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I've read that Hydrofluoric acid (HF) is extremely dangerous to touch, but what exactly makes it so toxic? It's weak acid ($K_a = 7.2 \times 10^{–4}$) and dissociates approximately 1/1000 as much as Hydrochloric acid does in water, so why is it so much more dangerous? Wouldn't a stronger acid cause more damage? What exactly happens when human skin comes into contact with Hydrofluoric acid that causes it to be so lethal?

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The confusion arises from the term weak, which has only to be interpreted in chemical terms.

Weak acid, as you say, just means that the acid does not readily dissociates, not that its effect are weak! Just to say one, HF corrodes glass, something that not even smoking HCl does.

Wikipedia has a nice summary of HF toxicity (see also the references in the article itself):

Hydrofluoric acid is a highly corrosive liquid and is a contact poison. It should be handled with extreme care, beyond that accorded to other mineral acids. Owing to its low dissociation constant, HF as a neutral lipid-soluble molecule penetrates tissue more rapidly than typical mineral acids. Because of the ability of hydrofluoric acid to penetrate tissue, poisoning can occur readily through exposure of skin or eyes, or when inhaled or swallowed. Symptoms of exposure to hydrofluoric acid may not be immediately evident. HF interferes with nerve function, meaning that burns may not initially be painful. Accidental exposures can go unnoticed, delaying treatment and increasing the extent and seriousness of the injury.
Once absorbed into blood through the skin, it reacts with blood calcium and may cause cardiac arrest. Burns with areas larger than 25 square inches (160 cm2) have the potential to cause serious systemic toxicity from interference with blood and tissue calcium levels. In the body, hydrofluoric acid reacts with the ubiquitous biologically important ions Ca2+ and Mg2+. Formation of insoluble calcium fluoride is proposed as the etiology for both precipitous fall in serum calcium and the severe pain associated with tissue toxicity.

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