According to Bohr's atomic theory,
I. kinetic energy of electron is proportional to Z^2/n^2
II. the

Q: Why is kinetic energy proportional to $Z^2/n^2$?

Kinetic energy is half the magnitude of potential energy, which is $-kZe^2/r$. Since $r \propto n^2/Z$, substituting this gives $KE \propto Z^2/n^2$.

Q: How do we remember all these proportionalities?

You only need to remember two: $r \propto n^2/Z$ and $v \propto Z/n$. Everything else (energy, frequency, force) can be derived from these two basic relations.

Q: Why is the frequency of revolution $v / (2\pi r)$?

Frequency is the number of revolutions per second. It is the reciprocal of the time period $T$. Since distance = speed $\times$ time, $2\pi r = v \times T$, so $f = 1/T = v / (2\pi r)$.

Q: Does the Coulombic force depend on the mass of the electron?

The expression for Coulombic force $F = kZe^2/r^2$ does not explicitly contain mass. However, the radius $r$ itself depends inversely on the mass of the electron.

Q: Are these proportionalities valid for multi-electron atoms?

No, Bohr's model and these specific proportionalities are strictly valid only for single-electron species like H, He$^+$, Li$^{2+}$, etc.

Question

According to Bohr's atomic theory,
I. kinetic energy of electron is $\propto \frac{Z^2}{n^2}$
II. the product of velocity ($v$) of electron and principal quantum number ($n$), '$vn$' $\propto Z^2$.
III. frequency of revolution of electron in an orbit is $\propto \frac{Z^3}{n^3}$
IV. coulombic force of attraction on the electron is $\propto \frac{Z^3}{n^4}$
Choose the most appropriate answer from the options given below.

Options:

(A) Only III

(B) Only I

(C) I, III and IV

(D) I and IV

Sigma Percentile · Accepted Answer

### The Foundation: Bohr's Postulates

Welcome to this beautiful problem on Bohr's Atomic Theory! We are given four statements about the proportionalities of various physical quantities of an electron, and we need to find the correct ones. To solve any problem related to Bohr's model, you only need to remember two fundamental proportionalities:

1. **Radius of the $n^{	ext{th}}$ orbit:** $r \propto \frac{n^2}{Z}$
2. **Velocity of the electron:** $v \propto \frac{Z}{n}$

Every other physical quantity—be it energy, frequency, or force—can be derived from these two basic building blocks. Let's systematically evaluate each statement.

### Analyzing Kinetic Energy and Velocity

Let's start with **Statement I**. It claims that the kinetic energy of an electron is proportional to $\frac{Z^2}{n^2}$. If we recall the energy expression in Bohr's model, the total energy is given by:

$$E_n = -13.6 \frac{Z^2}{n^2} 	ext{ eV}$$

The kinetic energy is simply the positive magnitude of this total energy ($KE = |E_n|$). So, yes, kinetic energy is indeed directly proportional to the square of the atomic number $Z$, and inversely proportional to the square of the principal quantum number $n$. Statement I is absolutely correct!

Now, let's examine **Statement II**. It talks about the product of velocity $v$ and the principal quantum number $n$. To check this, we use our fundamental proportionality for velocity:

$$v \propto \frac{Z}{n}$$

If we multiply both sides by $n$, we get:

$$v \cdot n \propto Z$$

However, the statement claims it is proportional to $Z^2$. That's a mismatch! Therefore, Statement II is incorrect.

### The Dance of the Electron: Frequency

Moving forward to **Statement III**, which is about the frequency of revolution. Imagine the electron racing around the nucleus. The frequency $f$ is the number of trips it makes in one second. Mathematically, frequency equals the velocity $v$ divided by the circumference of the orbit, $2\pi r$:

$$f = \frac{v}{2\pi r}$$

We already know $v \propto \frac{Z}{n}$ and $r \propto \frac{n^2}{Z}$. Let's substitute these into our frequency formula:

$$f \propto \frac{\left(\frac{Z}{n}ight)}{\left(\frac{n^2}{Z}ight)}$$

Simplifying this fraction, the $Z$ goes up to become $Z^2$, and the $n$ goes down to become $n^3$. So, frequency is proportional to $\frac{Z^2}{n^3}$. But look at the statement—it says $Z^3$! So, Statement III is also incorrect.

### The Invisible Thread: Coulombic Force

Finally, let's tackle **Statement IV**. This one is about the Coulombic force of attraction between the positively charged nucleus ($+Ze$) and the negatively charged electron ($-e$). According to Coulomb's law, the magnitude of this force $F_c$ is:

$$F_c = \frac{k(Ze)(e)}{r^2} \propto \frac{Z}{r^2}$$

Now, we substitute our trusty radius proportionality ($r \propto \frac{n^2}{Z}$):

$$F_c \propto \frac{Z}{\left(\frac{n^2}{Z}ight)^2} = \frac{Z}{\left(\frac{n^4}{Z^2}ight)}$$

When we flip the denominator up, the $Z^2$ multiplies with the existing $Z$ to give $Z^3$, and $n^4$ stays in the denominator. So, the force is proportional to $\frac{Z^3}{n^4}$. This matches Statement IV perfectly! It is correct.

### The Final Verdict

We have systematically analyzed all four statements. We found that Statement I correctly describes kinetic energy, and Statement IV correctly describes the Coulombic force. Statements II and III had incorrect powers of $Z$. Therefore, the combination of **I and IV** is the right answer, which corresponds to option (d).

The key takeaway here is to always remember the base proportionalities for radius and velocity; everything else can be derived from them!

Select Answer:

Visualized Solution

$K E \propto \frac{Z ^{2}}{n ^{2}}$

$v \cdot n \propto Z$

$f \propto \frac{Z ^{2}}{n ^{3}}$

$F_{c} \propto \frac{Z ^{3}}{n ^{4}}$

\text{Final Answer}

The Sigma Insight: Bohr's Model

The Foundation

Analyzing Kinetic Energy and Velocity

The Dance of the Electron

The Invisible Thread

The Final Verdict

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } KE∝n2Z2​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } v⋅n∝Z

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f∝n3Z2​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Fc​∝n4Z3​

\text{Final Answer}

The Sigma Insight: Bohr's Model

The Foundation

Analyzing Kinetic Energy and Velocity

The Dance of the Electron

The Invisible Thread

The Final Verdict

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$K E \propto \frac{Z ^{2}}{n ^{2}}$

$v \cdot n \propto Z$

$f \propto \frac{Z ^{2}}{n ^{3}}$

$F_{c} \propto \frac{Z ^{3}}{n ^{4}}$

The radius of the second Bohr orbit in terms of the Bohr radius, $a_{0}$ , in $Li^{2 +}$ is

The frequency of light emitted for the transition $n = 4$ to $n = 2$ of $He^{+}$ is equal to the transition in H atom corresponding to which of the following?