Question

38.5 electrons and matter waves
what is the de broglie wavelength of an electron that has a speed of $2.00 \times 10^4$ m/s?
\\(\bigcirc\\) $1.82 \times 10^{-3}$ m
\\(\bigcirc\\) $3.64 \times 10^{-17}$ m
\\(\bigcirc\\) $1.82 \times 10^{-10}$ m
\\(\bigcirc\\) $3.64 \times 10^{-8}$ m
\\(\bigcirc\\) $1.82 \times 10^{-26}$ m

Explanation:

Step1: Recall de Broglie wavelength formula

The de Broglie wavelength formula is \(\lambda=\frac{h}{p}\), where \(h = 6.626\times10^{-34}\space J\cdot s\) (Planck's constant) and \(p = mv\) (momentum, with \(m\) as mass and \(v\) as velocity). For an electron, \(m = 9.109\times10^{-31}\space kg\).

Step2: Calculate momentum \(p\)

Given \(v = 2.00\times10^{4}\space m/s\), \(p=mv=(9.109\times10^{-31}\space kg)(2.00\times10^{4}\space m/s)=1.8218\times10^{-26}\space kg\cdot m/s\)

Step3: Calculate wavelength \(\lambda\)

Using \(\lambda=\frac{h}{p}\), substitute \(h = 6.626\times10^{-34}\space J\cdot s\) and \(p = 1.8218\times10^{-26}\space kg\cdot m/s\):
\(\lambda=\frac{6.626\times10^{-34}}{1.8218\times10^{-26}}\approx3.64\times10^{-8}\space m\) Wait, no, wait, recalculate: Wait, \(h = 6.626\times 10^{-34}\), \(p=9.11\times 10^{-31}\times2\times 10^{4}=1.822\times 10^{-26}\). Then \(\lambda=\frac{6.626\times 10^{-34}}{1.822\times 10^{-26}}\approx3.64\times 10^{-8}\)? Wait, no, I think I made a mistake. Wait, no, let's check again. Wait, the electron mass is \(9.109\times 10^{-31}\space kg\), velocity \(2\times 10^{4}\space m/s\). So \(p = 9.109e - 31*2e4=1.8218e - 26\space kg\cdot m/s\). Then \(\lambda=h/p = 6.626e - 34/1.8218e - 26\approx3.64e - 8\)? Wait, but the options have 3.64e - 8? Wait, no, the options: 3.64×10⁻⁸ m is an option. Wait, but let me check the formula again. Wait, de Broglie wavelength is \(\lambda=\frac{h}{mv}\). So \(h = 6.626\times 10^{-34}\), \(m = 9.11\times 10^{-31}\), \(v = 2\times 10^{4}\). So \(\lambda=\frac{6.626\times 10^{-34}}{9.11\times 10^{-31}\times2\times 10^{4}}=\frac{6.626\times 10^{-34}}{1.822\times 10^{-26}}=\frac{6.626}{1.822}\times 10^{-8}\approx3.64\times 10^{-8}\space m\). So the correct option is 3.64×10⁻⁸ m.

Answer:

3.64×10⁻⁸ m (the option with 3.64×10⁻⁸ m)