Question

The innermost orbit of the hydrogen atom has a radius \(0.53 \mathrm{~A}\). what is radius of \(2^{\text {nd }}\) orbit is ? (A) \(2.12 \AA\) (B) \(1.06 \AA\) (C) \(21.2 \AA\) (D) \(10.6 \AA\)

Short answer

The short answer is: The radius of the second orbit is \(2.12 \AA\).

Step-by-step solution

Recall the formula for the radius of the nth orbit in Bohr's model

In the Bohr model, the radius of the nth orbit (denoted as \(r_n\)) is given by the formula: \[r_n = r_1 \cdot n^2\] where: - \(r_1\) represents the radius of the first orbit (also known as the Bohr radius) - \(n\) is the principal quantum number (nth orbit) In this problem we are given the radius of the first orbit and we need to find the radius of the second orbit (2nd orbit, so \(n=2\)).

Substitute the given values

We are given the radius of the first orbit (\(r_1\)) as 0.53 Å, and we need to find the radius of the second orbit (\(n=2\)). Let's substitute the values into the formula from Step 1: \[r_2 = r_1 \cdot n^2\] \[r_2 = 0.53 Å \cdot 2^2\]

Calculate the radius of the second orbit

Now we need to calculate the value of the expression from Step 2: \[r_2 = 0.53 Å \cdot 2^2\] \[r_2 = 0.53 Å \cdot 4\] \[r_2 = 2.12 Å\]

Identify the correct answer

The calculated radius of the second orbit is 2.12 Å, which corresponds to the answer choice (A) \(2.12 \AA\).