Assertion (A): The ratio of the speeds of the electron in the second to that in the third orbit in Bohr's hydrogen atom is \(\frac{3}{2}\).
Reason (R): The speed of the electron in \(\text{n}^{\text{th}}\) permitted orbit in hydrogen atom varies proportional to \(\frac{1}{\text{n}}\) i.e. \(v_{\text{n}} \propto \frac{1}{\text{n}}\).
Step 1 (Formula / Governing Law): [0.25 Mark]
According to Bohr's theory of the hydrogen atom, the speed of an electron in the \(\text{n}^{\text{th}}\) orbit is given by the relation:
where \(v_1\) is the speed of the electron in the first orbit and \(n\) is the principal quantum number. This confirms that \(v_n \propto \frac{1}{n}\).
Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
To find the ratio of the speed in the second orbit (\(n_2 = 2\)) to the speed in the third orbit (\(n_3 = 3\)):
Calculating the ratio \(\frac{v_2}{v_3}\):
$$\frac{v_2}{v_3} = \frac{v_1 / 2}{v_1 / 3} = \frac{v_1}{2} \times \frac{3}{v_1} = \frac{3}{2}$$Step 3 (Final Concluded Answer): [0.25 Mark]
Since the calculation yields \(\frac{3}{2}\), the Assertion (A) is True. The Reason (R) correctly states the proportionality \(v_n \propto \frac{1}{n}\), which is the fundamental law used to derive the result. Therefore, Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
💡 EL10 Concept Insight: Think of the electron's orbit like a race track; the closer you are to the center (the nucleus), the faster you have to run to stay in your lane. As you move to higher orbits (further away), the electron slows down, specifically following a "one-over-n" rule where doubling the orbit number cuts the speed in half.