Question

According to Bohr's atomic theory :- (A) Kinetic energy of electron is $\propto \frac{ Z ^{2}}{ n ^{2}}$. (B) The product of velocity (v) of electron and principal quantum number (n), 'vn' $\propto Z ^{2}$. (C) Frequency of revolution of electron in an orbit is $\propto \frac{ Z ^{3}}{ n ^{3}}$. (D) Coulombic force of attraction on the electron is $\propto \frac{ Z ^{3}}{ n ^{4}}$. Choose the most appropriate answer from the options given below:

• A. (C) Only
• B. (A) Only
• C. (A), (C) and (D) only
• D. (A) and (D) only

Answer 1

The Correct Option is D

Bohr's atomic theory provides a model for the hydrogen atom, where electrons orbit the nucleus just as planets orbit the sun. Here’s a detailed analysis of the given statements according to Bohr's theory:

1. Statement (A): Kinetic energy of electron is $\propto \frac{ Z ^{2}}{ n ^{2}}$
   • The kinetic energy (KE) of an electron in an orbit is directly proportional to the square of the atomic number \( Z \) and inversely proportional to the square of the principal quantum number \( n \). This statement is correct based on Bohr's theory.
2. Statement (B): The product of velocity (v) of electron and principal quantum number (n), 'vn' $\propto Z ^{2}$
   • According to Bohr's model, the velocity \( v \) of an electron in a given orbit is proportional to \( \frac{Z}{n} \). Therefore, the product \( vn \) would merely be proportional to \( Z \), not \( Z^2 \). Hence, this statement is incorrect.
3. Statement (C): Frequency of revolution of electron in an orbit is $\propto \frac{ Z ^{3}}{ n ^{3}}$
   • The frequency of revolution \( f \) is indeed given by the formula proportional to \( \frac{Z^2}{n^3} \). This statement incorrectly depicts the dependence on \( Z \) and thus is incorrect.
4. Statement (D): Coulombic force of attraction on the electron is $\propto \frac{ Z ^{3}}{ n ^{4}}$
   • The Coulombic force of attraction between the nucleus and the electron is proportional to \( \frac{Z^2}{r^2} \) (where \( r \) is the radius) and since \( r \propto \frac{n^2}{Z} \), we can rewrite \( F \propto \frac{Z^3}{n^4} \). Hence, this statement is correct.

Based on the analysis, the statements (A) and (D) are correct. Therefore, the most appropriate answer is: (A) and (D) only.

Answer 2

According to Bohr's theory :


(A) $KE =13.6 \frac{ z ^{2}}{ n ^{2}} \frac{ eV }{\text { atom }}$
$\Rightarrow KE \alpha \frac{ z ^{2}}{ n ^{2}}$


(B) speed of $e ^{-} \alpha \frac{ z }{ n }$
$\therefore v \times n \alpha z$


(C) Frequency of revolution of $e ^{-}=\frac{ v }{2 \pi r }$
$\therefore$ frequency $\alpha \frac{ z ^{2}}{ n ^{3}}$


(D) $F =\frac{ kq _{1} q _{2}}{ r ^{2}}=\frac{ kze ^{2}}{ r ^{2}}$
$\left\{ r \alpha \frac{ n ^{2}}{ z }\right.$
$\Rightarrow F \alpha \frac{ z }{\left(\frac{ n ^{2}}{ z }\right)^{2}}$
$\Rightarrow F \alpha \frac{ z ^{3}}{ n ^{4}}$