The rate of a reaction increases eight times when the concentration of the reactant increases four times. The order of the reaction is

Options

A. `0.5`
B. `1.5`
C. `2.5`
D. `2.0`

Explanation

To determine the order of the reaction based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between rate and concentration**:
The rate of a reaction can be expressed as:
\[ R = k[A]^n \]
where \( R \) is the rate of reaction, \( k \) is the rate constant, \( [A] \) is the concentration of the reactant, and \( n \) is the order of the reaction.

2. **Set up the initial conditions**:
- Let the initial concentration of the reactant be \( [A] = A \).
- The initial rate of reaction is \( R = R_1 \).

3. **Apply the changes in concentration and rate**:
- When the concentration is increased to \( 4A \), the new rate becomes \( R_2 = 8R_1 \).
- We can express this mathematically:
\[ R_1 = k[A]^n = kA^n \]
\[ R_2 = k[4A]^n = k(4A)^n = k \cdot 4^n \cdot A^n \]

4. **Relate the two rates**:
Since \( R_2 = 8R_1 \), we can write:
\[ k \cdot 4^n \cdot A^n = 8 \cdot k \cdot A^n \]

5. **Cancel out common terms**:
Dividing both sides by \( k \cdot A^n \) (assuming \( k \) and \( A \) are not zero), we get:
\[ 4^n = 8 \]

6. **Express 8 as a power of 4**:
We can rewrite \( 8 \) as \( 2^3 \) and \( 4 \) as \( 2^2 \):
\[ 4^n = 2^3 \implies (2^2)^n = 2^3 \implies 2^{2n} = 2^3 \]

7. **Equate the exponents**:
Since the bases are the same, we can set the exponents equal to each other:
\[ 2n = 3 \]

8. **Solve for \( n \)**:
Dividing both sides by 2 gives:
\[ n = \frac{3}{2} = 1.5 \]

### Conclusion:
The order of the reaction is \( \frac{3}{2} \) or \( 1.5 \).