Question

Gout-a condition that results in joint swelling and pain-is caused by the formation of sodium urate \(\left(\mathrm{NaC}_{5} \mathrm{H}_{3} \mathrm{~N}_{4} \mathrm{O}_{3}\right)\) crystals within tendons, cartilage, and ligaments. Sodium urate precipitates out of blood plasma when uric acid levels become abnormally high. This sometimes happens as a result of eating too many rich foods and consuming too much alcohol, which is why gout is sometimes referred to as the "disease of kings." If the sodium concentration in blood plasma is \(0.140 \mathrm{M},\) and \(K_{\mathrm{sp}}\) for sodium urate is \(5.76 \times 10^{-8}\), what minimum concentration of urate would result in precipitation?

Short answer

The minimum concentration of urate ions (\text{Urate}^-) required for precipitation is \(4.11 \times 10^{-7}\mathrm{M}\).

Step-by-step solution

Understanding the Question

Determine what the question is asking. Here, you're tasked with calculating the minimum concentration of urate that will cause precipitation to occur in blood plasma, given the sodium concentration and the solubility product constant, Ksp, for sodium urate.

Write the Precipitation Reaction

Set up the equation for the dissociation of sodium urate into its ions: \[\mathrm{NaC}_{5}\mathrm{H}_{3}\mathrm{N}_{4}\mathrm{O}_{3}(s) \rightleftharpoons \mathrm{Na}^+(aq) + \mathrm{C}_{5}\mathrm{H}_{3}\mathrm{N}_{4}\mathrm{O}_{3}^-(aq) \]

Express Ksp in Terms of Ion Concentrations

Use the expression for Ksp to relate the concentrations of the ions: \[ K_{\text{sp}} = [\mathrm{Na}^+][\mathrm{C}_{5}\mathrm{H}_{3}\mathrm{N}_{4}\mathrm{O}_{3}^-] \]

Plug in Given Concentrations

Substitute the given sodium concentration and the Ksp value into the Ksp expression to solve for the urate ion concentration: \[ 5.76 \times 10^{-8} = (0.140 \mathrm{M})[\mathrm{C}_{5}\mathrm{H}_{3}\mathrm{N}_{4}\mathrm{O}_{3}^-] \]

Solve for Urate Ion Concentration

Rearrange the Ksp expression to isolate and solve for the urate ion concentration: \[ [\mathrm{C}_{5}\mathrm{H}_{3}\mathrm{N}_{4}\mathrm{O}_{3}^-] = \frac{5.76 \times 10^{-8}}{0.140 \mathrm{M}} \]

Calculate the Minimum Urate Concentration

Perform the division to find the minimum concentration of urate ions that will cause precipitation: \[ [\mathrm{C}_{5}\mathrm{H}_{3}\mathrm{N}_{4}\mathrm{O}_{3}^-] = \frac{5.76 \times 10^{-8}}{0.140} = 4.11 \times 10^{-7}\mathrm{M} \]This is the minimum concentration of urate that must be present in the blood plasma for sodium urate to start precipitating.

Key concepts

Solubility Product Constant (Ksp)

The solubility product constant, or Ksp, is a fundamental concept in chemistry, particularly in understanding the solubility and precipitation of compounds in solution. Ksp is defined as the mathematical product of the molar concentrations of the ions of a sparingly soluble compound, each raised to the power of its coefficient in the balanced equation for the dissociation of the compound into its ions. A simpler way to think about Ksp is as a measure of how much a compound can dissolve in water before it starts to form a precipitate.

For example, in the case of sodium urate, which is relevant to gout, the equation used to represent its dissociation in water is: \[\begin{equation}\text{NaC}_5\text{H}_3\text{N}_4\text{O}_3(s) \rightleftharpoons \text{Na}^+(aq) + \text{C}_5\text{H}_3\text{N}_4\text{O}_3^-(aq)\end{equation}\]The corresponding Ksp expression would be: \[ K_{\text{sp}} = [\text{Na}^+][\text{C}_5\text{H}_3\text{N}_4\text{O}_3^-] \]The value of Ksp indicates the point at which a solution becomes saturated and excess solute will begin to form a solid, or a precipitate. Very low Ksp values indicate compounds that are only slightly soluble, such as sodium urate.

Urate Concentration in Gout

Gout is a painful condition associated with an increase in urate concentration in the blood. Uric acid is a normal waste product formed from the breakdown of purines, which are found in many foods. However, when uric acid levels become abnormally high, it can lead to the formation of urate crystals in the joints, causing inflammation and pain. This high level of uric acid in the blood is also known as hyperuricemia.

The human body attempts to manage high uric acid levels by dissolving it in the blood, after which the kidneys can excrete it. But when the body produces too much uric acid or the kidneys excrete too little, the equilibrium is disturbed. If the concentration of urate exceeds the solubility product, it will no longer remain in solution and will start to precipitate out as sodium urate crystals, triggering gout. Dietary habits, like consumption of rich foods and alcohol, can exacerbate this problem, leading to the title of 'disease of kings' often attributed to gout.

Precipitation Reaction

A precipitation reaction is a type of chemical reaction where a solid, known as a precipitate, forms and separates from a solution. It occurs when the product of the ion concentrations in a solution exceeds the Ksp of the compound they form upon combining. In such cases, the solution is supersaturated, and the excess solute will precipitate.

In our specific context, the precipitation of sodium urate occurs when the product of sodium ion (\( \text{Na}^+ \)) and urate ion (\( \text{C}_5\text{H}_3\text{N}_4\text{O}_3^- \)) concentrations surpasses the Ksp value for sodium urate (\( 5.76 \times 10^{-8} \)). At this point, the urate ions begin to form solid crystals, which can eventually lead to gout if they deposit in joints. Understanding the concept of precipitation reactions allows us to comprehend how and why these painful crystals form and potentially how to prevent their formation by controlling the concentration levels of reactive ions in the body.