Step-by-step explanation:
The angular momentum (L) of an electron in an orbit can be calculated using the formula:
\[L = n \cdot \frac{h}{2\pi}\]
Where:
- \(n\) is the principal quantum number of the electron, which is related to the orbit it's in.
- \(h\) is the Planck constant, approximately \(6.62607015 × 10^{-34}\) J·s.
- \(2\pi\) is a constant.
The de Broglie wavelength (\(\lambda\)) of an electron can be related to the radius of the orbit (\(r\)) using the following equation:
\[\lambda = \frac{h}{p}\]
Where:
- \(p\) is the momentum of the electron.
Now, the momentum of the electron in an orbit can be related to the angular momentum and the radius of the orbit:
\[p = \frac{L}{r}\]
Substitute this expression for \(p\) back into the de Broglie wavelength equation:
\[\lambda = \frac{h}{L/r}\]
Simplify:
\[\lambda = \frac{hr}{L}\]
Now, you can solve for \(L\):
\[L = \frac{hr}{\lambda}\]
So, the angular momentum of an electron with a de Broglie wavelength \(\lambda\) in an orbit of radius \(r\) is \(L = \frac{hr}{\lambda}\).