Given below are two statements.
Statement I: According to Bohr's model of an atom, qualitatively the

Q: Why does the electron not fall into the nucleus if the attraction is strong?

The electrostatic force provides the necessary centripetal force for circular motion, much like gravity keeps the Earth orbiting the Sun instead of falling into it.

Q: Is the velocity of an electron in Bohr's model close to the speed of light?

For hydrogen in the ground state, it's about $1/137$ times the speed of light. For high $Z$ elements, it can become relativistic, which is why Bohr's model has limitations.

Q: Does the mass of the nucleus affect the velocity?

In the simple Bohr model, we assume the nucleus is infinitely massive. If we consider reduced mass, there's a very tiny effect, but qualitatively $v \propto \frac{Z}{n}$ holds.

Q: What happens to velocity if both $Z$ and $n$ are doubled?

The velocity remains exactly the same, as the ratio $\frac{Z}{n}$ is constant.

Q: Why is Statement I counter-intuitive to some?

One might think less hold means it can fly faster, but in bound orbits, weaker hold means it must move slower to stay in that specific orbit without flying off.

Question

Given below are two statements.
Statement I: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in positive charges on the nucleus as there is no strong hold on the electron by the nucleus.
Statement II: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in principal quantum number.
In the light of the above statements, choose the most appropriate answer from the options given below.

Options:

(A) Both statement I and statement II are false.

(B) Both statement I and statement II are true.

(C) Statement I is false but statement II is true.

(D) Statement I is true but statement II is false.

Sigma Percentile · Accepted Answer

### The Planetary Dance of Electrons

Imagine an atom as a miniature solar system. At the very center lies a dense, positively charged nucleus, acting much like our Sun. Revolving around this nucleus are electrons, tracing out fixed, circular paths known as orbits. This elegant picture is the essence of **Bohr's Model of the Atom**.

But how fast are these electrons moving? Do they all travel at the same speed, or does their velocity depend on where they are and what atom they belong to? To answer this, we must look at the delicate balance of forces that keeps the electron in its orbit.

### The Master Equation for Velocity

For an electron to stay in a circular orbit, the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron must provide the necessary centripetal force. By combining this classical physics concept with Bohr's revolutionary postulate of quantized angular momentum ($mvr = \frac{nh}{2\pi}$), we can derive the exact velocity of an electron in any given orbit.

The resulting formula is a cornerstone of atomic physics:

$$v_n = 2.18 	imes 10^6 \left( \frac{Z}{n} ight) 	ext{ m/s}$$

From this equation, we can extract a beautiful, simplified proportionality:

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

Here, **$Z$** represents the atomic number (the number of protons, or the positive charge on the nucleus), and **$n$** is the principal quantum number (the orbit number).

### Analyzing Statement I: The Pull of the Nucleus

Let's put Statement I under the microscope. It claims that the velocity of an electron *increases* when the positive charge on the nucleus ($Z$) *decreases*. The reasoning provided is that a weaker hold allows the electron to move faster.

However, our master equation tells a different story. We know that $v \propto Z$. This means velocity is **directly proportional** to the nuclear charge. Physically, if the nucleus has a stronger positive charge (higher $Z$), it pulls the electron inward with greater force. To avoid crashing into the nucleus, the electron must travel faster to generate a sufficient outward centripetal effect. Conversely, if $Z$ decreases, the required velocity also decreases.

Therefore, Statement I is fundamentally **false**.

### Analyzing Statement II: The Size of the Orbit

Now, let's examine Statement II. It asserts that the velocity of an electron *increases* as the principal quantum number ($n$) *decreases*.

Looking back at our proportionality, $v \propto \frac{1}{n}$. This indicates an **inverse relationship** between velocity and the orbit number. When $n$ decreases, the electron is in an orbit closer to the nucleus. In these inner orbits, the electrostatic pull is incredibly strong due to the shorter distance. To maintain a stable orbit so close to the nucleus, the electron must zip around at a much higher speed.

Thus, a smaller $n$ indeed results in a higher velocity, making Statement II absolutely **true**.

### The Final Verdict

By systematically applying the mathematical principles of Bohr's model, we have successfully decoded the physical realities behind both statements. Statement I contradicts the direct proportionality between velocity and nuclear charge, while Statement II perfectly aligns with the inverse proportionality between velocity and orbit size.

Consequently, Statement I is false, and Statement II is true, leading us to the correct conclusion.

Select Answer:

Visualized Solution

$B o h r^{'} s M o d e l o f A t o m$

$V e l oc i t y o f E l ec t r o n$

$D e p e n d e n ce o n Z$

$D e p e n d e n ce o n n$

$C o n c l u s i o n$

The Sigma Insight: Bohr's Model

The Planetary Dance of Electrons

The Master Equation for Velocity

Analyzing Statement I

Analyzing Statement II

The Final Verdict

Similar Questions

Comprehension Passage

Which of the following options has the correct combination considering List-I and List-II ?

Which of the following options has the correct combination considering List-I and List-II ?

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to $\frac{h ^{2}}{x m a _{0}^{2}}$ . The value of $10 x$ is ……… . ( $a_{0}$ is radius of Bohr's orbit) (Nearest integer) [Given, $π = 3.14$ ]

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

Energy of an electron is given by $E = - 2.178 \times 1 0^{- 18} J (\frac{Z ^{2}}{n ^{2}})$ . Wavelength of light required to excite an electron in an hydrogen atom from level $n = 1$ to $n = 2$ will be ( $h = 6.62 \times 1 0^{- 34} J s$ and $c = 3.0 \times 1 0^{8} ms^{- 1}$ )

Which of the following is the energy of a possible excited state of hydrogen?

Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Bohr′s Model of Atom

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Velocity of Electron

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Dependence on Z

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Dependence on n

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

The Sigma Insight: Bohr's Model

The Planetary Dance of Electrons

The Master Equation for Velocity

Analyzing Statement I

Analyzing Statement II

The Final Verdict

Similar Questions

Comprehension Passage

Which of the following options has the correct combination considering List-I and List-II ?

Which of the following options has the correct combination considering List-I and List-II ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The radius of the second Bohr orbit in terms of the Bohr radius, a0​, in Li2+ is

Which of the following is the energy of a possible excited state of hydrogen?

$B o h r^{'} s M o d e l o f A t o m$

$V e l oc i t y o f E l ec t r o n$

$D e p e n d e n ce o n Z$

$D e p e n d e n ce o n n$

$C o n c l u s i o n$

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