If doubling the concentration of a reactant ‘A’ increases the rate 8 times and tripling the concentration of ‘A’ increases the rate 27 times, the rate is proportional to:

Options

A. Concentration of ‘A’
B. Square of concentration of ‘A’
C. Under root of the concentration of ‘A’
D. Cube of concentration of ‘A’

Answer

To solve the problem step-by-step, we need to analyze the relationship between the concentration of reactant A and the rate of the reaction based on the information provided.

### Step 1: Understand the relationship between concentration and rate
The rate of a reaction can be expressed as:
\[ \text{Rate} = k [A]^n \]
where:
- \( k \) is the rate constant,
- \( [A] \) is the concentration of reactant A,
- \( n \) is the order of the reaction with respect to A.

### Step 2: Set up the equations based on the given information
1. When the concentration of A is doubled (\( [A] \to 2[A] \)), the rate increases 8 times:
\[ R_1 = k (2[A])^n = 8R \]
This implies:
\[ 8R = k (2[A])^n \]

2. When the concentration of A is tripled (\( [A] \to 3[A] \)), the rate increases 27 times:
\[ R_2 = k (3[A])^n = 27R \]
This implies:
\[ 27R = k (3[A])^n \]

### Step 3: Divide the two equations
To eliminate \( k \) and \( R \), we can divide the first equation by the second:
\[ \frac{8R}{27R} = \frac{k (2[A])^n}{k (3[A])^n} \]
This simplifies to:
\[ \frac{8}{27} = \frac{(2[A])^n}{(3[A])^n} \]
Since \( [A] \) cancels out, we have:
\[ \frac{8}{27} = \left(\frac{2}{3}\right)^n \]

### Step 4: Solve for \( n \)
We can express \( 8 \) and \( 27 \) as powers:
\[ 8 = 2^3 \quad \text{and} \quad 27 = 3^3 \]
Thus:
\[ \frac{2^3}{3^3} = \left(\frac{2}{3}\right)^n \]
This leads to:
\[ \left(\frac{2}{3}\right)^3 = \left(\frac{2}{3}\right)^n \]
From this, we can conclude that:
\[ n = 3 \]

### Step 5: Determine the proportionality of the rate
Since we found that \( n = 3 \), we can express the rate as:
\[ \text{Rate} = k [A]^3 \]
This means that the rate is proportional to the cube of the concentration of A.

### Conclusion
The rate is proportional to \( [A]^3 \).