Question:
In a first order reaction, the reactant decomposes 25% of its initial concentration in 40 minutes. What is the value of rate constant of the reaction?
(Given: $\log 3 = 0.4771$, $\log 4 = 0.6021$)
In a first order reaction, the reactant decomposes 25% of its initial concentration in 40 minutes. What is the value of rate constant of the reaction?
(Given: $\log 3 = 0.4771$, $\log 4 = 0.6021$)
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For 1st order reactions, always use $k = \frac{2.303}{t} \log \frac{a}{a-x}$ and convert percentage directly into fraction form.
Updated On: Jul 18, 2026
- $7.19 \times 10^{-3} \, \text{min}^{-1}$
- $2.19 \times 10^{-3} \, \text{min}^{-1}$
- $5.19 \times 10^{-3} \, \text{min}^{-1}$
- $1.19 \times 10^{-3} \, \text{min}^{-1}$
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The Correct Option is A
Solution and Explanation
Step 1: Use first-order integrated rate law.
For a first-order reaction: \[ k = \frac{2.303}{t} \log \frac{a}{a-x} \] where \(a\) is initial concentration and \(a-x\) is remaining concentration after time \(t\).
Step 2: Interpret given data.
25% decomposed means: \[ a-x = 75\% = \frac{3}{4}a \] Time given: \[ t = 40 \, \text{min} \]
Step 3: Substitute in formula.
\[ k = \frac{2.303}{40} \log \left(\frac{a}{(3/4)a}\right) = \frac{2.303}{40} \log \left(\frac{4}{3}\right) \]
Step 4: Evaluate logarithm.
\[ \log\left(\frac{4}{3}\right) = \log 4 - \log 3 \] \[ = 0.6021 - 0.4771 = 0.1250 \]
Step 5: Calculate rate constant.
\[ k = \frac{2.303}{40} \times 0.125 \] \[ k = 0.057575 \times 0.125 = 0.00719 \]
Step 6: Final conclusion.
\[ \boxed{7.19 \times 10^{-3} \, \text{min}^{-1}} \]
For a first-order reaction: \[ k = \frac{2.303}{t} \log \frac{a}{a-x} \] where \(a\) is initial concentration and \(a-x\) is remaining concentration after time \(t\).
Step 2: Interpret given data.
25% decomposed means: \[ a-x = 75\% = \frac{3}{4}a \] Time given: \[ t = 40 \, \text{min} \]
Step 3: Substitute in formula.
\[ k = \frac{2.303}{40} \log \left(\frac{a}{(3/4)a}\right) = \frac{2.303}{40} \log \left(\frac{4}{3}\right) \]
Step 4: Evaluate logarithm.
\[ \log\left(\frac{4}{3}\right) = \log 4 - \log 3 \] \[ = 0.6021 - 0.4771 = 0.1250 \]
Step 5: Calculate rate constant.
\[ k = \frac{2.303}{40} \times 0.125 \] \[ k = 0.057575 \times 0.125 = 0.00719 \]
Step 6: Final conclusion.
\[ \boxed{7.19 \times 10^{-3} \, \text{min}^{-1}} \]
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