Question

What is the ratio of the first three radii of Bohr orbits?
(a) 1:05:3
(b) 1:2:3
(c) 1:4:9
(d) 1:8:27

Answer 1

Hint: It's very necessary to recall what is the Bohr orbital. So, this can be explained that in the Bohr model of the atom, electrons travel in defined circular orbits around the nucleus. Electrons can jump from one orbit to a different by emitting or absorbing energy.

Complete step by step solution:
Lets understand Bohr models and some of its main ideas:
Bohr’s model consists of a little nucleus (positively charged) surrounded by negative electrons traveling the nucleus in orbits. Bohr found that an electron located far away from the nucleus has more energy, and electrons on the brink of the nucleus have less energy.
Postulates of Bohr’s Model of an Atom
In an atom, electrons (negatively charged) revolve round the charged nucleus during a definite circular path called orbits or shells.
Each orbit or shell features a fixed energy and these circular orbits are referred to as orbital shells.
The energy levels are represented by an integer (n=1, 2, 3…) referred to as the quantum number. This range of quantum numbers starts from the nucleus side with n=1 having a rock bottom energy state, when an electron attains a rock bottom energy state, it's said to be within the state.
The electrons in an atom move from a lower energy state to a better energy state by gaining the specified energy and an electron moves from a better energy state to lower energy level by losing energy.
Atomic number is $ Z = 1 $
Hence the radius of nth orbital
 $ r = {n^2} \times 0.529 $
we can say that $ r \propto {n^2} $
For the first three orbits, n values are 1, 2 and 3
Therefore, the ratio of radii of first three orbit is given by
 $
  {r_1}:{r_2}:{r_3} = {1^2}:{2^2}:{3^2} \\
  {r_1}:{r_2}:{r_3} = 1:4:9 \\
  $
Hence (c) is the correct option.

Note:
Although Bohr model explained a lot but it still has some of the limitations. Bohr’s model of an atom failed to explain the Zeeman Effect. It also failed to explain the Stark effect. It violates the Heisenberg indeterminacy principle. It couldn't explain the spectra obtained from larger atoms.