Pseudogout, a condition with symptoms similar to those of gout (see Problem 126), is caused by the formation of calcium diphosphate (Ca₂P₂O₇) crystals within tendons, cartilage, and ligaments. Calcium diphosphate will precipitate out of blood plasma when diphosphate levels become abnormally high. If the calcium concentration in blood plasma is 9.2 mg/dL, and Ksp for calcium diphosphate is 8.64⨉10^-13, what minimum concentration of diphosphate results in precipitation?

Verified step by step guidance

1. Convert the concentration of calcium from mg/dL to moles per liter (M). Use the molar mass of calcium (40.08 g/mol) to perform this conversion.
2. Write the solubility product expression (Ksp) for calcium diphosphate, which is Ca2P2O7. The formula indicates that for every 1 formula unit that dissolves, 2 Ca^2+ ions and 1 P2O7^4- ion are produced. Thus, Ksp = [Ca^2+]^2[P2O7^4-].
3. Substitute the converted concentration of calcium into the Ksp expression. Since the stoichiometry of calcium in calcium diphosphate is 2:1, the concentration of Ca^2+ in the solution will be twice the concentration of Ca2P2O7 that dissolves.
4. Solve the Ksp expression for the concentration of P2O7^4- (diphosphate ion). Rearrange the expression to isolate [P2O7^4-] on one side of the equation.
5. The result from the previous step gives the minimum concentration of diphosphate required in the blood plasma for calcium diphosphate to begin precipitating.

Key Concepts

Solubility Product Constant (Ksp)

The solubility product constant (Ksp) is a numerical value that represents the equilibrium between a solid and its ions in a saturated solution. It is specific to a particular compound at a given temperature and is used to predict whether a precipitate will form in a solution. For calcium diphosphate, Ksp indicates the maximum product of the concentrations of its ions (Ca²⁺ and P₂O₇²⁻) that can exist in solution before precipitation occurs.

Precipitation Reaction

A precipitation reaction occurs when two soluble salts react in solution to form an insoluble solid, known as a precipitate. This process is driven by the formation of a compound that is less soluble than its constituent ions in the solution. In the case of pseudogout, the precipitation of calcium diphosphate crystals occurs when the concentrations of calcium ions and diphosphate ions exceed the Ksp value, leading to the formation of solid crystals.

Ion Concentration and Equilibrium

Ion concentration refers to the amount of a specific ion present in a solution, which can influence the solubility and precipitation of compounds. In equilibrium reactions, the concentrations of reactants and products remain constant over time. To determine the minimum concentration of diphosphate required for precipitation of calcium diphosphate, one must apply the Ksp expression, which relates the concentrations of calcium ions and diphosphate ions at equilibrium.