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

🧠 The Radius Rule Bohr's model states that the radius of an electron's orbit increases with the square of the principal quantum number. $r_n \propto n^2$ To find an incorrect statement, we just need to verify if the ratios of $n^2$ match the claims made in the options.

🗺️ The Ratio Audit

• Statement A: $r_3 / r_1$. Ratio $= 3^2 / 1^2 = 9$. (Correct)
• Statement B: $r_8 / r_4$. Ratio $= 8^2 / 4^2 = 64 / 16 = 4$. (Correct)
• Statement D: $r_4 / r_2$. Ratio $= 4^2 / 2^2 = 16 / 4 = 4$. (Correct)
• Statement C: $r_6 / r_4$. Ratio $= 6^2 / 4^2 = 36 / 16 = 2.25$. The statement claims it is "three times larger" ($3.0$). (Incorrect)

⚡ The Square Test Always think in terms of squares. $4 \to 8$ ($2\times$ $n$) means $4\times$ radius. $2 \to 4$ ($2\times$ $n$) means $4\times$ radius. $4 \to 6$ ($1.5\times$ $n$) means $2.25\times$ radius. $3$ is not a square of a simple $n$ ratio here.

⚠️ Linearity Trap Don't fall for the trap of thinking $r$ is proportional to $n$. If $r \propto n$, then $r_6$ would be $1.5$ times $r_4$. If $r \propto n^2$, it is $2.25$ times. The statement "three times" is wrong in both universes!

$\boxed{\text{Answer: (c)}}$