Animated Solution for Physics - Atoms and Nuclei: Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

The Bohr model of the atom provides a beautifully simple way to calculate the energy levels of hydrogen and hydrogen-like ions. In this problem, we are asked to find a specific energy level of a carbon ion that perfectly matches the ground state energy of a hydrogen atom. Let's break down the physics and the math behind this intriguing comparison.

The foundation of our solution is the general formula for the energy of an electron in the $n$-th orbit of a hydrogen-like species. According to Bohr's model, this energy is given by:

Here, $Z$ represents the atomic number (the number of protons in the nucleus), and $n$ is the principal quantum number, which tells us the specific energy level or orbit. The negative sign indicates that the electron is bound to the nucleus.

Let's start with the simpler of the two: the hydrogen atom. Hydrogen is the simplest element, with an atomic number $Z=1$.

The question specifies the "ground state" of hydrogen. The ground state is the lowest possible energy state, which corresponds to the first orbit, so $n=1$. Plugging these values into our master equation, we get:

This is a fundamental constant in atomic physics: the ionization energy of hydrogen is $13.6\text{ eV}$, meaning its ground state energy is $-13.6\text{ eV}$.

Now, let's turn our attention to the carbon ion. Carbon has an atomic number $Z=6$. We are looking for an unknown energy level, which we will simply call $n$.

Using the same Bohr formula, the energy of an electron in this $n$-th level of the carbon ion is:

The core condition given in the problem is that the energy of this specific level in the carbon ion is exactly equal to the ground state energy of the hydrogen atom. Therefore, we can set our two expressions equal to each other:

This equation looks intimidating at first glance, but it simplifies beautifully. The $-13.6$ term is present on both sides as a multiplier, so we can divide both sides by $-13.6$ to cancel it out completely. This leaves us with:

Rearranging this simple algebraic equation, we multiply both sides by $n^2$:

Taking the positive square root (since the principal quantum number $n$ must be a positive integer), we find:

Thus, the 6th energy level of the carbon ion has the exact same energy as the ground state of a hydrogen atom.

Before we conclude, it is important to address a subtle phrasing error in the original question. The problem asks about "single ionized carbon," which chemically refers to $C^{+}$. A neutral carbon atom has 6 electrons, so $C^{+}$ would still have 5 electrons.

However, the Bohr model formula $E_n=-13.6\frac{Z^2}{n^2}$ is strictly valid only for single-electron species (like $H$, $H{e}^{+}$, $L{i}^{2+}$, etc.). To apply this formula to carbon, we must be dealing with a hydrogen-like carbon ion, which is $C^{5+}$ (a carbon atom stripped of 5 of its 6 electrons).

In competitive exams like JEE, such minor language inaccuracies occasionally occur. The key is to recognize the mathematical intent of the examiner. The use of the Bohr model was clearly intended, so we confidently proceed with $Z=6$ and the standard formula, leading us to the correct answer of $n=6$.