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

The radius of the orbit in Bohr's model is proportional to \( \frac{n^2}{Z} \), where \( n \) is the orbit number and \( Z \) is the atomic number.

For the given radii: \[ \frac{r_2}{r_1} = \left( \frac{n_2}{n_1} \right)^2 \times \frac{Z_1}{Z_2}. \]

Substituting the values: \[ n_1 = 2, \quad n_2 = 4, \quad Z_1 = 2 \, (\text{for } \text{He}^+), \quad Z_2 = 4 \, (\text{for } \text{Be}^{3+}), \] we get: \[ \frac{r_2}{r_1} = \left( \frac{4}{2} \right)^2 \times \frac{2}{4}. \]

Simplify: \[ \frac{r_2}{r_1} = \left( 2 \right)^2 \times \frac{1}{2} = 4 \times \frac{1}{2} = 2. \]

Therefore, \( x = 2 \).