Question

Velocity of \( \text{I}^{\text{st}} \) orbit of hydrogen according to Bohr theory is \( v \), then what is the velocity of \( 3^{\text{rd}} \) orbit of hydrogen?

Hint:

Remember the key proportionalities for Bohr orbits:
Radius $r \propto \frac{n^2}{Z}$
Velocity $v \propto \frac{Z}{n}$
Energy $E \propto -\frac{Z^2}{n^2}$

Answer 1

Step 1: Understanding the Concept:
According to Bohr's model of the hydrogen atom, the velocity of an electron in a specific orbit is quantized. The velocity depends on the atomic number ($Z$) and the principal quantum number ($n$) of the orbit.

Step 2: Key Formula or Approach:
The formula for the velocity of an electron in the $n^{\text{th}}$ Bohr orbit is given by:
$v_n = \frac{2\pi k Z e^2}{nh} = v_0 \times \frac{Z}{n}$
Where $v_0$ is the velocity of the electron in the first orbit of hydrogen ($2.18 \times 10^6$ m/s).
This shows the direct proportionality: $v_n \propto \frac{Z}{n}$

Step 3: Detailed Explanation:
For a hydrogen atom, the atomic number $Z = 1$.
Therefore, the velocity in the $n^{\text{th}}$ orbit is inversely proportional to $n$:
$v_n \propto \frac{1}{n}$
Given that the velocity in the $1^{\text{st}}$ orbit ($n=1$) is $v$:
$v_1 = v$
We need to find the velocity in the $3^{\text{rd}}$ orbit ($n=3$), let's call it $v_3$.
Using the proportionality:
$\frac{v_3}{v_1} = \frac{1/3}{1/1} = \frac{1}{3}$
Substitute $v_1 = v$:
$\frac{v_3}{v} = \frac{1}{3}$
$v_3 = \frac{v}{3}$

Step 4: Final Answer:
The velocity in the $3^{\text{rd}}$ orbit is $\frac{v}{3}$.