What is the de Broglie wavelength of an electron traveling at 1.35 x 10^5 m/s?

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780134112831

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Identify the formula for the de Broglie wavelength: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant (6.626 x 10^{-34} \(\text{ m}\)^2 \(\text{ kg/s}\)), \( m \) is the mass of the electron (9.109 x 10^{-31} \(\text{ kg}\)), and \( v \) is the velocity of the electron.

Substitute the given velocity \( v = 1.35 \times 10^5 \text{ m/s} \) into the formula.

Substitute the known values for Planck's constant \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \) and the mass of the electron \( m = 9.109 \times 10^{-31} \text{ kg} \) into the formula.

Calculate the de Broglie wavelength by performing the division \( \lambda = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 1.35 \times 10^5} \).

Simplify the expression to find the de Broglie wavelength \( \lambda \).

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de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like behavior of particles. 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 concept illustrates that all matter exhibits both particle and wave characteristics, which is essential for understanding phenomena at the quantum level.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity (p = mv). In the context of the de Broglie wavelength, momentum is crucial because it directly influences the wavelength of a particle. For an electron, which has a small mass, even a relatively high speed can result in a significant momentum, thereby affecting its wave properties.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics, valued at approximately 6.626 x 10^-34 Js. It relates the energy of a photon to its frequency and plays a critical role in the de Broglie wavelength equation. Understanding Planck's constant is essential for calculating the wave properties of particles, as it bridges the gap between classical and quantum physics.