Multiple Choice
Calculate the energy of a neutron whose de Broglie wavelength is 0.7 Å. Assume the mass of the neutron is 1.675 x 10^-27 kg and use Planck's constant h = 6.626 x 10^-34 J·s.
9. Quantum Mechanics
De Broglie Wavelength
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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.
A
7.3 times the speed of light
B
0.0073 times the speed of light
C
0.073 times the speed of light
D
0.73 times the speed of light
Verified step by step guidance1
Start by recalling the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Js), \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.
Rearrange the formula to solve for the velocity \( v \): \( v = \frac{h}{m\lambda} \). This will allow us to find the speed of the electron.
Substitute the known values into the equation: \( h = 6.626 \times 10^{-34} \) Js, \( m = 9.11 \times 10^{-31} \) kg, and \( \lambda = 3.31 \times 10^{-10} \) m.
Calculate the velocity \( v \) using the substituted values. This will give you the speed of the electron in meters per second.
Finally, compare the calculated velocity \( v \) to the speed of light \( c = 3.00 \times 10^{8} \) m/s by dividing \( v \) by \( c \) to find how fast the electron is moving relative to the speed of light.
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