According to Bohr’s theory, which of the following correctly represents the variation of energy and radius of an electron in nth orbit of H-atom?

Options

A. `E_(n)prop(1)/(n^(2)),r prop (1)/(n^(2))`
B. `E_(n)prop(1)/(n^(2)),r prop n^(2)`
C. `E_(n)prop n^(2), r prop n^(2)`
D. `E_(n)prop n, r prop (1)/(n)`

Answer

To solve the question regarding the variation of energy and radius of an electron in the nth orbit of a hydrogen atom according to Bohr’s theory, we will follow these steps:

### Step 1: Understand the Energy Formula
According to Bohr’s theory, the energy (E) of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{k^2 \cdot 2\pi^2 \cdot m \cdot z^2 \cdot e^4}{n^2 \cdot h^2} \]
Where:
- \( k \) is a constant,
- \( m \) is the mass of the electron,
- \( z \) is the atomic number (which is 1 for hydrogen),
- \( e \) is the charge of the electron,
- \( n \) is the principal quantum number (the orbit number),
- \( h \) is Planck's constant.

### Step 2: Determine the Variation of Energy
From the formula, we can see that the energy \( E_n \) is inversely proportional to \( n^2 \):
\[ E_n \propto -\frac{1}{n^2} \]
This means as \( n \) increases, the energy becomes less negative (increases).

### Step 3: Understand the Radius Formula
The radius (R) of the nth orbit is given by the formula:
\[ R_n = \frac{n^2 \cdot h^2}{4\pi^2 \cdot m \cdot z \cdot e^2} \]
From this formula, we can see that the radius \( R_n \) is directly proportional to \( n^2 \):
\[ R_n \propto n^2 \]
This means as \( n \) increases, the radius increases.

### Step 4: Summarize the Relationships
- Energy \( E_n \) is inversely proportional to \( n^2 \): \( E_n \propto -\frac{1}{n^2} \)
- Radius \( R_n \) is directly proportional to \( n^2 \): \( R_n \propto n^2 \)

### Step 5: Analyze the Options
Now, we can analyze the options provided in the question:
1. Both energy and radius are inversely proportional.
2. Energy is inversely proportional and radius is directly proportional.
3. Energy is directly proportional and radius is directly proportional.
4. Energy is directly proportional and radius is inversely proportional.

From our analysis, the correct statement is:
- Energy is inversely proportional to \( n^2 \) and radius is directly proportional to \( n^2 \). Therefore, the correct answer is option 2.

### Final Answer
The correct representation of the variation of energy and radius of an electron in the nth orbit of a hydrogen atom according to Bohr's theory is:
**Option 2: Energy is inversely proportional to \( n^2 \) and radius is directly proportional to \( n^2 \).**