# Rate Law problem

## Homework Statement

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

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

## Homework Equations

$$Rate = k[A]^{m}^{n}$$

## The Attempt at a Solution

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

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

However, I don't really know where to move on from here. I tried to substitute in $$H_{2}SO_{4}$$ 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: $$Rate = k[H_{2}SO_{4}]^{2}[C_{6}H_{6}]$$

Any help is appreciated.