To find the de Broglie wavelength of an electron with a kinetic energy of 60 electron volts (eV), we can follow a series of physics equations and constants. Here is a detailed step-by-step solution:
1. Understanding the Concept:
- The de Broglie wavelength ([tex]\(\lambda\)[/tex]) is given by the equation:
[tex]\[
\lambda = \frac{h}{p}
\][/tex]
where [tex]\(h\)[/tex] is Planck's constant and [tex]\(p\)[/tex] is the momentum of the electron.
2. Kinetic Energy and Momentum:
- The kinetic energy ([tex]\(KE\)[/tex]) of the electron is related to its momentum ([tex]\(p\)[/tex]) by the formula:
[tex]\[
KE = \frac{p^2}{2m}
\][/tex]
where [tex]\(m\)[/tex] is the mass of the electron.
- Solving for [tex]\(p\)[/tex]:
[tex]\[
p = \sqrt{2m \cdot KE}
\][/tex]
3. Convert Kinetic Energy:
- Given the kinetic energy of the electron is 60 eV, we need to convert this energy into joules (J). The conversion factor is [tex]\(1 \, \text{eV} = 1.602176634 \times 10^{-19} \, \text{J}\)[/tex].
[tex]\[
KE = 60 \, \text{eV} \times 1.602176634 \times 10^{-19} \, \text{J/eV} = 9.6130598 \times 10^{-18} \, \text{J}
\][/tex]
4. Use Constants:
- Planck's constant ([tex]\(h\)[/tex]) is [tex]\(6.62607015 \times 10^{-34} \, \text{m}^2 \, \text{kg} \, \text{s}^{-1}\)[/tex].
- The mass of the electron ([tex]\(m\)[/tex]) is [tex]\(9.10938356 \times 10^{-31} \, \text{kg}\)[/tex].
5. Calculate Momentum:
- Plugging in the values into the momentum formula:
[tex]\[
p = \sqrt{2 \times 9.10938356 \times 10^{-31} \, \text{kg} \times 9.6130598 \times 10^{-18} \, \text{J}} = 5.184663682 \times 10^{-24} \, \text{kg} \, \text{m/s}
\][/tex]
6. Find the de Broglie Wavelength:
- Using the de Broglie wavelength formula:
[tex]\[
\lambda = \frac{h}{p} = \frac{6.62607015 \times 10^{-34} \, \text{m}^2 \, \text{kg} \, \text{s}^{-1}}{5.184663682 \times 10^{-24} \, \text{kg} \, \text{m/s}} = 1.583309126 \times 10^{-10} \, \text{m}
\][/tex]
Therefore, the de Broglie wavelength of an electron with a kinetic energy of 60 eV is approximately [tex]\(1.583309126 \times 10^{-10} \, \text{m}\)[/tex].
