Rate of decomposition of a substance increases by a factor 2.25 for 1.5 times increase in conentration at constant temperature . Calculate the order of reaction.

Answer

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

### Step 1: Understand the Rate Law
The rate of a reaction can be expressed by the rate law:
\[ \text{Rate} = k[A]^n \]
where:
- \( k \) is the rate constant,
- \( [A] \) is the concentration of the reactant,
- \( n \) is the order of the reaction.

### Step 2: Set Up the Initial and New Conditions
According to the problem, the rate of decomposition increases by a factor of 2.25 when the concentration increases by a factor of 1.5.

Let:
- Initial concentration = \( [A] \)
- New concentration = \( 1.5[A] \)
- Initial rate = \( \text{Rate}_1 = k[A]^n \)
- New rate = \( \text{Rate}_2 = k(1.5[A])^n \)

### Step 3: Write the Relationship Between Rates
From the information given, we can express the relationship between the new and initial rates:
\[ \frac{\text{Rate}_2}{\text{Rate}_1} = 2.25 \]
Substituting the expressions for the rates:
\[ \frac{k(1.5[A])^n}{k[A]^n} = 2.25 \]
This simplifies to:
\[ \frac{(1.5)^n}{1} = 2.25 \]

### Step 4: Solve for \( n \)
Now we need to solve the equation:
\[ (1.5)^n = 2.25 \]
We can express 2.25 as:
\[ 2.25 = (1.5)^2 \]
Thus, we have:
\[ (1.5)^n = (1.5)^2 \]
Since the bases are the same, we can equate the exponents:
\[ n = 2 \]

### Conclusion
The order of the reaction is \( n = 2 \).