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No, hydrogen (gas) doesn't spontaneously dissociate into H^+^ when dissolved in water because it's covalently bonded; it remains H~2~, just like nitrogen remains N~2~ when dissolved in water. Acidity is a measure of the concentration* of H^+^, so dissolving H~2~ doesn't impact the acidity.
*actually the activity
Thanks, I never understood - what's the practical difference between pH and activity?
pH is the negative of the (base 10) logarithm of the activity of H^+^: pH = -log~10~(a~H+~)
If you mean "what's the difference between concentration and activity," activity is the "effective concentration" of a species. For ideal solutions, activity is equal to concentration. For real solutions, interactions between the components in the solution may cause a species to "act" like it is more or less concentrated.
Dilute solutions at standard conditions are close to ideal: activity is about equal to concentration. But consider a concentrated solution of a salt: the activity will tend to be lower than the concentration because the cations and anions are not completely independent as in an ideal solution, but tend to "shield" each other due to electrostatic forces.
How do we then calculate the activity of concentrated or non-ideal systems?