Multiple Choice
The mass of an electron is 9.11 x 10^-31 kg. If the de Broglie wavelength for an electron in a hydrogen atom is 3.31 x 10^-10 m, how fast is the electron moving relative to the speed of light? The speed of light is 3.00 x 10^8 m/s.
9. Quantum Mechanics
De Broglie Wavelength
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What is the de Broglie wavelength of an electron moving at a velocity of 5.97 × 10^6 m/s? (Assume the mass of an electron is 9.11 × 10^-31 kg and Planck's constant is 6.63 × 10^-34 Js)
A
3.67 × 10^-10 m
B
4.89 × 10^-10 m
C
1.22 × 10^-10 m
D
2.45 × 10^-10 m
Verified step by step guidance1
Identify the formula for the de Broglie wavelength: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.
Substitute the given values into the formula: Planck's constant \( h = 6.63 \times 10^{-34} \) Js, the mass of the electron \( m = 9.11 \times 10^{-31} \) kg, and the velocity \( v = 5.97 \times 10^{6} \) m/s.
Calculate the product of the mass and velocity: \( mv = (9.11 \times 10^{-31} \text{ kg}) \times (5.97 \times 10^{6} \text{ m/s}) \).
Divide Planck's constant by the product of mass and velocity to find the de Broglie wavelength: \( \lambda = \frac{6.63 \times 10^{-34} \text{ Js}}{mv} \).
Ensure the units are consistent and simplify the expression to find the wavelength in meters.
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