Question

According to Bohr's model of hydrogen atom, which of the following statement is incorrect?

• A. Radius of 3rd orbit is nine times larger than that of 1st orbit.
• B. Radius of 8th orbit is four times larger than that of 4th orbit.
• C. Radius of 6th orbit is three times larger than that of 4th orbit.
• D. Radius of 4th orbit is four times larger than that of 2nd orbit.

Hint:

For Bohr's model, the radius of each orbit increases with the square of the principal quantum number \( n \). Thus, radius comparisons can be calculated using the formula \( r \propto n^2 \).

Answer 1

The Correct Option is C

We know that for Bohr’s model: \[ r \propto n^2 \] Where \( n \) is the principal quantum number. Hence, we have: \[ \frac{r_3}{r_1} = \left(\frac{3}{1}\right)^2 = 9, \quad \frac{r_8}{r_4} = \left(\frac{8}{4}\right)^2 = 4, \quad \frac{r_6}{r_4} = \left(\frac{6}{4}\right)^2 = 2.25, \quad \frac{r_4}{r_2} = \left(\frac{4}{2}\right)^2 = 4 \] Thus, the incorrect statement is option (3).

Answer 2

The problem asks to identify the incorrect statement among the given options regarding the radius of electron orbits in a hydrogen atom, based on Bohr's atomic model.

Concept Used:

According to Bohr's model for a hydrogen-like atom, the radius of the \(n\)-th orbit is given by the formula:

\[ r_n = a_0 \frac{n^2}{Z} \]

where:

• \( r_n \) is the radius of the \(n\)-th orbit.
• \( a_0 \) is the Bohr radius (the radius of the first orbit of hydrogen, approximately \( 0.529 \, \text{Å} \)).
• \( n \) is the principal quantum number of the orbit.
• \( Z \) is the atomic number.

For a hydrogen atom, \( Z = 1 \). Therefore, the formula simplifies to:

\[ r_n = a_0 n^2 \]

This implies that the radius of the \(n\)-th orbit is directly proportional to the square of the principal quantum number (\( r_n \propto n^2 \)). To compare the radii of two different orbits, say \( n_1 \) and \( n_2 \), we can use the ratio:

\[ \frac{r_{n_2}}{r_{n_1}} = \frac{n_2^2}{n_1^2} \]

Step-by-Step Solution:

We will evaluate each statement by calculating the ratio of the radii for the given orbits.

Step 1: Evaluate the statement "Radius of 3rd orbit is nine times larger than that of 1st orbit."

Here, \( n_2 = 3 \) and \( n_1 = 1 \).

\[ \frac{r_3}{r_1} = \frac{3^2}{1^2} = \frac{9}{1} = 9 \]

So, \( r_3 = 9 \times r_1 \). This statement is correct.

Step 2: Evaluate the statement "Radius of 8th orbit is four times larger than that of 4th orbit."

Here, \( n_2 = 8 \) and \( n_1 = 4 \).

\[ \frac{r_8}{r_4} = \frac{8^2}{4^2} = \frac{64}{16} = 4 \]

So, \( r_8 = 4 \times r_4 \). This statement is correct.

Step 3: Evaluate the statement "Radius of 6th orbit is three times larger than that of 4th orbit."

Here, \( n_2 = 6 \) and \( n_1 = 4 \).

\[ \frac{r_6}{r_4} = \frac{6^2}{4^2} = \frac{36}{16} = \frac{9}{4} = 2.25 \]

The radius of the 6th orbit is 2.25 times larger than that of the 4th orbit, not three times. Therefore, this statement is incorrect.

Step 4: Evaluate the statement "Radius of 4th orbit is four times larger than that of 2nd orbit."

Here, \( n_2 = 4 \) and \( n_1 = 2 \).

\[ \frac{r_4}{r_2} = \frac{4^2}{2^2} = \frac{16}{4} = 4 \]

So, \( r_4 = 4 \times r_2 \). This statement is correct.

The only incorrect statement among the options is that the radius of the 6th orbit is three times larger than that of the 4th orbit.