In a hydrogen-like species, the net force acting on a revolving electron in an orbit as given by the Bohr model is proportional to [n `rarr` principal quantum numbe, Z `rarr` atomic number]

Options

A. `(Z^(3))/(n^(4))`
B. `(n^(3))/(Z^(2))`
C. `(Z^(2))/(n^(3))`
D. `(n^(4))/(Z^(3))`

Answer

To solve the problem of determining the net force acting on a revolving electron in a hydrogen-like species according to the Bohr model, we can follow these steps:

### Step 1: Understand the Forces Acting on the Electron
In the Bohr model, the electron revolves around the nucleus due to the electrostatic force of attraction between the positively charged nucleus (with atomic number Z) and the negatively charged electron. This force can be equated to the centripetal force required to keep the electron in circular motion.

### Step 2: Write the Expression for Centripetal Force
The centripetal force \( F_c \) acting on the electron can be expressed as:
\[ F_c = \frac{m v^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron,
- \( r \) is the radius of the orbit.

### Step 3: Determine the Velocity of the Electron
From the Bohr model, the velocity of the electron in the nth orbit is given by:
\[ v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 4: Determine the Radius of the Orbit
The radius of the nth orbit is given by:
\[ r_n = 0.53 \frac{n^2}{Z} \text{ Å} \]
This can also be expressed in meters by converting angstroms to meters.

### Step 5: Substitute the Velocity and Radius into the Centripetal Force Equation
Substituting the expressions for \( v_n \) and \( r_n \) into the centripetal force equation:
\[ F_c = \frac{m \left(2.18 \times 10^6 \frac{Z}{n}\right)^2}{0.53 \frac{n^2}{Z}} \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
1. Calculate \( v^2 \):
\[ v^2 = \left(2.18 \times 10^6 \frac{Z}{n}\right)^2 = (2.18^2 \times 10^{12}) \frac{Z^2}{n^2} \]
2. Substitute into the force equation:
\[ F_c = \frac{m (2.18^2 \times 10^{12}) \frac{Z^2}{n^2}}{0.53 \frac{n^2}{Z}} = \frac{m (2.18^2 \times 10^{12}) Z^3}{0.53 n^4} \]

### Step 7: Identify the Proportionality
From the final expression, we can see that the net force \( F_c \) is proportional to:
\[ F_c \propto \frac{Z^3}{n^4} \]

### Conclusion
Thus, the net force acting on a revolving electron in a hydrogen-like species is proportional to \( \frac{Z^3}{n^4} \).

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