Problem 61 - Chemistry The Science in Context

Question

Tooth enamel contains the mineral hydroxyapatite. Hydroxyapatite reacts with fluoride ion in toothpaste to form fluorapatite. The equilibrium constant for the reaction between hydroxyapatite and fluoride ion is \(K=8.48 .\) Write the equilibrium constant expression for the following reaction. In which direction does the equilibrium lie? $$\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3}(\mathrm{OH})(s)+\mathrm{F}^{-}(a q) \rightleftharpoons \mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3}(\mathrm{F})(s)+\mathrm{OH}^{-}(a q)$$

Step-by-Step Solution

Answer

Question: Write the equilibrium constant expression for the reaction between hydroxyapatite and fluoride ion, and determine whether the equilibrium lies to the left or right based on the given equilibrium constant, \(K = 8.48\). Answer: The equilibrium constant expression is \(K=\frac{[\mathrm{OH}^{-}]}{[\mathrm{F}^{-}]}\), and the equilibrium lies to the right, favoring the formation of fluorapatite and hydroxide ion.

Step 1: Write the balanced reaction

The balanced reaction is given in the exercise: $$\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3}(\mathrm{OH})(s)+\mathrm{F}^{-}(a q) \rightleftharpoons \mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3}(\mathrm{F})(s)+\mathrm{OH}^{-}(a q)$$

Step 2: Write the equilibrium constant expression

The equilibrium constant expression can be written as: $$K=\frac{[\mathrm{OH}^{-}]}{[\mathrm{F}^{-}]}$$ where \([\mathrm{OH}^{-}]\) and \([\mathrm{F}^{-}]\) represent the equilibrium concentrations of the hydroxide and fluoride ions, respectively.

Step 3: Determine the direction of the equilibrium

Since the value of \(K = 8.48\), this means that the concentration of OH- is larger than the concentration of F- at equilibrium. Therefore, the equilibrium favors the products (right side) of the reaction. In conclusion, the equilibrium constant expression for this reaction is \(K=\frac{[\mathrm{OH}^{-}]}{[\mathrm{F}^{-}]}\), and the equilibrium lies to the right, favoring the formation of fluorapatite and hydroxide ion.

Key Concepts

Equilibrium Constant

The equilibrium constant, denoted by the symbol \(K\), is a crucial concept in chemical reactions that reach a state where the rate of the forward reaction equals the rate of the reverse reaction. It provides the ratio of the concentrations of products to reactants for a given reaction at equilibrium at a specific temperature.
The equilibrium constant expression depends on the balanced chemical equation of a reaction. For the formation of fluorapatite from hydroxyapatite, the equilibrium constant \(K\) is expressed as:\[K = \frac{[\mathrm{OH}^{-}]}{[\mathrm{F}^{-}]}\]The concentration of the solids like hydroxyapatite and fluorapatite are not included in the equilibrium expression because their concentrations remain constant. Instead, only aqueous species such as \(\mathrm{F}^{-}\) and \(\mathrm{OH}^{-}\) are considered.
In this specific equation, if \(K > 1\), the equilibrium position is shifted towards the products, indicating a larger concentration of products compared to reactants. Thus, here with \(K = 8.48\), it is evident that fluorapatite and hydroxide ions are favored at equilibrium.

Le Châtelier's Principle

Le Châtelier's Principle is a fundamental concept used to predict the effect of stress on a system at equilibrium. According to this principle, if a dynamic equilibrium is disrupted by changing the conditions, the position of equilibrium shifts to counteract the change and restore a new equilibrium state.
In the reaction of hydroxyapatite with fluoride ions to form fluorapatite, varying conditions like concentration, temperature, or pressure can shift the equilibrium. For example:

• Increasing the concentration of fluoride ions (\([\mathrm{F}^{-}]\)) shifts the equilibrium towards the products, increasing fluorapatite formation.


• Similarly, if hydroxide ions (\([\mathrm{OH}^{-}]\)) are removed, the equilibrium shifts right to produce more \(\mathrm{OH}^{-}\).

Understanding Le Châtelier’s Principle helps in controlling industrial and laboratory chemical reactions by optimizing conditions to favor desired products.

Hydroxyapatite and Fluorapatite

Hydroxyapatite and fluorapatite are two minerals closely related to dental health. Hydroxyapatite, with the formula \(\mathrm{Ca}_{5}(\mathrm{PO}_{4})_{3}(\mathrm{OH})\) is a natural component of human teeth enamel, providing hardness and structural integrity.
Fluorapatite, formed when fluoride ions react with hydroxyapatite, is expressed as \(\mathrm{Ca}_{5}(\mathrm{PO}_{4})_{3}(\mathrm{F})\). It is more resistant to acid attacks, which helps in reducing the risk of cavities. Fluoride in toothpaste promotes fluorapatite production, making teeth more resistant to decay.
Reaction to form fluorapatite:\[\mathrm{Ca}_{5}(\mathrm{PO}_{4})_{3}(\mathrm{OH})(s) + \mathrm{F}^{-}(aq) \rightleftharpoons \mathrm{Ca}_{5}(\mathrm{PO}_{4})_{3}(\mathrm{F})(s) + \mathrm{OH}^{-}(aq)\]This reaction highlights the balance between these two mineral forms and underscores why fluoride treatment is a central practice in dentistry to maintain tooth integrity and health.