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The reaction stoichiometry states that 1 mole of A will react with 2 moles of B to produce 1 mole of AB2. Since we need 2 moles of B to completely react with 1 mole of A, we will run out of B first (there is only one mole of B available). Whatever reactant runs out first is called the limiting reactant or limiting reagent. Thus B is the limiting reagent in the above scenerio. The reaction stops after the limiting reagent runs out. In the above case 1 mole of B reacts with 0.5 mol of A to produce 0.5 mole of AB2 with 0.5 mol of A unreacted.
The reaction is spontaneous because its Gibbs free energy is lower after the reaction than before; the entropy at the end of the process is greater than the enthalpy. The reaction takes heat from the water, so it is endothermic. University of Oxford: Spontaneous Reactions.