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Chapter 39: Problem 14

At what speed is an electron's de Broglie wavelength (a) \(1.0 \mathrm{pm},\) (b) \(1.0 \mathrm{nm},\) (c) \(1.0 \mu \mathrm{m},\) and (d) \(1.0 \mathrm{mm} ?\)

Short Answer

Expert verified
The calculated speeds of the electron for the respective wavelengths are: (a) \(v = 7.27 *10^6 \mathrm{m/s}\) for \(1.0 \mathrm{pm}\),(b) \(v = 7.27 *10^3 \mathrm{m/s}\) for \(1.0 \mathrm{nm}\),(c) \(v = 7.27 \mathrm{m/s}\) for \(1.0 \mu \mathrm{m}\), (d) \(v = 7.27 *10^{-3} \mathrm{m/s}\) for \(1.0 \mathrm{mm}\).

Step by step solution

01

Identify Known Values

We know that \(h = 6.626 * 10^-34 \mathrm{Js}\), the mass \(m\) of an electron is \(9.11 * 10^-31 \mathrm{kg}\), and the value of the given wavelengths \(\lambda\) in meters.
02

Convert Wavelengths to Meters

Convert the wavelengths to meters: (a) \(1.0 \mathrm{pm} = 1.0 * 10^-12 \mathrm{m}\), (b) \(1.0 \mathrm{nm} = 1.0 * 10^-9 \mathrm{m}\), (c) \(1.0 \mu \mathrm{m} = 1.0 * 10^-6 \mathrm{m}\), (d) \(1.0 \mathrm{mm} = 1.0 * 10^-3 \mathrm{m}\).
03

Calculate Velocity using the Formula

Substitute the known values into the formula \(v = h/(m*\lambda)\) and calculate the speed for each given wavelength.
04

Repeat Calculations for all Wavelengths

Repeat Step 3 for all given wavelengths to obtain the speeds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Mass
The electron mass is a fundamental constant in physics that describes the mass of an electron. This value is approximately \(9.11 \times 10^{-31} \) kilograms.
It is an essential value in the calculation of various physical phenomena, including the de Broglie wavelength.
Electrons are subatomic particles with a very small mass.
Although it might seem negligible, their mass plays a crucial role in the equations that describe electronic behavior and interactions.
  • The rest mass of an electron is always \(9.11 \times 10^{-31} \) kg.
  • It is essential for calculations in quantum mechanics and particle physics.
  • Mass affects how particles behave at high speeds or in high-precision measurements.
Planck Constant
The Planck constant is another fundamental constant crucial for understanding the quantum world.
Denoted by \( h \), its value is approximately \(6.626 \times 10^{-34} \mathrm{Js}\).
It appears in many key equations in physics, such as the equation for energy of a photon, \( E = h u \), and the de Broglie wavelength equation.
  • Planck's constant relates to the behavior of particles at very small scales.
  • It helps link the energy of the particles with their frequency.
  • Understanding \( h \) is foundational for fields like quantum mechanics and photonics.
Wavelength Conversion
Converting units of wavelength is vital when calculating de Broglie wavelengths.
Wavelengths are often given in various units such as picometers, nanometers, micrometers, and millimeters.
For calculations, wavelengths typically need to be converted into meters, the standard SI unit for length.
For example:
  • 1 picometer = \(1.0 \times 10^{-12}\) meters.
  • 1 nanometer = \(1.0 \times 10^{-9}\) meters.
  • 1 micrometer = \(1.0 \times 10^{-6}\) meters.
  • 1 millimeter = \(1.0 \times 10^{-3}\) meters.
These conversions ensure that the dimensions match across different parts of your calculations.
Velocity Calculation
To find the velocity of a particle such as an electron, using its de Broglie wavelength, we use the formula:
\[ v = \frac{h}{m \cdot \lambda} \]
Here, \( v \) is the velocity, \( h \) is the Planck constant, \( m \) is the mass of the electron, and \( \lambda \) is the wavelength.
  • This formula shows how wavelength is inversely proportional to velocity.
  • For a smaller wavelength, the velocity of the electron is higher.
  • Different wavelengths will result in varying velocities, showcasing the wave-particle duality of matter.
By substituting the known values of the electron's mass and Planck constant, you can calculate the electron's speed for any given wavelength.
This allows you to understand particle behavior at microscopic scales.

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Most popular questions from this chapter

a. Calculate the orbital radius and the speed of an clectron in both the \(n=99\) and the \(n=100\) states of hydrogen. b. Determine the orbital frequency of the electron in each of these states. c. Calculate the frequency of a photon emitted in a \(100 \rightarrow 99\) transition. d. Compare the photon frequency of part c to the average of your two orbital frequencies from part b. By what percent do they differ?

A 100 W lightbulb emits about 5 W of visible light. (The other \(95 \mathrm{W}\) are emitted as infrared radiation or lost as heat to the surroundings.) The average wavelength of the visible light is about \(600 \mathrm{nm},\) so make the simplifying assumption that all the light has this wavelength. a. What is the frequency of the emitted light? b. How many visible-light photons does the bulb emit per second?

Consider an electron undergoing cyclotron motion in a magnetic ficld. According to Bohr, the electron's angular momentum must be quantized in units of \(\hbar\) a. Show that allowed radii for the clectron's orbit are given by \(r_{n}=(n \hbar / e B)^{1 / 2},\) where \(n=1,2,3, \ldots\) b. Compute the first four allowed radii in a 1.0 T magnetic field. c. Find an expression for the allowed cnergy levels \(E_{n}\) in terms of \(\hbar\) and the cyclotron frequency \(f_{q \sigma}\)

What is the quantum number of an electron confined in a 3.0-nm-long one- dimentional box if the electron's de Broglie wavelength is \(1.0 \mathrm{nm} ?\)

Show that Planck's constant \(\hbar\) has units of angular momentum.
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