`2N_2O_5 rarr 4 NO_2 +O_2`
If `-(D[N_2O_5])/(dt) =k_1[N_2O_5]`
`(d[NO_2])/(dt) =k_2[N_2O_5]`
` ([O_2])/(dt) =k_3[N_2O_5]`
What is the relation between ` k_1, k_2` and ` k_3 ?`.

Options

A. `k_2=2k_1 "and" k_3 = 1/2 k_1 `
B. `k_1=2k_1 " and " k_3 =2k_1`
C. `k_1=k_2 =k_3`
D. `k_1=2k_2=3k_2`

Explanation

`underset(2 "moles") (2 N_(2)O_(5)) to underset(4 "moles") (4 NO_(2)) + underset(1 "moles")(O_(2))`
`therefore ` Rate = `(-1)/(2) (d [N_(2) O_(5)])/(dt) = (1)/(4) (d[NO_(2)])/(dt) = (d[O_(2)])/(dt)`
`therefore - (d [N_(2) O_(5)])/(dt) = (1)/(2) (d[NO_(2)])/(dt) = (2d[O_(2)])/(dt)`
`therefore k_(1) [ N_(2) O_(5)] = (k_(2))/(2) [ NO_(2)] = 2 k_(3) [ O_(2)]`
`therefore 2 k_(1) = k_(2) = 4k_(3)`