A type of kidney stone is a precipitate of calcium oxalate (CaC2O4, Ksp = 2.3 x 10^-9). A urine sample has a Ca2+ concentration of 2.3 x 10^-3 M and an oxalic acid (H2C2O4, Ka1 = 5.9 x 10^-2, Ka2 = 6.4 x 10^-5) concentration of 1.1 x 10^-4 M. (a) A typical pH for urine is 5.5. Will a precipitate of calcium oxalate form under these conditions? (b) A vegetarian diet results in a higher pH for urine, typically greater than 7. Would kidney stones be more or less likely to form in urine with higher pH?

Verified step by step guidance

1. Step 1: Calculate the concentration of oxalate ions (C2O4^2-) in the urine at pH 5.5. Start by determining the concentration of hydrogen ions [H+] using the pH value: [H+] = 10^(-pH).
2. Step 2: Use the second dissociation constant (Ka2) of oxalic acid to find the concentration of oxalate ions. The equilibrium expression for the second dissociation is: Ka2 = [H+][C2O4^2-]/[HC2O4^-]. Assume [HC2O4^-] is approximately equal to the initial concentration of oxalic acid, since the first dissociation is much stronger.
3. Step 3: Calculate the ion product (Q) for calcium oxalate using the concentrations of Ca^2+ and C2O4^2-: Q = [Ca^2+][C2O4^2-]. Compare this value to the solubility product constant (Ksp) of calcium oxalate.
4. Step 4: Determine if a precipitate will form by comparing Q to Ksp. If Q > Ksp, a precipitate will form; if Q < Ksp, no precipitate will form.
5. Step 5: Consider the effect of a higher pH on the formation of kidney stones. At higher pH, the concentration of oxalate ions (C2O4^2-) increases due to the increased deprotonation of oxalic acid, potentially increasing the likelihood of precipitation if the ion product exceeds Ksp.

Key Concepts

Solubility Product Constant (Ksp)

The solubility product constant (Ksp) is an equilibrium constant that applies to the solubility of ionic compounds. It quantifies the extent to which a compound can dissolve in water, represented by the product of the molar concentrations of its ions, each raised to the power of their coefficients in the balanced equation. For calcium oxalate, Ksp = 2.3 x 10^-9 indicates that at equilibrium, the product of the concentrations of Ca2+ and C2O4^2- ions must not exceed this value for the compound to remain dissolved.

pH and Acid-Base Equilibrium

pH is a measure of the acidity or basicity of a solution, with lower values indicating higher acidity. In the context of urine, the pH affects the ionization of weak acids like oxalic acid, which can dissociate into oxalate ions (C2O4^2-) that participate in the formation of calcium oxalate precipitate. The degree of ionization is influenced by the pH, and a lower pH (like 5.5) favors the formation of oxalate ions, potentially leading to precipitation if the ion concentrations exceed the Ksp.

Impact of Diet on Urine pH

Diet can significantly influence urine pH, with vegetarian diets typically resulting in more alkaline urine (pH > 7). An increase in pH can affect the solubility of certain compounds; for calcium oxalate, a higher pH may reduce the concentration of oxalate ions due to decreased ionization of oxalic acid. Consequently, this could lower the likelihood of calcium oxalate precipitation, making kidney stone formation less probable in alkaline urine compared to more acidic conditions.