The decomposition of `N_(2)O_(5)` according to the equation
`2N_(2)O_(5) (g) to 4NO_(2) (g) +O_(2)(g)`
is a first - order reaction . After 30 minutes from the start of the decomposition in a closed vessel , the total pressure developed is found to be `284.5` mmHg and an complete decomposition the total pressure is `584.5` mmHg . Calculate the rate constant of the reaction .

Explanation

`{:(,a,,,),(,2N_(2)O_(5)to,4NO_(2),+O_(2),),("After 30 min",a-x,2x,x//2,):}`
` :. (a-x) +2x +x/2 = 284.5`
or ` a + (3x)/2 = 284.5" " …(1)`
After complete decomposition of `N_(2)O_(5)`
`{:(2N_(2)O_(5)to,4NO_(2)+,O_(2)),(0,2a,a//2):}`
` :. 2a + a/2 = 584.5" " ...(2)`
or ` (5a)/2 = 584.5 " " ...(2)`
From (1) and (2) we get
a = 233.5 , x = 34
Thus , for a first reaction of the type `2A to ` products
` k = 1/(2t) log. a/(a-x)`
` k = (2.303)/(2xx30) log. (233.5)/((233.5 -34))`
` = 2.625 xx10^(-3) min^(-1)`