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Consider the SN1 reaction shown below:

https://imgur.com/a/EuMnG

Would different products be formed if $\ce{CH3SH}$ was an additional reagent?

https://imgur.com/a/BCBM9

Superimpose both energy diagrams on top of each other.

For the first question, yes, two products would be formed. I'm confused about the second question though. I know The rate determining step would remain unchanged, as the rate depends on the concentration of the substrate not the nucleophile (SN1 is unimolecular). However, methanethiol is a better nucleophile than methanol. Would the activation energy for the second transition state be less for methanethiol?

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Obviously saying "the rate is same but the activation barrier is different" is a contradiction, as both are directly related via the arrhenius equation. The rate is predominantly determined by the rate determining step. If there are several rate determining steps (aka several steps with similar activation energies), the overall rate will be a product of all steps.

So what does that mean for your question?

You stated that methanethiol is a better nucleophile than methanol. This would suggest that the activation barrier in the last step is slightly lower in the case of methanethiol. However, the rate determining first step is unaffected. Hence, the overall rate will be only slighter smaller for methanol.

The ratio of both products will be determined by the last step.

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