Animated Solution for Chemistry - Atomic Structure: The radius of the second Bohr orbit in terms of the Bohr radius, $a_0$, in ${\text{Li}}^{2+}$ is

When Niels Bohr introduced his atomic model in 1913, it was a monumental leap in our understanding of the microscopic world. By blending classical mechanics with the nascent ideas of quantum theory, Bohr provided a framework that perfectly explained the emission spectrum of hydrogen. But the true beauty of the Bohr model lies in its scalability. It doesn't just work for hydrogen; it works for any hydrogen-like species—an atom or ion that has been stripped of all but one electron.

This brings us to the fascinating case of the ${\text{Li}}^{2+}$ ion. Lithium normally has three electrons, but if we remove two of them, we are left with a single electron orbiting a nucleus with three protons. This is a perfect playground for Bohr's equations.

In the Bohr model, the radius of an electron's orbit is determined by a delicate balance between the electrostatic force pulling the electron inward and the centripetal requirement of its circular motion, constrained by the quantization of angular momentum. When the dust of the derivation settles, we are left with a beautifully simple scaling law:

Let's break down the anatomy of this equation. The term $a_0$ is the Bohr radius, a fundamental physical constant representing the radius of the first orbit of a hydrogen atom. Its value is approximately $0.529\text{ A˚}$.

The variable $n$ is the principal quantum number, representing the orbit's energy level ($n=1,2,3,\dots$). Notice that the radius scales with $n^2$. This means as you move to higher orbits, the distance from the nucleus increases quadratically. The second orbit is four times larger than the first, the third is nine times larger, and so on.

The variable $Z$ is the atomic number, representing the number of protons in the nucleus. The radius is inversely proportional to $Z$. This makes perfect physical sense: a nucleus with more protons exerts a stronger electrostatic pull on the electron, drawing the orbit closer in.

Now, let's apply this master equation to our specific problem. We are asked to find the radius of the second Bohr orbit of the ${\text{Li}}^{2+}$ ion.

First, we identify our parameters. Since we are dealing with Lithium, we look at the periodic table and find that its atomic number is $3$. Therefore, our nuclear charge parameter is:

Next, the problem specifies the "second Bohr orbit". This directly gives us our principal quantum number:

With our parameters locked in, the rest is a straightforward substitution. We take our master equation and plug in the values for $n$ and $Z$:

Squaring the principal quantum number gives us $4$. We then multiply this by the Bohr radius $a_0$ and divide by the atomic number $3$:

And there we have it! The radius of the second Bohr orbit for the ${\text{Li}}^{2+}$ ion is exactly $\frac{4}{3}$ times the fundamental Bohr radius. This perfectly matches option (b).

This problem is a classic example of how powerful scaling laws are in physics. By understanding how a property (like radius) depends on fundamental variables (like $n$ and $Z$), we can effortlessly calculate values for a wide range of physical systems without having to re-derive the complex mechanics from scratch.