Question

Ratio of centripetal acceleration for an electron revolving in 3rd and 5th Bohr orbit of hydrogen atom is

• A. $425 : 18$
• B. $625 : 81$
• C. $125 : 27$
• D. $221 : 36$

Hint:

Instead of memorizing the full expression for acceleration, just remember the fundamental scaling factors: radius grows as $n^2$ and velocity drops as $1/n$. Combining them always shows that any acceleration parameters scale extremely quickly as an inverse fourth power ($1/n^4$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem requires computing the ratio of centripetal accelerations ($a_c$) experienced by a revolving electron moving between two distinct energy levels—the 3rd ($n_1 = 3$) and 5th ($n_2 = 5$) Bohr orbits—of a hydrogen atom.

Step 2: Key Formula or Approach:
The basic definition of centripetal acceleration is: $$a = \frac{v^2}{r}$$ According to Bohr's atomic model, the orbital velocity $v$ and orbital radius $r$ of an electron scale with the principal quantum number $n$ according to these proportionalities: $$v \propto \frac{1}{n} \quad \text{and} \quad r \propto n^2$$ Substituting these orbital proportions into the acceleration formula yields: $$a \propto \frac{\left(\frac{1}{n}\right)^2}{n^2} \implies a \propto \frac{1}{n^4}$$ This allows us to set up an inverse fourth-power ratio: $\frac{a_3}{a_5} = \left(\frac{5}{3}\right)^4$.

Step 3: Detailed Explanation:
Set up the explicit proportionality ratio using our derived relationship: $$\frac{a_3}{a_5} = \left(\frac{n_2}{n_1}\right)^4$$ Substitute $n_1 = 3$ and $n_2 = 5$ into the expression: $$\frac{a_3}{a_5} = \left(\frac{5}{3}\right)^4$$ Expand the fourth-power calculation for both the numerator and the denominator independently: $$5^4 = 5 \times 5 \times 5 \times 5 = 625$$ $$3^4 = 3 \times 3 \times 3 \times 3 = 81$$ Combining these results gives the ratio: $$\frac{a_3}{a_5} = \frac{625}{81} \implies 625 : 81$$ This maps to option (B).

Step 4: Final Answer:
The ratio of the centripetal accelerations is $625 : 81$, which corresponds to option (B).