radius of 2nd 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___.
Show 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.
Correct Answer: 2
Solution and Explanation
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, \( x = 2 \).
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