According to Bohr's theory, the variation of perimeter(s) of the electronic orbit with the order of orbit (n) in a particular atom is

Options

A. Linear
B. Parabolic
C. Exponential
D. Rectangular hyperbolic

Answer

To solve the problem of how the perimeter of the electronic orbit varies with the order of the orbit (n) according to Bohr's theory, we can follow these steps:

### Step 1: Understand the Concept of Perimeter
The perimeter of an electronic orbit in an atom is essentially the circumference of the circular orbit. The formula for the circumference (perimeter) of a circle is given by:
\[ \text{Perimeter} = 2\pi r \]
where \( r \) is the radius of the orbit.

**Hint:** Remember that the perimeter of a circle is directly related to its radius.

### Step 2: Relate the Radius to the Principal Quantum Number (n)
According to Bohr's theory, the radius of the nth orbit (denoted as \( R_n \)) is given by the formula:
\[ R_n = \frac{n^2}{Z} \cdot a_0 \]
where:
- \( n \) is the principal quantum number,
- \( Z \) is the atomic number,
- \( a_0 \) is the Bohr radius (approximately \( 0.529 \, \text{Å} \)).

**Hint:** The radius increases with the square of the principal quantum number.

### Step 3: Substitute the Radius into the Perimeter Formula
Substituting the expression for \( R_n \) into the perimeter formula, we get:
\[ \text{Perimeter} = 2\pi R_n = 2\pi \left(\frac{n^2}{Z} \cdot a_0\right) \]
This simplifies to:
\[ \text{Perimeter} = \frac{2\pi a_0}{Z} \cdot n^2 \]

**Hint:** Notice that the perimeter is proportional to \( n^2 \).

### Step 4: Identify the Variation Type
From the equation \( \text{Perimeter} = k \cdot n^2 \) (where \( k = \frac{2\pi a_0}{Z} \)), we can see that the perimeter varies with the square of the principal quantum number \( n \). This indicates a parabolic relationship.

**Hint:** A relationship of the form \( y = kx^2 \) represents a parabolic variation.

### Conclusion
Thus, according to Bohr's theory, the variation of the perimeter of the electronic orbit with the order of the orbit \( n \) is parabolic.

**Final Answer:** The perimeter varies parabolically with the order of the orbit \( n \).

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