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

Answer: (C)

Verified step by step guidance

1. 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).
2. 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).
3. Substitute \(p = mv\) into the de Broglie equation to solve for velocity \(v\): \(\lambda = \frac{h}{mv} \implies v = \frac{h}{m\lambda}\)
4. 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}}$
5. 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.