JEE Main Problem: Decomposition Time for 80% Completion

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Daily Challenge JEE Main Practice

Decomposition Time for 80% Completion

Use the given half-life to calculate the time required for a specific percentage of decomposition.

Question:

At $30^\circ\text{C}$, the half life for the decomposition of $AB_2$ is $200 \text{ s}$ and is independent of the initial concentration of $AB_2$.

The time required for $80\%$ of the $AB_2$ to decompose is:

[Given: $\log 2 = 0.30$, $\log 3 = 0.48$]

Select the correct option:

Detailed Solution

Concept Check: The problem states that the half-life is "independent of the initial concentration". This immediately tells us that the reaction follows First-Order Kinetics.

Step 1: Calculate the Rate Constant ($k$)

For a first-order reaction, the relationship between the rate constant and half-life is:

$$ k = \frac{0.693}{t_{1/2}} $$ $$ k = \frac{2.303 \log 2}{200} $$

Tip: Keep it in this form for now. It often helps terms cancel out later!

Step 2: Set up the First-Order Rate Equation

The integrated rate law for a first-order reaction is:

$$ t = \frac{2.303}{k} \log \left( \frac{[A]_0}{[A]_t} \right) $$

We are looking for the time ($t$) when $80\%$ of $AB_2$ has decomposed.

• Let initial concentration $[A]_0 = 100$
• Amount decomposed = $80\%$ of $100 = 80$
• Amount remaining, $[A]_t = 100 - 80 = \mathbf{20}$

Step 3: Substitute and Calculate

Substitute the values of $k$, $[A]_0$, and $[A]_t$ into the rate equation:

$$ t = \frac{2.303}{\left(\frac{2.303 \log 2}{200}\right)} \log \left( \frac{100}{20} \right) $$

Notice how the $2.303$ terms cancel out beautifully:

$$ t = \frac{200}{\log 2} \log (5) $$

We need the value of $\log 5$. We can find this using the given $\log 2 = 0.30$:

$$ \log 5 = \log \left(\frac{10}{2}\right) = \log 10 - \log 2 = 1 - 0.30 = 0.70 $$

Now plug this back into our equation for $t$:

$$ t = \frac{200}{0.30} \times 0.70 $$ $$ t = 200 \times \frac{7}{3} $$ $$ t = \frac{1400}{3} \approx 466.67 \text{ s} $$

Final Answer: Option (C) 467 s

Review First Order Kinetics

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