According to Bohr's atomic theory: (A) Kinetic energy of electron Z2n2 (B) The product of velocity (v) of electron and…

🧠 The Fundamental Constants Audit Everything in the Bohr model—velocity, energy, force, frequency—is a power function of $Z$ and $n$. We need to verify the exponents for four key properties.

🗺️ The Exponent Scan

• (A) KE $\propto Z^2/n^2$: Energy is always based on the square of $Z$ over the square of $n$. Correct.
• (B) Product $|vn| \propto Z^2$: $v \propto Z/n$, so $v \cdot n \propto (Z/n) \cdot n = Z$. This claims $Z^2$. Incorrect.
• (C) Frequency $f \propto Z^3/n^3$: $f = v/2\pi r$. Ratio $(Z/n) / (n^2/Z) = Z^2/n^3$. This claims $Z^3/n^3$. Incorrect.
• (D) Coulombic Force $F \propto Z^3/n^4$: $F \propto Z/r^2$. Ratio $Z / (n^2/Z)^2 = Z^3/n^4$. Correct.

Valid statements are (A) and (D).

⚡ The "Third Power Filter" Notice that only the Force reaches the $Z^3$ power because it depends on $r^2$. Most properties in Bohr (Energy, $a_0$) keep $Z$ at first or second power. The $Z^3/n^4$ pattern for force is a unique fingerprint worth memorizing.

⚠️ Common Traps Mixing up Frequency and Velocity! Students often think everything about "motion" ($v$ and $f$) must have the same ratio. But frequency accounts for the circumference ($r$), introducing another factor of $n^2/Z$.

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