According to Bohr's atomic theory, which of the following is correct ?
(a)Potential energy of electron `propZ^(2)/n^(2)`
(b)The product of velocity of electron and principal quantum number (n) `prop-Z^(2)`
(c)Frequency for revolution of electron in an orbit `propZ^(2)/n^(3)`
(d)Coulombic force of attraction on the electron `propZ^(2)/n^(2)`

Options

A. Potential energy of electron `propZ^(2)/n^(2)`
B. The product of velocity of electron and principal quantum number (n) `prop-Z^(2)`
C. Frequency fo revolution of electron in an orbit `propZ^(2)/n^(3)`
D. Coulombic force of attraction on the electron `propZ^(2)/n^(2)`

Answer

To solve the question regarding Bohr's atomic theory and determine which of the given statements is correct, we will analyze each option step by step.

### Step 1: Analyze Potential Energy of Electron
The potential energy (PE) of an electron in a hydrogen-like atom can be expressed as:
\[ PE \propto -\frac{Z^2}{n^2} \]
Where:
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number.

**Conclusion for Option (a)**: The expression given as \( PE \propto \frac{Z^2}{n^2} \) is incorrect because it should be negative. Thus, Option (a) is **not correct**.

### Step 2: Analyze Product of Velocity of Electron and Principal Quantum Number
The velocity \( v \) of an electron in a Bohr orbit is given by:
\[ v \propto \frac{Z}{n} \]
If we multiply the velocity by the principal quantum number \( n \):
\[ v \cdot n \propto Z \]
The statement claims that \( v \cdot n \propto -Z^2 \), which is incorrect.

**Conclusion for Option (b)**: The expression given is incorrect. Thus, Option (b) is **not correct**.

### Step 3: Analyze Frequency of Revolution of Electron
The frequency \( f \) of an electron in an orbit can be derived from the expression:
\[ f \propto \frac{Z^2}{n^3} \]
This means that frequency is directly proportional to \( Z^2 \) divided by \( n^3 \).

**Conclusion for Option (c)**: The expression \( f \propto \frac{Z^2}{n^3} \) is correct. Thus, Option (c) is **correct**.

### Step 4: Analyze Coulombic Force of Attraction on the Electron
The Coulombic force \( F \) acting on the electron can be expressed as:
\[ F \propto \frac{Z^2}{n^4} \]
The statement claims that \( F \propto \frac{Z^2}{n^2} \), which is incorrect.

**Conclusion for Option (d)**: The expression given is incorrect. Thus, Option (d) is **not correct**.

### Final Conclusion
Based on the analysis:
- Option (a) is incorrect.
- Option (b) is incorrect.
- Option (c) is correct.
- Option (d) is incorrect.

Therefore, the correct answer according to Bohr's atomic theory is **Option (c)**.

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