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When two isotopes interact, a phenomenon known as isotope exchange can occur. In the context of our exercise, we're dealing with hydrogen isotopes in heavy water (D₂O) and ordinary water (H₂O). Heavy water contains deuterium, which is a heavier isotope of hydrogen, denoted as ²H. When these isotopes come in contact, an exchange happens between hydrogen atoms. For the reaction \( \mathrm{H}_2\mathrm{O} + \mathrm{D}_2\mathrm{O} \rightleftharpoons 2 \mathrm{HDO} \), the isotopes are exchanging places to form a new compound, HDO.

• Hâ‚‚O splits into two hydrogen atoms and an oxygen atom.
• Similarly, Dâ‚‚O splits into two deuterium atoms and an oxygen atom.
• These atoms then recombine to form the mixed isotope product, HDO.

This reaction is dynamic, meaning it can go forward or backward until it reaches a point where the rate of exchange in both directions is balanced, known as equilibrium.

In chemical reactions, like our isotope exchange, the equilibrium constant, \(K_c\), is crucial. The equilibrium constant tells us how much of each substance is present at equilibrium. It's a ratio of the concentration of the products to the reactants raised to the power of their coefficients in the balanced reaction equation. For the reaction \( \mathrm{H}_{2} \mathrm{O} + \mathrm{D}_{2} \mathrm{O} \rightleftharpoons 2 \mathrm{HDO} \), the expression for \( K_c \) is:\[K_c = \frac{[\mathrm{HDO}]^2}{[\mathrm{H}_{2} \mathrm{O}][\mathrm{D}_{2} \mathrm{O}]}\]Here, \([\mathrm{HDO}]\) is the concentration of the product and \([\mathrm{H}_{2} \mathrm{O}]\) and \([\mathrm{D}_{2} \mathrm{O}]\) are the concentrations of the reactants at equilibrium. A higher \( K_c \) value, like 3.86 in our example, suggests that the reaction favors the products (HDO in this case). However, if \( K_c \) were less than 1, the reactants would dominate. Equilibrium is not about equal amounts but rather a point of balance in terms of concentration.

This exercise also relates to nuclear chemistry due to the involvement of isotopes. In nuclear chemistry, isotopes of an element can have vastly different properties. For instance, deuterium (²H) is a stable isotope of hydrogen with one neutron, distinguishing it from the common hydrogen isotope, which has no neutrons. Heavy water \(\mathrm{D}_{2} \mathrm{O}\) is particularly important in nuclear reactors because:

• It acts as a neutron moderator, slowing down neutrons to sustain a controlled nuclear chain reaction.
• Due to its increased mass, deuterium is less likely to absorb neutrons compared to ordinary hydrogen.

Understanding isotopes and their roles in reactions like these is central to exploring nuclear chemistry applications. The ability of heavy water to facilitate nuclear reactions without becoming radioactive itself makes it invaluable in this field.