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 min. from the start of the decomposition in a closed vessel, the total pressure developed is found to be 284.5 mm of Hg and on complete decomposition, the total pressure is 584.5 mm of Hg. Calculate the rate constant for the reaction.

Answer

`k = 5.2 xx 10^(-3) min^(-1)`

Explanation

`2N_(2)O_(5)(g) rarr 4NO_(2)(g) + O_(2)(g)`
OR
`N_(2)O_(5)(g) rarr 2NO_(2)(g) + (1)/(2)O_(2)(g)`
`{:("Initial pressure",a,0,0,),("At time t",a-x,2x,x//2,),("At complete",,,,),("dissociation",0,2a,a//2,):}`
Total number of moles at time `t = a-x+2x + (x)/(2)`
`= a+(3)/(2)x`
Total number of moles at complete dissociation
`= 2a + (a)/(2) = (5)/(2)a`
`:. (5)/(2)a prop 584.5 mm Hg`
and `a+(3)/(2)x prop 284.5 mm Hg`
Solving: `a prop 233.8 mm Hg` and `x prop 33.8 mm Hg`.
Substituting the value in first order equation:
`k = (2.303)/(t) log.(a)/(a-x) = (2.303)/(30 min) log.(233.8)/((233.8 - 33.8))`
`= 5.21 xx 10^(-3) min^(-1)`