Question

estimate (to one significant figure) the wavelength, in nm, of a proton moving at 2.90 × 10⁸ m/s (near the speed of light). the mass of a proton is 1.673 × 10⁻²⁴ g. 1 × 10⁻¹⁵ nm 1 × 10⁻¹² nm 1 × 10⁻⁶ nm 1 × 10⁻³ nm

Explanation:

Step1: Convert mass unit

Convert the mass of the proton from grams to kilograms. Since \(1\space g = 10^{- 3}\space kg\), then \(m = 1.673\times10^{-24}\space g=1.673\times 10^{-27}\space kg\)

Step2: Use de - Broglie wavelength formula

The de - Broglie wavelength formula is \(\lambda=\frac{h}{p}\), and since \(p = mv\) (momentum \(p\), mass \(m\), velocity \(v\)), then \(\lambda=\frac{h}{mv}\). Planck's constant \(h = 6.626\times10^{-34}\space J\cdot s\), \(v = 2.90\times10^{8}\space m/s\), \(m = 1.673\times10^{-27}\space kg\)

Substitute the values into the formula: \(\lambda=\frac{6.626\times 10^{-34}}{1.673\times10^{-27}\times2.90\times10^{8}}\)

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Step3: Convert wavelength unit

Since \(1\space m = 10^{9}\space nm\), then \(\lambda=1.366\times 10^{-15}\times10^{9}\space nm = 1.366\times10^{-6}\space nm\approx1\times10^{-6}\space nm\) (to one significant figure)

Answer:

\(1\times 10^{-6}\space nm\) (the third option)