 J is required to produce the photoelectric effect in chromium metal. (b) Light with a wavelength of 2.50⨉10^-7 m falls on a piece of chromium in an evacuated glass tube. What is the minimum de Broglie wavelength of the emitted electrons? (Note that the energy of the incident light must be conserved; that is, the photon's energy must equal the sum of the energy needed to eject the electron plus the kinetic energy of the electron.)

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Calculate the energy of the incident photon using the formula: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (6.626 \(\times\) 10^{-34} \(\text{ J s}\)), \( c \) is the speed of light (3.00 \(\times\) 10^8 \(\text{ m/s}\)), and \( \lambda \) is the wavelength of the light (2.50 \(\times\) 10^{-7} \(\text{ m}\)).

Determine the kinetic energy of the emitted electron by subtracting the minimum energy required to eject the electron (7.21 \(\times\) 10^{-19} \(\text{ J}\)) from the energy of the incident photon calculated in the previous step.

Use the kinetic energy of the electron to find its velocity using the formula: \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of an electron (9.11 \(\times\) 10^{-31} \(\text{ kg}\)). Solve for \( v \).

Calculate the de Broglie wavelength of the emitted electron using the formula: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant and \( v \) is the velocity of the electron found in the previous step.

Ensure all units are consistent and check calculations for any errors.

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Photoelectric Effect

The photoelectric effect refers to the phenomenon where electrons are emitted from a material when it absorbs light of sufficient energy. The energy of the incoming photons must exceed a certain threshold, known as the work function, for electrons to be ejected. This effect demonstrates the particle nature of light, as photons carry quantized energy proportional to their frequency, described by the equation E = hf, where E is energy, h is Planck's constant, and f is frequency.

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Photoelectric Effect

Photon Energy and Wavelength

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. In the context of the photoelectric effect, the energy of the incident light must be calculated to determine if it can overcome the work function of the material. If the photon energy exceeds the work function, the excess energy contributes to the kinetic energy of the emitted electrons.

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Photon Energy Formulas

de Broglie Wavelength

The de Broglie wavelength is a concept that describes the wave-like behavior of particles, such as electrons. It is given by the equation λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. For emitted electrons, their momentum can be derived from their kinetic energy, allowing us to calculate their de Broglie wavelength after they are ejected from the chromium metal.

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De Broglie Wavelength Formula

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