Question

Why might a reaction with a very negative gibbs free energy not happen, such as hydrogen reacting with oxygen? And what is the concentration of a reactant after 3 half-lives, if the reaction in the question is first order

Answer 1

Because gibbs free energy is not the only thing affecting a reaction. There is kinetics and thermodynamics. Thermodynamics tells us that the reaction is posible, like this case. But kinetics tells us about the speed of that reaction. For example the reaction of carbon to become diamond is spontaneous, i mean thermodynamics allow the reaction to occur, but the kinetic is really really slow, so we see the carbon as carbon, but eventually it will become diamond (it will take a lot of time). That is why a reaction can have a negative DG but is "not happening".

About half life:

At first half life the compound will decrease 50% of the initial amount. So, if initial amount is 100%, at the end of the first half life we have 50%. The second half life consume the half of the 50%, so, we will have 25% of the original amount, and finally at the third half life we will have 12,5% of the original amount.