Question

An optically active compound $A$ undergoes fist order conversion to another optically active compounds $B$ and $C$. From the following data of optical rotation, calculate the time from the start at which the solution will become a racemic mixture. $A\to B+C$

time (min)	0	10	$\infty$
optical rotation ${(}^{\circ })$	${15}^{\circ }$	$5^{\circ }$	$-5^{\circ }$

• A. $20min$
• B. $30min$
• C. $5min$
• D. $100min$

Answer 1

The correct option is A $20min$

$k=\frac{1}{t}ln\frac{{\alpha }_{\infty }-{\alpha }_0}{{\alpha }_{\infty }-{\alpha }_t}$

${\alpha }_{\infty }$=rotation at $t=\infty$

${\alpha }_0$= rotation at $t=0$

$a_t$= rotation at time $t$

$k=\frac{1}{10}ln\frac{-5-15}{-5-5}=\frac{1}{10}ln2$....(1)

Now, when solution will be racemic mixture, ${\alpha }_t=0$

$t=\frac{1}{k}ln\frac{-5-15}{-5-0}=\frac{1}{k}ln4$

$=\frac{2}{k}ln2$

From (1), $t=\frac{2ln2}{ln2}\times 10$ $=20$ min