Animated Solution for Chemistry - Atomic Structure: Comprehension Passage

Welcome to the fascinating world of quantum mechanics! Today, we are going to dive deep into one of the most elegant and historically significant models in physics and chemistry: Bohr's Model of the Atom. Imagine you are stepping back in time to 1913. Scientists are baffled by the stability of atoms and the mysterious spectral lines of hydrogen. Enter Niels Bohr, who brilliantly combined classical mechanics with the emerging quantum theory to give us a picture of the atom that is both intuitive and mathematically beautiful.

In this problem, we are tasked with exploring how various physical quantities of an electron—such as its radius, angular momentum, kinetic energy, and potential energy—depend on the principal quantum number, $n$. This is a classic matrix match problem that tests your fundamental understanding of Bohr's postulates. Let's break it down step by step and uncover the hidden symmetries of the atom!

Before we jump into the equations, let's visualize the setup. Picture a central, positively charged nucleus. It's massive and practically stationary. Around it, an electron is whizzing in perfectly circular orbits. But here is the catch: the electron cannot just orbit anywhere. It is restricted to specific, quantized paths, much like a person can only stand on the steps of a staircase, not in between them.

These allowed paths are denoted by the principal quantum number, $n$, where $n=1,2,3,\dots$ and so on. As $n$ increases, the electron moves further away from the nucleus. Our goal is to figure out exactly how these properties scale with $n$.

Let's start with the first quantity in List-I: the radius of the $n^{\text{th}}$ orbit. How does the distance of the electron from the nucleus change as we move to higher orbits?

From Bohr's theory, the radius $r_n$ of the $n^{\text{th}}$ orbit for a hydrogen-like species is given by the famous formula:

Here, $Z$ is the atomic number, which represents the number of protons in the nucleus. For a given atom, $Z$ is a constant. The number $0.529\text{ A˚}$ is known as the Bohr radius, often denoted as $a_0$.

Look closely at the relationship between $r_n$ and $n$. The radius is directly proportional to the square of the principal quantum number!

This means if you move from the first orbit ($n=1$) to the second orbit ($n=2$), the radius doesn't just double; it quadruples! The orbits get spaced further and further apart. Matching this with List-II, we see that (I) corresponds perfectly with (T).

Next up is the angular momentum of the electron. In classical physics, angular momentum can take any continuous value. But Bohr introduced a radical idea: the angular momentum is quantized.

He postulated that the angular momentum $L$ of an electron in a circular orbit is an integral multiple of $\frac{h}{2\pi }$, where $h$ is Planck's constant. Mathematically, this is expressed as:

In this equation, $m$ is the mass of the electron, $v_n$ is its tangential velocity, and $r_n$ is the orbital radius. Since $h$ and $2\pi$ are fundamental constants, the angular momentum is directly proportional to $n$ to the power of one.

It's a beautifully simple linear relationship. As you step up to the next orbit, the angular momentum increases by exactly one unit of $\frac{h}{2\pi }$. Looking at our lists, this means (II) matches flawlessly with (S).

Kinetic and Potential

Now, let's talk about energy. The electron possesses kinetic energy due to its motion and potential energy due to its electrostatic attraction to the nucleus.

First, the kinetic energy ($K_n$). The formula for the kinetic energy of an electron in the $n^{\text{th}}$ orbit is:

Notice the $n^2$ in the denominator. As the electron moves to higher orbits (larger $n$), it slows down, and its kinetic energy decreases. The relationship is inversely proportional to the square of $n$.

This tells us that (III) matches with (P).

What about the potential energy ($U_n$)? The electrostatic potential energy is negative because it's an attractive bound state. The formula is:

A fascinating property of the Bohr model (and a consequence of the Virial Theorem) is that the potential energy is exactly twice the magnitude of the kinetic energy, but with a negative sign. So, $U_n=-2K_n$.

Despite the negative sign, we are looking for the functional dependence on $n$. Just like kinetic energy, the potential energy has $n^2$ in the denominator.

Therefore, (IV) also matches with (P).

The Grand Match

We have successfully decoded the dependencies for all four quantities. Let's summarize our master matches:

(I) Radius: Proportional to $n^2⟹$ (T) (II) Angular Momentum: Proportional to $n^1⟹$ (S) (III) Kinetic Energy: Proportional to $n^{-2}⟹$ (P) (IV) Potential Energy: Proportional to $n^{-2}⟹$ (P)

Now, let's conquer the two questions based on this passage.

Solving Question 1: The question asks for the correct combination from the given options. Let's evaluate them: (A) (II), (R): Incorrect. Angular momentum is proportional to $n^1$, not $n^0$. (B) (I), (P): Incorrect. Radius is proportional to $n^2$, not $n^{-2}$. (C) (I), (T): Correct! Radius is indeed proportional to $n^2$. (D) (II), (Q): Incorrect. Angular momentum is proportional to $n^1$, not $n^{-1}$.

Thus, the correct answer for the first question is Option (C).

Solving Question 2: Let's evaluate the options for the second question: (A) (III), (S): Incorrect. Kinetic energy is proportional to $n^{-2}$, not $n^1$. (B) (IV), (Q): Incorrect. Potential energy is proportional to $n^{-2}$, not $n^{-1}$. (C) (IV), (U): Incorrect. Potential energy is proportional to $n^{-2}$, not $n^{1/2}$. (D) (III), (P): Correct! Kinetic energy is indeed proportional to $n^{-2}$.

Thus, the correct answer for the second question is Option (D).

And there you have it! By systematically applying the fundamental formulas of Bohr's model, we've not only solved the problem but also reinforced our understanding of atomic physics. Remember, physics isn't just about memorizing formulas; it's about seeing the interconnectedness of nature's laws. Keep questioning, keep exploring, and you'll master these concepts in no time!