Question

If the radius of the 3^rd Bohr’s orbit of hydrogen atom is r₃ and the radius of 4^th Bohr’s orbit is r₄. Then

• A. \(r_4=\frac{9}{16}r_3\)
• B. \(r_4=\frac{16}{9}r_3\)
• C. \(r_4=\frac{3}{4}r_3\)
• D. \(r_4=\frac{4}{3}r_3\)

Answer 1

The Correct Option is B

To solve this problem, we need to understand the Bohr model of the hydrogen atom, which describes the electron's motion in discrete circular orbits around the nucleus. The radius of these orbits, according to Bohr's theory, is given by the formula:

r_n = n^2 \times a_0

where:

• r_n is the radius of the n^{th} orbit,
• n is the principal quantum number (orbit number),
• a_0 is the Bohr radius (approximately 0.529 \ Å).

In this question, we need to find the relation between the radius of the 3^rd orbit (r_3) and the radius of the 4^th orbit (r_4).

According to the formula:

For the 3^rd orbit: r_3 = 3^2 \times a_0 = 9a_0

For the 4^th orbit: r_4 = 4^2 \times a_0 = 16a_0

To find the ratio \frac{r_4}{r_3}:

\frac{r_4}{r_3} = \frac{16a_0}{9a_0} = \frac{16}{9}

Thus, the radius of the 4^th Bohr's orbit is \frac{16}{9} times the radius of the 3^rd Bohr's orbit.

Hence, the correct answer is r_4=\frac{16}{9}r_3.