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Sketch the interference pattern that results from the diffraction of electrons passing through two closely spaced slits.
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Ch.7 - Quantum-Mechanical Model of the Atom
Tro4th EditionChemistry: A Molecular ApproachISBN: 9780134112831
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TOC.tableOfContents
- Ch.1 - Matter, Measurement & Problem Solving136
- Ch.2 - Atoms & Elements112
- Ch.3 - Molecules, Compounds & Chemical Equations159
- Ch.4 - Chemical Quantities & Aqueous Reactions140
- Ch.5 - Gases118
- Ch.6 - Thermochemistry106
- Ch.7 - Quantum-Mechanical Model of the Atom62
- Ch.8 - Periodic Properties of the Elements103
- Ch.9 - Chemical Bonding I: The Lewis Model104
- Ch.10 - Chemical Bonding II: Molecular Shapes & Valence Bond Theory89
- Ch.11 - Liquids, Solids & Intermolecular Forces73
- Ch.12 - Solids and Modern Material59
- Ch.13 - Solutions110
- Ch.14 - Chemical Kinetics140
- Ch.15 - Chemical Equilibrium78
- Ch.16 - Acids and Bases155
- Ch.17 - Aqueous Ionic Equilibrium176
- Ch.18 - Free Energy and Thermodynamics108
- Ch.19 - Electrochemistry118
- Ch.20 - Radioactivity and Nuclear Chemistry82
- Ch.21 - Organic Chemistry150
Tro4th EditionChemistry: A Molecular ApproachISBN: 9780134112831Non è quello che usi tu?Cambia libro di testo
Capitolo 7, Problema 49
The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.20 nm in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.20 nm?
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Identify the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of an electron, and \( v \) is the velocity of the electron.
Rearrange the formula to solve for velocity \( v \): \( v = \frac{h}{m\lambda} \).
Substitute the known values into the equation: Planck's constant \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \), the mass of an electron \( m = 9.109 \times 10^{-31} \text{ kg} \), and the desired wavelength \( \lambda = 0.20 \text{ nm} = 0.20 \times 10^{-9} \text{ m} \).
Calculate the velocity \( v \) using the substituted values.
Ensure the units are consistent and check the calculation for any possible errors.
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Resolution Limit
The resolution limit of a microscope refers to the smallest distance between two points that can still be distinguished as separate entities. In optical microscopes, this limit is primarily determined by the wavelength of light used; shorter wavelengths allow for higher resolution. In electron microscopy, the resolution can be significantly improved due to the much shorter de Broglie wavelength of electrons compared to visible light.
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Limiting Reagent Concept
de Broglie Wavelength
The de Broglie wavelength is a concept from quantum mechanics that describes the wave-like behavior of particles, such as electrons. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This relationship implies that faster-moving particles have shorter wavelengths, which is crucial for achieving high resolution in electron microscopy.
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00:58De Broglie Wavelength Formula
Momentum and Speed of Electrons
Momentum (p) is defined as the product of mass (m) and velocity (v) of an object, expressed as p = mv. In the context of electrons in an electron microscope, their speed must be calculated to achieve a specific de Broglie wavelength for desired resolution. By manipulating the speed of the electrons, one can control their momentum and thus their wavelength, allowing for the fine-tuning of the microscope's resolution capabilities.
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