Consider Bohr's theory for hydrogen atom. The magnitude of angular momentum, orbit radius and frequency of the electron in `n^(th)` energy state in a hydrogen atom are l, r &f respectively. Find out the value of `x`. If (frl) is directly proportional to `n^(x)`.

Explanation

To solve the problem, we need to analyze the relationships between the frequency (f), radius (r), and angular momentum (l) of an electron in the nth energy state of a hydrogen atom according to Bohr's theory.

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- According to Bohr's theory:
- The frequency \( f \) of the electron is inversely proportional to the principal quantum number \( n \):
\[ f \propto \frac{1}{n} \]
- The radius \( r \) of the electron's orbit is directly proportional to the square of the principal quantum number \( n \):
\[ r \propto n^2 \]
- The angular momentum \( l \) of the electron is directly proportional to the principal quantum number \( n \):
\[ l \propto n \]

2. **Combining the Relationships**:
- Now, we can express \( f \), \( r \), and \( l \) in terms of \( n \):
- Let \( f = k_1 \cdot \frac{1}{n} \) (where \( k_1 \) is a constant),
- Let \( r = k_2 \cdot n^2 \) (where \( k_2 \) is a constant),
- Let \( l = k_3 \cdot n \) (where \( k_3 \) is a constant).

3. **Finding the Product \( f \cdot r \cdot l \)**:
- Now, we calculate the product \( f \cdot r \cdot l \):
\[ f \cdot r \cdot l = \left(k_1 \cdot \frac{1}{n}\right) \cdot \left(k_2 \cdot n^2\right) \cdot \left(k_3 \cdot n\right) \]
- Simplifying this expression:
\[ f \cdot r \cdot l = k_1 \cdot k_2 \cdot k_3 \cdot \frac{n^2 \cdot n}{n} = k_1 \cdot k_2 \cdot k_3 \cdot n^2 \]

4. **Establishing the Proportionality**:
- From the above expression, we see that:
\[ f \cdot r \cdot l \propto n^2 \]
- This means that \( f \cdot r \cdot l \) is directly proportional to \( n^2 \).

5. **Comparing with the Given Expression**:
- The problem states that \( f \cdot r \cdot l \) is directly proportional to \( n^x \).
- From our analysis, we have:
\[ f \cdot r \cdot l \propto n^2 \]
- Therefore, by comparing the exponents, we find:
\[ x = 2 \]

### Final Answer:
The value of \( x \) is \( 2 \).

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