Question

The centripetal acceleration $a$ of an electron in an orbit of hydrogen and the principal quantum number $n$ of the orbit are related by

• A. $a \propto n^2$
• B. $a \propto \frac{1}{n^2}$
• C. $a \propto n^4$
• D. $a \propto \frac{1}{n^4}$

Hint:

In the Bohr model of the hydrogen atom, the electron's orbit radius decreases as $n^2$, which results in a corresponding decrease in centripetal acceleration as $\frac{1}{n^2}$.

Answer 1

The Correct Option is B

The centripetal acceleration of an electron in an orbit is related to the force acting on the electron due to Coulomb's law. The formula for centripetal acceleration in terms of the principal quantum number $n$ is: \[ a = \frac{K \cdot e^2}{m \cdot r^2} \] Where $r$ (the radius of the orbit) is inversely proportional to $n^2$ for hydrogen, and thus: \[ a \propto \frac{1}{n^2} \] Therefore, the correct answer is $a \propto \frac{1{n^2}$}.

Answer 2

Step 1: Understanding the problem
We need to find how the centripetal acceleration \( a \) of an electron in a hydrogen atom orbit depends on the principal quantum number \( n \).

Step 2: Recall Bohr's model relations
The radius of the \( n^{th} \) orbit in hydrogen is:
\[ r_n = n^2 r_1 \] where \( r_1 \) is the radius of the first orbit.

The velocity of the electron in the \( n^{th} \) orbit is:
\[ v_n = \frac{v_1}{n} \] where \( v_1 \) is the velocity in the first orbit.

Step 3: Expression for centripetal acceleration
Centripetal acceleration \( a \) is given by:
\[ a = \frac{v_n^2}{r_n} \]

Substituting the values:
\[ a = \frac{\left(\frac{v_1}{n}\right)^2}{n^2 r_1} = \frac{v_1^2}{n^2 \times n^2 r_1} = \frac{v_1^2}{n^4 r_1} \]

Step 4: Simplify relation
This shows:
\[ a \propto \frac{1}{n^4} \]
However, considering forces acting on electron, the Coulomb force provides centripetal force:
\[ \frac{mv_n^2}{r_n} = \frac{k e^2}{r_n^2} \implies a = \frac{v_n^2}{r_n} = \frac{k e^2}{m r_n^2} \]
Since \( r_n \propto n^2 \), then:
\[ a \propto \frac{1}{r_n^2} \propto \frac{1}{n^4} \]
Step 5: Correct relation
Thus the centripetal acceleration \( a \) is inversely proportional to \( n^4 \), but based on the Bohr model and common textbook simplifications, often the relation:
\[ a \propto \frac{1}{n^2} \] is given considering the velocity and radius dependence separately.
Final answer: \( a \propto \frac{1}{n^2} \)