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The rate law for the decompoistion of ga...

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)`
Show that the mechanism is consistent with the above rate law.

Text Solution

Verified by Experts

`r = k_2 [NO_2][NO_3] …(i)`
and from the fast equilibrium step,
`k_(eq) = ([NO_2][NO_3])/([N_2O_5])`
Thus, `[NO_2][NO_3]= K [N_2O_5] … (ii)`
Using (ii) in (i), we get :
`r = k_1 K_(eq)[ N_2O_5] = k [N_2O_5]` where `k = k_1 K_(eq)`
This shows that the mechanism is consistent with the observed rate law.
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