Rate Law problem

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Homework Statement


Sulfonation of benzene has the following mechanism:
(1) [tex]2H_{2}SO_{4} \rightarrow H_{3}O^{+} + HSO_{4}^{-} + SO_{3} [/tex] [fast]
(2) [tex]SO_{3} + C_{6}H_{6} \rightarrow H(C_{6}H_{5}^{+})SO_{3}^{-}[/tex] [slow]
(3) [tex] H(C_{6}H_{5}^{+})SO_{3}^{-} + HSO_{4}^{-} \rightarrow C_{6}H_{5}SO_{3}^{-} + H_{2}SO_{4}[/tex] [fast]
(4) [tex] C_{6}H_{5}SO_{3}^{-} + H_{3}O^{+} \rightarrow C_{6}H_{5}SO_{3}H + H_{2}O[/tex] [fast]

Write the overall rate law for the initial rate of reaction.

Homework Equations


[tex]Rate = k[A]^{m}^{n}[/tex]


The Attempt at a Solution


First, I found the overall rate of reaction to be: [tex]H_{2}SO_{4} + C_{6}H_{6} \rightarrow C_{6}H_{5}SO_{3}H + H_{2}O[/tex]

For the rate determining step I got [tex]Rate = k[SO_{3}][C_{6}H_{6}][/tex]

However, I don't really know where to move on from here. I tried to substitute in [tex]H_{2}SO_{4}[/tex] so that I it can correspond to the overall equation for the reaction, but it didn't work out.

I do know that the answer is: [tex]Rate = k[H_{2}SO_{4}]^{2}[C_{6}H_{6}][/tex]

Any help is appreciated.
 

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