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Difficulty: MediumWave-Particle Duality and de Broglie Wavelength

An electron of mass 9.1×10−31 kg9.1 \times 10^{-31}\text{ kg} is moving with a velocity of 2.0×106 m s−12.0 \times 10^6\text{ m s}^{-1}. What is the de Broglie wavelength of the electron? (Planck's constant h=6.63×10−34 J sh = 6.63 \times 10^{-34}\text{ J s})

  1. 3.64×10−10 m3.64 \times 10^{-10}\text{ m}Answer
  2. B
    1.21×10−57 m1.21 \times 10^{-57}\text{ m}
  3. C
    2.75×109 m2.75 \times 10^9\text{ m}
  4. D
    3.64×10−40 m3.64 \times 10^{-40}\text{ m}

Answer

The de Broglie wavelength of the electron is 3.64×10−10 m3.64 \times 10^{-10}\text{ m}.
According to de Broglie's wave-particle duality hypothesis, the wavelength λ\lambda associated with a moving particle of mass mm and velocity vv is given by λ=hmv\lambda = \frac{h}{m v}. Substituting h=6.63×10−34 J sh = 6.63 \times 10^{-34}\text{ J s}, m=9.1×10−31 kgm = 9.1 \times 10^{-31}\text{ kg}, and v=2.0×106 m s−1v = 2.0 \times 10^6\text{ m s}^{-1} yields λ=6.63×10−341.82×10−24=3.64×10−10 m\lambda = \frac{6.63 \times 10^{-34}}{1.82 \times 10^{-24}} = 3.64 \times 10^{-10}\text{ m}.

Step-by-Step Solution

1
Calculate the linear momentum (pp) of the electron
p=mv=(9.1×10−31 kg)×(2.0×106 m s−1)=1.82×10−24 kg m s−1p = m v = (9.1 \times 10^{-31}\text{ kg}) \times (2.0 \times 10^6\text{ m s}^{-1}) = 1.82 \times 10^{-24}\text{ kg m s}^{-1}
The de Broglie wavelength depends directly on the momentum of the moving particle.
2
Apply the de Broglie wavelength formula λ=hp\lambda = \frac{h}{p}
λ=6.63×10−34 J s1.82×10−24 kg m s−1=3.64285...×10−10 m\lambda = \frac{6.63 \times 10^{-34}\text{ J s}}{1.82 \times 10^{-24}\text{ kg m s}^{-1}} = 3.64285... \times 10^{-10}\text{ m}
Substitute Planck's constant and the calculated momentum to find the matter wavelength.
3
Express the answer in standard scientific notation with appropriate significant figures
λ=3.64×10−10 m\lambda = 3.64 \times 10^{-10}\text{ m}
Matching standard exam accuracy and precision.

Key Concept

de Broglie Wavelength of Matter Waves
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Topics

Wave-Particle Duality and de Broglie Wavelength

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