The radius of the $2^{\text {nd }}$ orbit of $Li ^{2+}$ is $x$. The expected radius of the $3^{\text {rd }}$ orbit of $Be ^{3+}$ is
The radius of the $2^{\text {nd }}$ orbit of $Li ^{2+}$ is $x$. The expected radius of the $3^{\text {rd }}$ orbit of $Be ^{3+}$ is
- $\frac{16}{27} \pi$
- $\frac{4}{9} x$
- $\frac{9}{4} x$
- $\frac{27}{16} x$
The Correct Option is D
Solution and Explanation
Using the formula for the radius of the nth orbit: \[ r_n = k \cdot \frac{n^2}{Z} \] For Li$^{2+}$ (\textit{Z} = 3, \textit{n} = 2): \[ r_2 = k \cdot \frac{2^2}{3} = \frac{4k}{3} \] For Be$^{3+}$ (\textit{Z} = 4, \textit{n} = 3): \[ r_3 = k \cdot \frac{3^2}{4} = \frac{9k}{4} \] The ratio of radii: \[ \frac{r_3}{r_2} = \frac{\frac{9k}{4}}{\frac{4k}{3}} = \frac{27}{16} \] Thus, the radius of the 3$^{rd}$ orbit of Be$^{3+}$ is $\frac{27}{16}x$.
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Concepts Used:
Molecular Orbital Theory
The Molecular Orbital Theory is a more sophisticated model of chemical bonding where new molecular orbitals are generated using a mathematical process called Linear Combination of Atomic Orbitals (LCAO).
Molecular Orbital theory is a chemical bonding theory that states that individual atoms combine together to form molecular orbitals. Due to this arrangement in MOT Theory, electrons associated with different nuclei can be found in different atomic orbitals. In molecular orbital theory, the electrons present in a molecule are not assigned to individual chemical bonds between the atoms. Rather, they are treated as moving under the influence of the atomic nuclei in the entire molecule.
