In the reaction `A rarr `products, time required to complete `50%` reaction was found to increase 9 times when the initial concentration of the reactant was decreased to one third. The rate law equation is `:`

Options

A. `-(d(A))/(dt)=K(A)^(1//2)`
B. `-(d(A))/(dt)=K(A)`
C. `-(d(A))/(dt)=K(A)^(2)`
D. `-(d(A))/(dt)=K(A)^(3)`

Explanation

To solve the problem, we need to analyze the relationship between the time required to complete 50% of the reaction (T_half) and the concentration of the reactant (A).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that when the initial concentration of the reactant (A) is decreased to one-third, the time required to complete 50% of the reaction increases by 9 times.
- Let the initial concentration be \( [A_0] \) and when decreased, it becomes \( [A_0/3] \).

2. **Establishing the Relationship**:
- The time required to reach 50% completion (T_half) for a reaction can be expressed in terms of the rate constant (k) and the concentration of the reactant.
- For a general reaction, T_half can be related to the concentration as follows:
\[ T_{1/2} \propto \frac{1}{[A]^{n-1}} \]
- This means that T_half is inversely proportional to the concentration raised to the power of \( (n-1) \), where n is the order of the reaction.

3. **Setting Up the Equation**:
- When the concentration is \( [A_0] \), let the time be \( T_{1/2} \).
- When the concentration is \( [A_0/3] \), the time becomes \( 9T_{1/2} \).
- Therefore, we can write:
\[ 9T_{1/2} \propto \frac{1}{([A_0/3])^{n-1}} \]
- From this, we can set up the equation:
\[ 9T_{1/2} = k \cdot \frac{1}{(A_0/3)^{n-1}} \]
- And for the original concentration:
\[ T_{1/2} = k \cdot \frac{1}{[A_0]^{n-1}} \]

4. **Relating the Two Expressions**:
- Dividing the two expressions gives:
\[ \frac{9T_{1/2}}{T_{1/2}} = \frac{[A_0]^{n-1}}{(A_0/3)^{n-1}} \]
- This simplifies to:
\[ 9 = \left(\frac{A_0}{A_0/3}\right)^{n-1} = (3)^{n-1} \]

5. **Solving for n**:
- We can now solve for \( n \):
\[ 3^{n-1} = 9 \implies 3^{n-1} = 3^2 \]
- Therefore, \( n - 1 = 2 \) or \( n = 3 \).

6. **Conclusion**:
- The reaction is third order, and the rate law can be expressed as:
\[ R = k[A]^3 \]
- Thus, the rate law equation is \( R = k[A]^3 \).