Select the incorrect graph for velocity of `e^(-)` in an orbit vs. Z, `1/n` and n :

Options

A.
B.
C.
D.

Answer

To solve the problem of selecting the incorrect graph for the velocity of an electron in an orbit versus Z, \( \frac{1}{n} \), and \( n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between velocity, Z, and n**:
The velocity \( v \) of an electron in an orbit is derived from the balance of forces acting on it. The centrifugal force is equal to the electrostatic force. This can be expressed as:
\[ mv^2/r = \frac{kZe^2}{4\pi \epsilon_0 r^2} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, \( r \) is the radius of the orbit, \( Z \) is the atomic number, \( e \) is the charge of the electron, and \( k \) is Coulomb's constant.

2. **Use angular momentum quantization**:
The angular momentum of the electron is quantized:
\[ mvr = n\frac{h}{2\pi} \]
where \( n \) is a principal quantum number and \( h \) is Planck's constant.

3. **Derive the formula for velocity**:
By substituting \( r \) from the angular momentum equation into the force balance equation, we can derive that:
\[ v = \frac{2\pi^2 k e^2 Z}{n h} \]
This shows that the velocity \( v \) is proportional to \( \frac{Z}{n} \):
\[ v \propto \frac{Z}{n} \]

4. **Analyze the graphs**:
- **Graph A (v vs n)**: Since \( v \) is inversely proportional to \( n \), this graph should be a rectangular hyperbola. This is correct.
- **Graph B (v vs \( \frac{1}{n} \))**: Since \( v \) is directly proportional to \( \frac{1}{n} \), this graph should be a straight line passing through the origin. This is correct.
- **Graph C (v vs Z)**: Since \( v \) is directly proportional to \( Z \), this graph should also be a straight line passing through the origin. This is correct.
- **Graph D (v vs n)**: This graph is claimed to be a straight line with a negative slope, which contradicts the derived relationship that \( v \) is inversely proportional to \( n \) (should be a rectangular hyperbola). This is incorrect.

5. **Conclusion**:
The incorrect graph is **Graph D**, which incorrectly represents the relationship between velocity and \( n \).

### Final Answer:
The incorrect graph is **D**.