Multiple Choice
What is the de Broglie wavelength of an electron traveling at 1.12 x 10^5 m/s, given that the mass of an electron is 9.11 x 10^-31 kg?
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9. Quantum Mechanics
De Broglie Wavelength
Multiple Choice
At what speed must an electron travel for its de Broglie wavelength to be 1.0 nm?
A
1.1 × 10^4 m/s
B
2.2 × 10^6 m/s
C
7.3 × 10^5 m/s
D
3.0 × 10^7 m/s
Verified step by step guidance1
Recall the de Broglie wavelength formula, which relates a particle's wavelength \(\lambda\) to its momentum \(p\):
\(\lambda = \frac{h}{p}\)
where \(h\) is Planck's constant (\(6.626 \times 10^{-34}\) J·s).
Express the momentum \(p\) of the electron in terms of its mass \(m\) and velocity \(v\):
\(p = mv\)
where \(m\) is the mass of the electron (\(9.109 \times 10^{-31}\) kg).
Substitute \(p = mv\) into the de Broglie equation to solve for velocity \(v\):
\(\lambda = \frac{h}{mv} \implies v = \frac{h}{m\lambda}\)
Plug in the known values for \(h\), \(m\), and the given wavelength \(\lambda = 1.0\) nm (which is \(1.0 \times 10^{-9}\) m) into the velocity equation:
\(v = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 1.0 \times 10^{-9}}\)
Calculate the velocity \(v\) from the above expression to find the speed at which the electron must travel to have a de Broglie wavelength of 1.0 nm.
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