form the following data for the decompoistion of `N_(2)O_(5)` in carbon tertrachloride solution at `321 K`, show that the reaction is of the first order and calculate the rate constant.

Explanation

If the reaction is of the first order, it must obey the equation :
`k=(2.303)/(t)log""(a)/(a-x)=(2.303)/(t)log""(V_(oo))/(V_(oo)-V_(t))`
In the present case, we are given that `V_(oo)=34.75" cm"^(3)`
The value of k at each instant can be calculated as follows :
`{:("t(min)",,,V_(t),,,V_(oo)-V_(t),,,k=(2.303)/(t)log""(V_(oo))/(V_(oo)-V_(t))),(10,,,6.30,,,34.75-6.30=28.45,,,k=(2.303)/(10min)log""(34.75)/(28.45)=0.01997min^(-1)),(15,,,8.95,,,34.75-8.95=25.80,,,k=(2.303)/(15min)log""(34.75)/(25.80)=0.01985min^(-1)),(20,,,11.40,,,34.75-11.40=23.35,,,k=(2.303)/(20min)log""(34.75)/(23.35)=0.01987min^(-1)),(25,,,13.50,,,34.75-13.50=21.25,,,k=(2.303)/(25min)log""(34.75)/(21.25)=0.01967min^(-1)):}`
Since the value of k comes to be nearly constant, hence given reaction is of the first order. The average value of the constant `=0.01984" minute"^(-1)`