The thermal decomposition of a compound is of first order. If `50%` of a sample of the compound is decomposed in 120 minutes, how long it take for `90%` of the compounds to decompose.

Options

A. Nearly 240 minutes
B. Nearly 480 minutes
C. Nearly 450 minutes
D. Nearly 400 minutes

Answer

To solve the problem of determining how long it takes for 90% of a compound to decompose in a first-order reaction, we can follow these steps:

### Step 1: Understand the given data
We know that:
- The reaction is first-order.
- 50% of the compound decomposes in 120 minutes.

### Step 2: Determine the half-life
The half-life (t₁/₂) for a first-order reaction is given as:
\[ t_{1/2} = 120 \text{ minutes} \]

### Step 3: Calculate the rate constant (k)
For a first-order reaction, the rate constant (k) can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} \]
Substituting the half-life:
\[ k = \frac{0.693}{120} \]
\[ k \approx 0.005775 \text{ min}^{-1} \]

### Step 4: Set up the equation for 90% decomposition
If 90% of the compound decomposes, then 10% remains. If we assume the initial concentration (A₀) is 100%, then the concentration at time t (A) is:
\[ A = 10 \text{ (since 90% is decomposed)} \]

### Step 5: Use the first-order rate equation
The first-order rate equation is given by:
\[ k = \frac{2.303}{t} \log\left(\frac{A_0}{A}\right) \]
Substituting the values we have:
\[ 0.005775 = \frac{2.303}{t} \log\left(\frac{100}{10}\right) \]

### Step 6: Simplify the logarithm
Calculate the logarithm:
\[ \log\left(\frac{100}{10}\right) = \log(10) = 1 \]
So the equation simplifies to:
\[ 0.005775 = \frac{2.303}{t} \cdot 1 \]

### Step 7: Solve for time (t)
Rearranging the equation to solve for t:
\[ t = \frac{2.303}{0.005775} \]
Calculating this gives:
\[ t \approx 398.78 \text{ minutes} \]

### Step 8: Round the answer
Rounding off, we find:
\[ t \approx 400 \text{ minutes} \]

Thus, it takes approximately **400 minutes** for 90% of the compound to decompose.

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