\(A\longrightarrow 2B\)
t = 0
A =1
t = 100
A \(= 1 -0.1 = 0.9\)
B \(=2\times0.01=0.2\)
\(k=\frac{1}{t}ln\frac{[A_0]}{[A_t]}\)
\(k=\frac{1}{t}\;ln\;\frac{1.0}{0.9}\) \(\bigg[k=\frac{0.693}{t_{1/2}}\bigg]\)
\(\frac{0.693}{t_{1/2}}=\frac{1}{t}\;[ln\;10-ln\;9]\)
\(0.693\times t=t_{1/2}[ln\;10-2\;ln\;3]\) \(\bigg[ln\;9\Rightarrow ln\;3^2=2\;ln\;3\bigg]\)
\(0.693\times t=t_{1/2}[2.3-2\times1.1]\)
\(0.693\times t=t_{1/2}[2.3-2.2]\)
\(0.693\times t=t_{1/2}[0.1]\)
\(t_{1/2}=\frac{0.693\times t}{0.1}\)
\(=\frac{0.693\times 100}{0.1}\)
\(t_{1/2}=693\;minute\)