The angular speed of electron in the nth orbit of hydrogen atom is

Options

A. Directly proportional to `n^(2)`
B. Directly proportional to n
C. Inversely proportional to `n^(3)`
D. Inversely proportional to n

Answer

To find the angular speed of an electron in the nth orbit of a hydrogen atom, we can use Bohr's model of the hydrogen atom. Let's break down the solution step by step.

### Step 1: Understand Angular Speed
Angular speed (ω) is defined as the rate of change of angular displacement and can be expressed as:
\[ \omega = \frac{v}{r} \]
where \(v\) is the linear velocity of the electron and \(r\) is the radius of the orbit.

**Hint:** Remember that angular speed relates linear velocity and radius.

### Step 2: Velocity of the Electron
According to Bohr's model, the velocity of the electron in the nth orbit is given by:
\[ v_n \propto \frac{1}{n} \]
This means that the speed of the electron decreases as the principal quantum number \(n\) increases.

**Hint:** Recall that in Bohr's model, higher energy levels correspond to lower speeds.

### Step 3: Radius of the Orbit
The radius of the nth orbit is given by:
\[ r_n \propto n^2 \]
This indicates that the radius increases with the square of the principal quantum number \(n\).

**Hint:** Higher orbits are further away from the nucleus, hence the square relationship.

### Step 4: Substitute into Angular Speed Formula
Now, substituting the expressions for \(v_n\) and \(r_n\) into the angular speed formula:
\[ \omega_n = \frac{v_n}{r_n} \propto \frac{\frac{1}{n}}{n^2} = \frac{1}{n^3} \]
This shows that the angular speed is inversely proportional to the cube of the principal quantum number \(n\).

**Hint:** Combine the relationships for velocity and radius carefully to derive the angular speed.

### Step 5: Conclusion
From the above derivation, we conclude that the angular speed of an electron in the nth orbit of a hydrogen atom is:
\[ \omega \propto \frac{1}{n^3} \]
Thus, the correct answer is that the angular speed is inversely proportional to \(n^3\).

**Final Answer:** The angular speed of an electron in the nth orbit of a hydrogen atom is inversely proportional to \(n^3\) (Option 3).

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