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Chemistry Question on Chemical Kinetics

Consider the following reaction, the rate expression of which is given below:
A+B→C\text{A} + \text{B} \to \text{C}
rate=k[A]1/2[B]1/2\text{rate} = k [\text{A}]^{1/2} [\text{B}]^{1/2}
The reaction is initiated by taking 1M concentration of A and B each. If the rate constant (kk) is 4.6×10−2 s−14.6 \times 10^{-2} \, \text{s}^{-1}, then the time taken for A to become 0.1 M is ______ sec. (nearest integer)

Answer

For this reaction:
K=2.303tlog⁡[A]0[A]K = \frac{2.303}{t} \log \frac{[A]_0}{[A]}
Given: k=4.6×10−2 s−1, [A]0=1 M, [A]=0.1 Mk = 4.6 \times 10^{-2} \, \text{s}^{-1}, \, [A]_0 = 1 \, \text{M}, \, [A] = 0.1 \, \text{M}
4.6×10−2=2.303tlog⁡10.14.6 \times 10^{-2} = \frac{2.303}{t} \log \frac{1}{0.1}
4.6×10−2=2.303t×14.6 \times 10^{-2} = \frac{2.303}{t} \times 1
t=2.3034.6×10−2≈50 sec.t = \frac{2.303}{4.6 \times 10^{-2}} \approx 50 \, \text{sec.}