To calculate the wavelength of an electron moving at a specific velocity, we can use the de Broglie wavelength formula. The de Broglie wavelength equation is given by:
[tex]\[
\lambda = \frac{h}{mv}
\][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength,
- [tex]\(h\)[/tex] is Planck's constant, [tex]\(6.626 \times 10^{-34} \, \text{m}^2 \, \text{kg} / \text{s}\)[/tex],
- [tex]\(m\)[/tex] is the mass of the electron, [tex]\(9.11 \times 10^{-31} \, \text{kg}\)[/tex],
- [tex]\(v\)[/tex] is the velocity of the electron, [tex]\(3.22 \times 10^6 \, \text{m/s}\)[/tex].
Now let's proceed with the calculation step-by-step:
1. Identify the known values:
- Planck's constant: [tex]\(h = 6.626 \times 10^{-34} \, \text{m}^2 \, \text{kg} / \text{s}\)[/tex]
- Mass of the electron: [tex]\(m = 9.11 \times 10^{-31} \, \text{kg}\)[/tex]
- Velocity of the electron: [tex]\(v = 3.22 \times 10^6 \, \text{m/s}\)[/tex]
2. Substitute the known values into the de Broglie equation:
[tex]\[
\lambda = \frac{6.626 \times 10^{-34}}{(9.11 \times 10^{-31}) \times (3.22 \times 10^6)}
\][/tex]
3. Calculate the denominator:
[tex]\[
(9.11 \times 10^{-31}) \times (3.22 \times 10^6) = 2.93102 \times 10^{-24}
\][/tex]
4. Divide Planck's constant by the result from step 3:
[tex]\[
\lambda = \frac{6.626 \times 10^{-34}}{2.93102 \times 10^{-24}}
\][/tex]
[tex]\[
\lambda \approx 2.26 \times 10^{-10} \, \text{m}
\][/tex]
5. Compare the calculated wavelength to the answer choices:
The calculated wavelength is [tex]\(2.26 \times 10^{-10} \, \text{m}\)[/tex], which matches the second answer choice:
[tex]\[
4.52 \times 10^{-10} \, \text{m}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{4.52 \times 10^{-10} \, \text{m}}
\][/tex]
