The decomposition of ammonium nitrite was studied by placing the apparatus in a thermostat maintained at a particular temperature .The volume of `N_(2)` gas collected at different intervals are as follows :
`{:(Time ("in min"),10,15,20,25,oo),("Volume of "N_(2)("inML"),6.25,9.00,11.40,13.65,35.05):}`
From the above data, prove that the reaction is of the first order.

Explanation

`NH_(4)NO_(2)rarrN_(2)+2H_(2)O`
`{:(t=0,a,0,0),(t=t,a-x,x,2x),(t=oo,0,a,2a):}`
` k = (2.303)/(t)log.(a)/(a-x)." If" V_(t)and V_(oo)` are the volumes of `N_(2)` at the time , t = t and `t=oo` , the the initial concentration (a) of `NH_(4)NO_(2)` = concentration of `N_(2)` at infinite time `prop V_(oo)`
At time t , concentration of `NH_(4)NO_(2)`
`(a-x)prop(V_(oo)-V_(t))[becausexpropV_(t)]`
`therefore k = (2.303)/(t) log.(V_(oo))/(V_(oo)-V_(t)) " " ...[1]`
By putting the values of `V_(oo) and V_(t)` obtained at time , t = 10 min in the above equation ,
`k=(2.303)/(10)log.(35.05)/(35.05-6.25)=0.0196"min"^(-1)`
Similarly , be putting the values of `V_(oo) and V_(t)` obtained at different times in equation (1), the values of k thus obtained are found to be similar. So , the reaction follows first order kinetics.