Speed `(V_(n))` of electron in nth orbit versus principal quantum number (n) graphs is

Options

A.
B.
C.
D.

Answer

To solve the question regarding the speed \( V_n \) of an electron in the nth orbit versus the principal quantum number \( n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
The speed of an electron in the nth orbit is given by the formula:
\[ V_n \propto \frac{Z}{n} \]
where \( Z \) is the atomic number of the element. For a given atom, \( Z \) remains constant.

2. **Express the Relationship Mathematically**:
Since \( Z \) is constant, we can simplify the relationship to:
\[ V_n \propto \frac{1}{n} \]
This indicates that the speed \( V_n \) is inversely proportional to the principal quantum number \( n \).

3. **Graphical Representation**:
When two quantities are inversely proportional, the graph of one quantity versus the other will be a hyperbola. Specifically, if we plot \( V_n \) on the y-axis and \( n \) on the x-axis, the graph will decrease as \( n \) increases.

4. **Sketch the Graph**:
- As \( n \) increases, \( V_n \) decreases.
- The graph starts from a higher value of \( V_n \) when \( n = 1 \) and approaches zero as \( n \) increases indefinitely.

5. **Identify the Correct Option**:
Based on the graphical representation of the relationship \( V_n \propto \frac{1}{n} \), we can conclude that the correct option for the graph of \( V_n \) versus \( n \) is the one that shows a hyperbolic decline.

### Conclusion:
The correct graph for the speed \( V_n \) of an electron in the nth orbit versus the principal quantum number \( n \) is a hyperbola that decreases as \( n \) increases.