The velocity of an electron in single electron atom in an orbit

Options

A. is independent of the atomic number of the element
B. increases with increases in atomic number
C. decreases with increases in atomic number
D. increases with increases in quantum number

Explanation

To find the velocity of an electron in a single electron atom in an orbit, we can follow these steps:

### Step 1: Understand the formula for velocity
The velocity of an electron in the nth orbit can be expressed using the formula:
\[ v_n = \sqrt{\frac{k \cdot Z \cdot e^2}{n \cdot h^2}} \]
where:
- \( v_n \) = velocity of the electron in the nth orbit
- \( k \) = Coulomb's constant (related to the electrostatic force)
- \( Z \) = atomic number (number of protons in the nucleus)
- \( e \) = charge of the electron
- \( n \) = principal quantum number (indicating the energy level)
- \( h \) = Planck's constant

### Step 2: Analyze the relationship between variables
From the formula, we can see that:
- The velocity \( v_n \) is directly proportional to the atomic number \( Z \).
- The velocity \( v_n \) is inversely proportional to the principal quantum number \( n \).

This means:
- If \( Z \) increases, \( v_n \) increases.
- If \( n \) increases, \( v_n \) decreases.

### Step 3: Evaluate the options based on the relationship
Given the options:
1. Independent of the atomic number of the element.
2. Increases with the increase in atomic number.
3. Decreases with increase in atomic number.
4. Increases when quantum number decreases.

From our analysis:
- Option 1 is incorrect because the velocity depends on \( Z \).
- Option 2 is correct since \( v_n \) increases with \( Z \).
- Option 3 is incorrect because \( v_n \) does not decrease with an increase in \( Z \).
- Option 4 is also correct since \( v_n \) increases when \( n \) decreases.

### Conclusion
The correct option is **B: Increases with the increase in atomic number**.

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