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Step 1: Identify the type of reaction and given data
This is a first-order reaction with the rate constant
k = 20\,\text{min}^{-1} . We want the time required for the reactant concentration
to decrease from its initial value C_0 to C_0/32 .
Step 2: Write the integrated rate law for a first-order reaction
For a first-order reaction, the concentration at time t is given by
C(t) = C_0 \, e^{-kt}.
We want C(t) = \frac{C_0}{32}. Substituting into the equation:
\frac{C_0}{32} = C_0 \, e^{-kt}.
Step 3: Solve for the time t
Divide both sides by C_0 to get
\frac{1}{32} = e^{-kt}.
Taking the natural logarithm on both sides:
\ln\!\biggl(\frac{1}{32}\biggr) = -kt.
Note that \frac{1}{32} = 2^{-5} , so
\ln(2^{-5}) = -5\,\ln(2).
Therefore,
-5\,\ln(2) = -k\,t \quad\Longrightarrow\quad t = \frac{5\,\ln(2)}{k}.
Step 4: Substitute numerical values
Given k = 20\,\text{min}^{-1} and \ln(2) \approx 0.693 ,
t = \frac{5 \times 0.693}{20}
= \frac{3.465}{20}
= 0.17325\,\text{min}.
Step 5: Convert to the requested format and round
Since 0.17325\,\text{min} = 17.325 \times 10^{-2}\,\text{min} , rounding to the nearest integer for the coefficient in front of 10^{-2} gives
\boxed{17 \times 10^{-2}\,\text{min}}.
