Question

If 10.0 g of substance A decomposes for 100 min, the mass of A remaining unreacted is found to be 1.25g. What is the half life of this reaction if it follows first order kinetics?

Answer 1

A -------> product

At the beginning of the reaction , the mass of A is 'a' gm . At time 't' the amount of A that has been decomposed is 'x' gm.

From the first order reaction , we can write

$K = \frac{1}{t}ln\left ( \frac{a}{a-x} \right )$ .,..,.............. (1)

Here K = rate constant

Here given that

a = initial mass = 10.0 g

t =time = 100 min

(a-x) = unreacted mass at time t = 1.25 g

Putting this value  

$K = \frac{1}{100min}ln\left ( \frac{10g}{1.25g} \right )$

K = 0.0208 min ^-1

For half life , we know the the decomposition of half mass .

Hence x= ( a/2) when t = t_{​​​​​​1/2}

Putting this value in equation (1) , we get

$K = \frac{1}{t_{1/2}}ln\left ( \frac{a}{a-a/2} \right )$

$K = \frac{1}{t_{1/2}}ln\left ( \frac{a}{a/2} \right )$

$K = \frac{1}{t_{1/2}}ln2$

${t_{1/2}}=\frac{1}{K} ln2$

Putting the k value

${t_{1/2}}=\frac{1}{0.0208 min^{-1}} ln2$

Hence half of life of this reaction is 33.32 min .