Problem 63 - Chemistry: An Atoms-Focused Approach

Question

Tooth enamel is actually a composite material containing both hydroxyapatite and a calcium phosphate, \(\mathrm{Ca}_{8}\left(\mathrm{HPO}_{4}\right)_{2}\left(\mathrm{PO}_{4}\right)_{4} \cdot 6 \mathrm{H}_{2} \mathrm{O}\left(K_{\mathrm{sp}}=1.1 \times 10^{-47}\right)\) a. Is this calcium mineral more or less soluble than hydroxyapatite \(\left(K_{\mathrm{sp}}=2.3 \times 10^{-59}\right) ?\) b. Calculate the solubility in moles per liter of hydroxyapatite, \(\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3}(\mathrm{OH}), K_{\mathrm{sp}}=2.3 \times 10^{-59} \mathrm{in}\) water at \(25^{\circ} \mathrm{C}\) "c. Explain why the production of weak acids by bacteria on teeth and gums increases the solubility of hydroxyapatite.

Step-by-Step Solution

Answer

How does the production of weak acids by bacteria influence the solubility of hydroxyapatite in tooth enamel? Answer: The calcium mineral is more soluble than hydroxyapatite, as it has a higher solubility constant value. The production of weak acids by bacteria increases the solubility of hydroxyapatite by reacting with hydrogen ions, causing the equilibrium to shift and resulting in tooth decay.

Step 1: a. Comparison of solubilities

To determine if the calcium mineral is more or less soluble than hydroxyapatite, we need to compare their solubility constant values (\(K_{sp}\)). Calcium mineral: \(K_{sp} = 1.1 \times 10^{-47}\) Hydroxyapatite: \(K_{sp} = 2.3 \times 10^{-59}\) If a compound has a higher \(K_{sp}\) value, then it is more soluble. In this case, the calcium mineral has a higher \(K_{sp}\) value than hydroxyapatite, so it is more soluble.

Step 2: b. Solubility calculation for hydroxyapatite

To calculate the solubility of hydroxyapatite in moles per liter at 25°C, we need to write its dissociation equation: \(\mathrm{Ca}_{5}\left(\mathrm{PO}_{4}\right)_{3} \mathrm{(OH)} \rightleftharpoons 5 \mathrm{Ca}^{2+} + 3 \mathrm{PO}_{4}^{3-} + \mathrm{OH}^{-}\) Now we can use the solubility constant expression to set up the equation: \(K_{sp} = [\mathrm{Ca}^{2+}]^5 [\mathrm{PO}_{4}^{3-}]^3 [\mathrm{OH}^-]\) The solubility constant of hydroxyapatite, \(K_{sp} = 2.3 \times 10^{-59}\). Let \(s\) be the solubility of hydroxyapatite in moles per liter: \([\mathrm{Ca}^{2+}] = 5s\) \([\mathrm{PO}_{4}^{3-}] = 3s\) \([\mathrm{OH}^-] = s\) Plugging the concentrations into the solubility constant expression: \(K_{sp} = (5s)^5 (3s)^3 (s)\) Solve for \(s\): \(2.3 \times 10^{-59} = (5s)^5 (3s)^3 (s)\) \(s \approx 1.4 \times 10^{-15}\) So, the solubility of hydroxyapatite in water at 25°C is approximately \(1.4 \times 10^{-15}\) moles per liter.

Step 3: c. Solubility of hydroxyapatite and weak acids

When weak acids are produced by bacteria on teeth and gums, they can react with the hydroxyapatite present in tooth enamel. This reaction increases the concentration of hydrogen ions (\(H^+\)) in the environment. Since hydroxyapatite is a basic compound, reacting with \(H^+\) ions will cause the equilibrium to shift to the left, increasing the dissolution of hydroxyapatite. So, the production of weak acids by bacteria increases the solubility of hydroxyapatite, resulting in tooth decay.

Key Concepts

Hydroxyapatite

Hydroxyapatite is a naturally occurring mineral that plays a crucial role in the composition of human teeth and bones. It has a chemical formula of \( \mathrm{Ca}_{5}(\mathrm{PO}_{4})_{3}(\mathrm{OH}) \). This mineral is responsible for the hardness and strength of tooth enamel.

• It is considered poorly soluble in water, as indicated by its very low solubility product constant \(K_{sp}\) of \(2.3 \times 10^{-59}\).
• This low solubility is beneficial as it provides endurance against everyday wear.

When discussing solubility, a low \(K_{sp}\) means that hydroxyapatite does not easily dissolve, which is essential for maintaining strong teeth. However, the interaction with acids can increase its solubility, leading to potential tooth decay.

Calcium Phosphate

Calcium phosphate compounds are prevalent in various biological structures. One such compound is represented by the formula \( \mathrm{Ca}_{8}(\mathrm{HPO}_{4})_{2}(\mathrm{PO}_{4})_{4} \cdot 6 \mathrm{H}_{2} \mathrm{O} \). This compound is more soluble than hydroxyapatite, thanks to its higher \(K_{sp}\) value of \(1.1 \times 10^{-47}\).

• The higher \(K_{sp}\) indicates greater solubility, making calcium phosphate more likely to dissolve in bodily fluids.
• This characteristic can be useful or detrimental, depending on the context. In the case of comparing materials' solubility, calcium phosphate is noticeably more soluble than hydroxyapatite.

The solubility disparity between these minerals highlights how changes in \(K_{sp}\) can dramatically affect physical properties and applications.

Acid-Base Equilibrium

The concept of acid-base equilibrium is essential in understanding chemical reactions involving hydroxyapatite. When weak acids are produced by bacteria, they release hydrogen ions \(H^+\), affecting the surrounding environment.

• These \(H^+\) ions can increase the solubility of hydroxyapatite by reacting with its components, leading to further dissolution.
• This process demonstrates Le Chatelier’s principle where the system shifts to counterbalance the increased \(H^+\) concentration.

In simpler terms, when acids attack the teeth, they disrupt the stable balance, causing hydroxyapatite to dissolve and leading to cavities. Understanding this equilibrium is crucial for explaining why maintaining oral hygiene is vital to prevent tooth decay.