Question

Calculate the radius of Bohr’s fifth orbit for hydrogen atom in nm.

Answer 1

Hint: To answer this question, you must recall the formula for finding the radius of an orbit of an atom given by the Bohr’s model of the atom. The Bohr’s model of atom was one of the first models of atoms that explained the position of the emission and absorption lines shown by hydrogen atom and hydrogen like single electron species successfully.
 $ {{\text{r}}_{\text{n}}} = 0.0529\dfrac{{{{\text{n}}^2}}}{{\text{Z}}} $
Where, $ {{\text{r}}_{\text{n}}} $ represents the radius of the nth orbit calculated in nm.
 $ {\text{n}} $ represents the orbit under consideration
And, $ {\text{Z}} $ represents the atomic number of the atom which for hydrogen is 1.

Complete step by step solution
In his model, Bohr formulated three postulates for the explanation of the structure of atom:
In an atom, the electrons revolve around the nucleus in fixed circular energy levels known as orbitals.
These fixed energy levels are denoted by integral values known as the principal quantum number. An electron is said to be present in its ground state when it is in the lowest energy orbital. An electron present in a higher energy orbital moves to a lower energy orbital by the loss of energy and an electron present in a lower energy orbital moves to a higher energy orbital by absorbing energy.
The third postulate of Bohr's atomic theory tells that the angular momentum of an electron in its orbit is fixed and quantized.
To find the radius of the fifth orbit of hydrogen atom, we use the formula: $ {{\text{r}}_{\text{n}}} = 0.0529\dfrac{{{{\text{n}}^2}}}{{\text{Z}}} $
Substituting the values, we get $ {{\text{r}}_{\text{5}}} = 0.0529 \times {\left( 5 \right)^2} $
 $ \Rightarrow {{\text{r}}_5} = 1.3225{\text{ nm}} $

Note
Although the Bohr’s model of atom was successful in explaining many concepts but it failed as it could not be applied to all atoms. Bohr’s model of atom is applicable only for one electron containing species or hydrogen atom like species. Although this model paved the way for further theories that led to a successful determination of the structure of the atom.