The rate law for the decompoistion of gaseous `N_(2)O_(5)`,
`N_(2)O_(5)(g) rarr 2NO_(2)(g) + (1)/(2)O_(2)(g)` is observed to be:
`r = (-d[N_(2)O_(5)])/(dt) = k[N_(2)O_(5)]`
A reaction machanism which has been suggested to be conisstent with this rate law is
`N_(2)O_(5)(g) overset(k_(eq))hArr NO_(2)(g)+NO_(3)(g) ("fast equilibrium")`
`NO_(2) (g) + NO_(3)(g) overset(k_(1))rarr NO_(2)(g) + NO(g) + O_(2)(g)` (slow)
`NO(g) + NO_(3)(g) overset(k_(2))rarr 2NO_(2)(g)` (fast)

Explanation

`r = k_(1)[NO_(2)][NO_(3)]` …(i)
and form the fast equilibrium step,
`k_(eq) = ([NO_(2)][NO_(3)])/([N_(2)O_(5)]`
Thus, `[NO_(2)] [NO_(3)] = k[N_(2)O_(5)]` …(ii)
Uisng Eq. (ii) in Eq. (i), we get
`r = k_(1)k_(eq)[N_(2)O_(5)] = k[N_(2)O_(5)]` where `k = k_(1)k_(eq)`
This shows that the mechanism is conisstent with the observed rate law.