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NEET levelSingle correct optionDifficulty 1 of 3ck113

For a first-order reaction followed polarimetrically, with observed rotations $r_{0}$ , $r_{t}$ and $r_{\infty}$ at time zero, $t$ and infinity, the rate constant is

A $k = \frac{1}{t} ln \frac{r _{t} - r _{\infty}}{r _{0} - r _{\infty}}$
B $k = \frac{1}{t} ln \frac{r _{0} - r _{\infty}}{r _{0} - r _{t}}$
C $k = \frac{1}{t} ln \frac{r _{0} - r _{\infty}}{r _{t} - r _{\infty}}$
D $k = \frac{1}{t} ln \frac{r _{0}}{r _{t}}$

Answer: C

The quantity

r_{t} - r_{\infty}

is proportional to the concentration of reactant left, so

[A]_{0} \propto r_{0} - r_{\infty}

and

[A]_{t} \propto r_{t} - r_{\infty}

. Then

k = \frac{1}{t} ln \frac{[ A ] _{0}}{[ A ] _{t}}

gives the first option.

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