Question

radius of 2^nd orbit of He⁺ of Bohr's model is r₁ and that of fourth orbit of Be³⁺ is represented as r₂. Now the ratio\(\frac{ r_2}{r_1} \)is x: 1. The value of x is___.

Hint:

The radius of an orbit in Bohr's model is proportional to \( \frac{n^2}{Z} \). Use this relationship to compare radii for different atoms and orbit numbers.

Answer 1

Correct Answer: 2

To determine the ratio of the radii of the orbits, we use Bohr's model. The radius of the n^th orbit for a hydrogen-like atom is given by the formula: \( r_n = \frac{n^2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot Z} \), where \( h \) is Planck's constant, \( m \) is the electron mass, \( e \) is the elementary charge, and \( Z \) is the atomic number.

For He⁺ (helium ion), \( Z = 2 \). Thus, the radius of the 2^nd orbit is:

\( r_1 = \frac{4 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot 2} = \frac{2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2} \).

For Be³⁺ (beryllium ion), \( Z = 4 \). Thus, the radius of the 4^th orbit is:

\( r_2 = \frac{16 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2 \cdot 4} = \frac{4 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2} \).

The ratio \(\frac{r_2}{r_1}\) is then:

\( \frac{r_2}{r_1} = \frac{\frac{4 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2}}{\frac{2 \cdot h^2}{4 \pi^2 \cdot m \cdot e^2}} = \frac{4}{2} = 2 \).

Thus, the ratio is x:1, with x equal to 2, fitting the expected range of \(2,2\).