Solve Rate Law Problem: H2SO4 + C6H6 → C6H5SO3H + H2O

[Precursor]

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

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

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.

 

[nate808]

Hello! It looks like you're on the right track. The overall rate law for this reaction can be determined by considering the slowest step, which in this case is the second step. This step involves both SO3 and C6H6, so the rate law would be rate = k[SO3][C6H6]. However, since the first step produces SO3, we can substitute in the rate of the first step for [SO3], giving us rate = k(H2SO4)^2[C6H6]. This makes sense because the first step is fast and reversible, so the concentration of SO3 will be proportional to the concentration of H2SO4 squared. I hope this helps!

 

[Blueshift5]

Thank you.

I would suggest starting by writing out the rate law for each step of the mechanism. Then, using the rate-determining step (the slowest step) as a basis, write out the overall rate law by substituting in the concentrations of the reactants and products from the other steps.

In this case, the rate-determining step is step 2, which involves the reactants SO3 and C6H6. So the rate law for this step would be Rate = k[SO3][C6H6].

Now, looking at the other steps, we can see that step 1 and step 3 both involve the reactant H2SO4, and step 4 involves the product C6H5SO3H. We can use these to substitute into the rate law for step 2.

Starting with step 1, we know that the concentration of H2SO4 is equal to the concentration of H3O+ because the reaction is a strong acid. So we can substitute [H2SO4] for [H3O+] in the rate law for step 2.

Next, looking at step 3, we can see that the concentration of H2SO4 is equal to the concentration of HSO4^- because the reaction is a strong acid. So we can substitute [H2SO4] for [HSO4^-] in the rate law for step 2.

Finally, looking at step 4, we can see that the concentration of C6H5SO3H is equal to the concentration of C6H5SO3^- because the reaction is also a strong acid. So we can substitute [C6H5SO3^-] for [C6H5SO3H] in the rate law for step 2.

Putting all of this together, we get the overall rate law: Rate = k[H2SO4]^2[C6H6] as shown in the given answer.