The enthalpies of neutralization of a weak base \(A O H\) and a strong base \(B O H\) by \(H C l\) are \(-12250 \mathrm{cal} / \mathrm{mol}\) and \(-13000 \mathrm{cal} / \mathrm{mol}\) respectively. When one mole of \(\mathrm{HCl}\) is added to a solution containing 1 mole of \(A \mathrm{OH}\) and 1 mole of \(B O H\), the enthalpy change was \(-12500 \mathrm{cal} /\) mol. In what ratio is the acid distribution between \(A O H\) and \(B O H ?\) (a) \(2: 1\) (b) \(2: 3\) (c) \(1: 2\) (d) None of these

At its most fundamental level, chemical thermodynamics is concerned with the science of energy transfer within chemical reactions. A key aspect of this field is the study of enthalpy, which is a measure of the total heat content in a system at constant pressure. Enthalpy changes are crucial for understanding whether a reaction is exothermic (releases heat) or endothermic (absorbs heat).
An important application of these principles occurs in the study of neutralization reactions, where an acid and a base react to form water and a salt, typically accompanied by a change in enthalpy. This change, known as the enthalpy of neutralization, is a critical factor for scientists and engineers when designing chemical processes and can help predict the energetics of reactions in a range of scientific fields.

A neutralization reaction is a type of chemical reaction in which an acid and a base react together to produce a salt and water. The main feature of these reactions is the combination of H+ ions from the acid and OH- ions from the base to form H₂O. The reactions between strong acids and strong bases typically release more heat, resulting in higher negative enthalpy changes, compared to reactions involving weak acids or bases.
During neutralization, the temperature change of the solution, which is directly related to the heat evolved or absorbed, helps us quantify the enthalpy change. In the context of thermodynamics, understanding such energy changes helps students predict and control the outcomes of chemical reactions in various scientific or industrial applications.

The process of enthalpy change calculations involves determining the heat change associated with a chemical reaction. It's essential to understand that the magnitude and sign of enthalpy change tell us whether a reaction is exothermic (negative value) or endothermic (positive value). When calculating the enthalpy of neutralization, it's often based on the presumption that reactants and products are in their standard states. Additionally, it assumes that the only reaction occurring is the formation of water from the acid and base.
In the provided exercise, we are given the enthalpy changes for two different bases reacting with an acid. To find the distribution of the acid between the two bases, the enthalpy change of a mixture is treated as the weighted average of the separate reactions. Through setting up and solving an algebraic equation that represents this weighted average, the ratio of how the acid is divided between the two bases can be accurately identified, as demonstrated in the step-by-step solution.