Naive question on the SN1 mechanism

I was teaching the $$S_N1$$ mechanism to a friend. He asked a question a subtle question I can't shake off. Take a tertiary alkyl halide and aqueous sodium hydroxide. The first step is the formation of the carbocation. My friend asked how did the alkyl halide just do that all of a sudden. What about the presence of the $$\ce{NaOH}$$ made this compound do this if it doesn't even interact with it ( in the first step)? I first thought maybe it was just that the first step is something that is a feasible process, and that the alkyl halide would do (to an appreciable extent) that even without the $$\ce{NaOH}$$, but I am not sure.

Is my idea correct?

• The solvation of the halide part led to energy being released which compensated for the bond being broken(which apparently seems "spontaneous") – Yusuf Hasan Mar 31 at 9:33
• So this would have happened even without the NaOH present? – Sal_99 Mar 31 at 12:25
• Heard of the term solvolysis if not look it up – Aditya Garg Mar 31 at 13:33