In a first order reaction A `to` B, 60% of a given sample of a compound decomposes in 45 mins. What is hal life of reaction? Also write the rate law equation for above first order reaction.

Answer

To solve the problem, we need to determine the half-life of a first-order reaction where 60% of a compound decomposes in 45 minutes. We will also write the rate law equation for the reaction.

### Step 1: Understand the Reaction
In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The reaction can be represented as:
\[ A \rightarrow B \]

### Step 2: Determine Remaining Concentration
If 60% of the sample decomposes, then 40% remains. Assuming we start with 100 units of the sample:
- Initial concentration, \( [A_0] = 100 \)
- Decomposed concentration, \( x = 60 \)
- Remaining concentration, \( [A] = 100 - 60 = 40 \)

### Step 3: Use the First-Order Rate Equation
The integrated rate law for a first-order reaction is given by:
\[ k \cdot t = 2.303 \cdot \log \left( \frac{[A_0]}{[A]} \right) \]
Where:
- \( k \) = rate constant
- \( t \) = time (in minutes)
- \( [A_0] \) = initial concentration
- \( [A] \) = concentration at time \( t \)

Substituting the values:
- \( t = 45 \) minutes
- \( [A_0] = 100 \)
- \( [A] = 40 \)

### Step 4: Calculate the Rate Constant \( k \)
Substituting into the equation:
\[ k \cdot 45 = 2.303 \cdot \log \left( \frac{100}{40} \right) \]
\[ k \cdot 45 = 2.303 \cdot \log(2.5) \]

Calculating \( \log(2.5) \):
\[ \log(2.5) \approx 0.39794 \]
Now substituting this value:
\[ k \cdot 45 = 2.303 \cdot 0.39794 \]
\[ k \cdot 45 \approx 0.916 \]
Now, solving for \( k \):
\[ k \approx \frac{0.916}{45} \]
\[ k \approx 0.02036 \, \text{min}^{-1} \]

### Step 5: Calculate the Half-Life \( t_{1/2} \)
The half-life for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
Substituting the value of \( k \):
\[ t_{1/2} = \frac{0.693}{0.02036} \]
\[ t_{1/2} \approx 34.1 \, \text{minutes} \]

### Step 6: Write the Rate Law Equation
For a first-order reaction, the rate law can be expressed as:
\[ \text{Rate} = k[A] \]
Where \( [A] \) is the concentration of the reactant.

### Final Answer
- The half-life of the reaction is approximately **34.1 minutes**.
- The rate law equation is **Rate = k[A]**.