Calculate the radius of bohr's third orbit in hydrogen atom.

Explanation

To calculate the radius of Bohr's third orbit in a hydrogen atom, we can use the formula for the radius of the nth orbit in a hydrogen-like atom:

\[ R_n = \frac{n^2 h^2}{4 \pi^2 m e^2} \cdot \frac{1}{z} \]

Where:
- \( R_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (for the third orbit, \( n = 3 \)),
- \( h \) is Planck's constant,
- \( m \) is the mass of the electron,
- \( e \) is the charge of the electron,
- \( z \) is the atomic number of the atom (for hydrogen, \( z = 1 \)).

From the derivation, we can simplify the formula for hydrogen:

\[ R_n = 0.529 \cdot \frac{n^2}{z} \text{ angstroms} \]

Now, substituting the values for the third orbit (\( n = 3 \) and \( z = 1 \)):

1. Calculate \( n^2 \):
\[ n^2 = 3^2 = 9 \]

2. Substitute \( n^2 \) and \( z \) into the formula:
\[ R_3 = 0.529 \cdot \frac{9}{1} \text{ angstroms} \]

3. Calculate \( R_3 \):
\[ R_3 = 0.529 \cdot 9 = 4.761 \text{ angstroms} \]

Thus, the radius of Bohr's third orbit in a hydrogen atom is approximately **4.761 angstroms**.