According to Solomons & Fryhle 10$^{th}$ edition Ch 11.8,
Because the chloride ion is a weaker nucleophile than bromide or iodide ions, hydrogen chloride doesn't react with primary/secondary alcohols unless ZnCl$_2$ or a similar lewis acid is added to the mixture as well.
However, we also know that symmetric ethers can be formed by an $S_N2$ reaction between two alcohols, such as the synthesis of diethyl ether from two molecules of ethanol under dilute acidic conditions, despite the fact that alcohols are generally bad nucleophiles.
With regards to the Mayr-Patz equation, ethanol has $N = 6.68, s_N=0.85$ and chloride has $N = 13.00, s_N = 0.6$ (both in 80% Ethanol, 20% water albeit chloride's $s_N$ is estimated). Unfortunately, I don't know where to find the E value for a protonated ethanol.
Given this information, I'm now wondering if ethanol is necessarily always a worse nucleophile than chloride (considering it has a higher $s_N$ value), and if it is worse, how is it possible for symmetric ether synthesis to be feasible with just acid, while $S_N2$ happens at a negigible rate between ethanol and HCl?
