Question

The radius of innermost orbit of a hydrogen atom is 5.3x10^-11 m. What is the radius of the third allowed orbit of the hydrogen atom?

• A. 1.06Å
• B. 1.59Å
• C. 4.77Å
• D. 0.53Å

Answer 1

The Correct Option is C

To determine the radius of the third allowed orbit of a hydrogen atom, we use Bohr's model of the hydrogen atom. The model states that the radius of the n-th orbit is proportional to the square of the principal quantum number, \(n\).

The formula for the radius of an orbit in a hydrogen atom is given by:

\(r_n = n^2 \times r_0\)

where:

• \(r_n\) is the radius of the n-th orbit.
• \(r_0\) is the radius of the innermost orbit (or the first orbit), which is given as \(5.3 \times 10^{-11}\) m.
• \(n\) is the principal quantum number.

For the third orbit, \(n = 3\). Thus, the radius of the third orbit \((r_3)\) is:

\(r_3 = 3^2 \times r_0 = 9 \times 5.3 \times 10^{-11}\) m

Calculating this gives:

\(r_3 = 47.7 \times 10^{-11}\) m = \(4.77 \times 10^{-10}\) m

To convert this to angstroms (Å), knowing that \(1 \, \text{Å} = 10^{-10} \, \text{m}\):

\(r_3 = 4.77 \, \text{Å}\)

Therefore, the radius of the third allowed orbit of the hydrogen atom is 4.77 Å.

Correct Answer: 4.77 Å