Assume that the radius of the first Bohr orbit of hydrogen atom is 0.6 A°. The radius of the third Bohr orbit of He\(^+\) is _________ picometer. (Nearest Integer)
Detailed Step-by-Step Solution
This numerical tests your understanding of the formula for the radius of a Bohr orbit in hydrogen-like species.
Step 1: The General Formula
The radius of the \( n^{\text{th}} \) Bohr orbit for a hydrogen-like species is given by the formula:
Where \( r_1(\text{H}) \) is the radius of the first orbit of Hydrogen, \( n \) is the orbit number, and \( Z \) is the atomic number.
Step 2: Identify given values for He\(^+\)
- Radius of 1st Bohr orbit of H, \( r_1(\text{H}) = 0.6 \text{ A°} \) (as given in the question)
- Atomic number of Helium, \( Z = 2 \)
- Target orbit for He\(^+\), \( n = 3 \)
Step 3: Calculate the Radius
Substitute the values into the formula:
\( r_3(\text{He}^+) = 0.6 \times \frac{9}{2} \)
\( r_3(\text{He}^+) = 0.6 \times 4.5 \)
\( r_3(\text{He}^+) = 2.7 \text{ A°} \)
Step 4: Convert to Picometers (pm)
The question requires the answer in picometers. We know the unit conversion:
\( 1 \text{ pm} = 10^{-12} \text{ m} \)
Therefore, \( 1 \text{ A°} = 100 \text{ pm} \)
Now, convert our result:
Conclusion: The calculated radius is exactly 270 pm. Since it is already an integer, the final answer is 270.
Acing JEE Main Numerical Types
In JEE Main, integer/numerical value questions test your precision and unit conversion skills. A common mistake here is forgetting to convert Angstroms (A°) to Picometers (pm). To practice more high-yield questions on the Bohr Model and atomic radii, study our extensive notes on the Structure of Atom Class 11 Chemistry.
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