[Sophia7X]

Having trouble with a rate law question on a mechanism...

Step 1: N₂H₂O₂   N₂HO₂^- + H⁺ (fast equilibrium)
Step 2: N₂HO₂^-   N₂O + OH^- (slow)
Step 3: H⁺ + OH^-   H₂O (fast)

According to the answer key, it is:
rate = k[N₂H₂O₂]/[H⁺]
But I am leaning toward: rate = k[N₂H₂O₂]
Arguing with an answer key, yeah I know that's futile lol, but I'm skeptical.

I tried this two ways.

First way...
I combined steps 2 and 3. OH^- is an intermediate, right? So my combined equation is:  N₂HO₂^- + H⁺ N₂O + H₂O
Then I used the fast equilibrium to substitute in the reactants for the above equation.
Final equation: N₂H₂O₂   N₂O + H₂O

Is this acceptable?



Second way...
I made an equilibrium expression using the fast equilibrium step.
K_eq = [N₂HO₂^-][H⁺]/[N₂H₂O₂]
so [N₂HO₂^-] = K_eq [N₂H₂O₂]/[H⁺]
(slow step 2) rate =k[N₂HO₂^-]
I plugged in for [N₂HO₂^-]... Rate = k(K_eq [N₂H₂O₂]/[ H⁺])

So why is the answer rate = k[N₂H₂O₂]/[H⁺]?
Isn't k, the constant for rate laws, completely different from the equilibrium constant K_eq?
What am I missing here? Please help. Sorry if this is a little on the longer side

[UG]

Second way...
I made an equilibrium expression using the fast equilibrium step.
K_eq = [N₂HO₂^-][H⁺]/[N₂H₂O₂]
so [N₂HO₂^-] = K_eq [N₂H₂O₂]/[H⁺]
(slow step 2) rate =k[N₂HO₂^-]
I plugged in for [N₂HO₂^-]... Rate = k(K_eq [N₂H₂O₂]/[ H⁺])

So why is the answer rate = k[N₂H₂O₂]/[H⁺]?
Isn't k, the constant for rate laws, completely different from the equilibrium constant K_eq?
What am I missing here? Please help. Sorry if this is a little on the longer side

This is the way I would have gone about answering this problem. Often in these type of problems, they just make the value 'k' equal to 'k x K_eq' ie, they substituted kK_eq with k into the equation rate = k(K_eq [N₂H₂O₂]/[ H⁺]). Although they may generally call it k' or something other than k to show that it has been 'modified'

[vladi307]

Combine the reactions 1 and 2 but you must have on your mind that the total reaction  speed of N2HO2- is zero (from intermediate complex theory). By this way, we have:
k1[N2H2O2]-k-1[N2HO2-][H+]-k2[N2HO2]=0. Then, we found the [N2HO2-] equal to:
[N2HO2-]=k1[N2H2O2]/k-1[H+]+k2. Step 2 is the slow step, so k2<<k-1[H+]. Then:
[N2HO2-]=k1[N2H2O2]/k-1[H+]=K[N2H2O2]/[H+], where K=k1/k-1 (equilibrium constant). The rate law will include the slower step from the mechanism. So:
rate=k2[N2HO2-]=Kk2[N2H2O2]/[H+]=k'[N2H2O2]/[H+]

[Sophia7X]

Thanks to both of you. Cleared up some confusion.