Question

In Bohr's theory of hydrogen atom if speed of an electron in the first orbit is v, then the speed in 3rd orbit is

Hint:

Key proportionalities for Bohr orbits to memorize:
Radius: $r \propto n^2 / Z$ (gets much bigger)
Velocity: $v \propto Z / n$ (gets slower)
Energy: $E \propto -Z^2 / n^2$ (gets less negative/closer to zero)

Answer 1

Step 1: Understanding the Concept:
Niels Bohr's model of the hydrogen atom postulates that electrons travel in discrete, quantized circular orbits. The velocity of an electron in these orbits depends on the principal quantum number ($n$) and the atomic number ($Z$).

Step 2: Key Formula or Approach:
The theoretical derivation of the Bohr model yields the formula for the velocity ($v_n$) of an electron in the $n$-th orbit:
\[ v_n = \frac{Z e^2}{2 \epsilon_0 n h} \]
From this equation, for a specific atom where $Z$ is constant, the velocity is inversely proportional to the principal quantum number $n$:
\[ v_n \propto \frac{1}{n} \]

Step 3: Detailed Explanation:
Let the speed in the first orbit ($n=1$) be $v_1 = v$.
We want to find the speed in the third orbit, so $n=3$, let's call it $v_3$.
Using the inverse proportionality:
\[ \frac{v_3}{v_1} = \frac{1/3}{1/1} \]
\[ \frac{v_3}{v_1} = \frac{1}{3} \]
Substitute $v_1 = v$:
\[ \frac{v_3}{v} = \frac{1}{3} \]
\[ v_3 = \frac{v}{3} \]

Step 4: Final Answer:
The speed in the 3rd orbit is $v/3$.