Question

Assertion : In Bohr model of hydrogen atom, the angular momentum of an electron in \( n \)th orbit is proportional to the square root of its orbit radius \( r_n \)

Reason (R): According to Bohr model, electron can jump to its nearest orbits only.

• A. If both Assertion and Reason (R) are true and Reason (R) is the correct explanation of Assertion .
• B. If both Assertion and Reason (R) are true but Reason (R) is not the correct explanation of Assertion .
• C. If Assertion is true but Reason (R) is false.
• D. If both Assertion and Reason (R) are false.

Hint:

Do not treat $L_n \propto n$ and $r_n \propto n^2$ as separate, unrelated facts -- combine them ($n \propto \sqrt{r_n}$) whenever a question links angular momentum to orbit radius. Separately, remember the Bohr model permits jumps between any two orbits, not just adjacent ones -- that is a common trap in Reason statements.

Answer 1

The Correct Option is C

We are given an assertion (A) and a reason (R) related to Bohr's model of the hydrogen atom. Let's analyze both statements:

1. Assertion (A): "In Bohr's model of the hydrogen atom, the angular momentum of an electron in the \(n\)-th orbit is proportional to the square root of its orbit radius \(r_n\)."

This statement is incorrect. According to Bohr's model, the angular momentum (\(L\)) of an electron in the \(n\)-th orbit is quantized and is given by the equation:

\[ L = n \hbar \]

where:

• \(L\) is the angular momentum of the electron,
• \(n\) is the principal quantum number (the orbit number), and
• \(\hbar\) is the reduced Planck’s constant.

The radius \(r_n\) of the \(n\)-th orbit is proportional to \(n^2\), i.e.,

\[ r_n \propto n^2 \]

Therefore, the angular momentum is proportional to the quantum number \(n\), not to the square root of the radius. The assertion is therefore incorrect.

2. Reason (R): "According to Bohr's model, the electron can jump to its nearest orbits only."

This statement is correct. In Bohr's model, when an electron absorbs or emits energy, it jumps from one orbit to another. However, the electron can only jump between specific orbits that correspond to the allowed energy levels. These energy levels are quantized, and the electron can only transition between these discrete orbits (energy levels) in response to energy absorption or emission. This transition typically occurs between orbits that are closest to each other in energy. Therefore, the reason is true.

3. Evaluation of the Assertion and Reason:

While the reason (R) is correct, the assertion (A) is incorrect because, according to Bohr’s model, the angular momentum is proportional to \(n\), not the square root of the radius. Therefore, the assertion is false, even though the reason is true.

4. Conclusion:

The correct answer is: Assertion is false, and Reason is true.

Answer 2

Concept:

• Combine the two standard Bohr-model results, $L_n \propto n$ and $r_n \propto n^2$, directly to check the Assertion, instead of treating them as unrelated facts.

Step 1: Recall the two standard Bohr results.
For the $n$th orbit in the Bohr model: the angular momentum $L_n = \frac{nh}{2\pi}$, so $L_n \propto n$. Also, the orbit radius follows $r_n \propto n^2$.

Step 2: Combine the two relations to test the Assertion.
From $r_n \propto n^2$, taking the square root of both sides gives $n \propto \sqrt{r_n}$. Substituting this into $L_n \propto n$ gives directly $L_n \propto \sqrt{r_n}$. So the claim that angular momentum is proportional to the square root of the orbit radius follows immediately by combining the two standard results -- the Assertion is true.

Step 3: Test the Reason.
The Bohr model allows an electron to jump between any two allowed orbits, not only neighbouring ones, as long as the absorbed or emitted photon carries exactly the energy difference between those two orbits. For example, the Lyman series in the hydrogen spectrum includes transitions from $n=2,3,4,...$ all the way down to $n=1$, which are not jumps to the nearest orbit. So the Reason is false.

Step 4: Combine both results.
Since the Assertion is true and the Reason is false, the correct choice is that Assertion (A) is true but Reason (R) is false.

Final Answer: Assertion (A) is true, Reason (R) is false.