Problem 53 - Chemistry: An Atoms-Focused Approach

Question

What are the likely signs of \(\Delta S\) and \(\Delta G\) for the dissolution of tooth enamel?

Step-by-Step Solution

Answer

Answer: Yes, the dissolution of tooth enamel is a spontaneous process under these conditions, as evidenced by a positive change in entropy (ΔS) and a negative Gibbs free energy (ΔG).

Step 1: Identifying the reaction and its components

Tooth enamel is composed mainly of hydroxyapatite, which can be represented by the following chemical formula: $$Ca_{10}(PO_4)_6(OH)_2$$ The dissolution of tooth enamel takes place in the presence of acids (H+ ions). The general reaction can be represented as: $$10Ca_{10}(PO_4)_6(OH)_2 + 50H^+ \rightarrow 50Ca^{2+} + 30HPO_4^{2-} + 20H_2O$$ Now that we have the reaction, we can analyze the changes in entropy (ΔS) and Gibbs free energy (ΔG).

Step 2: Determine the sign of ΔS

Entropy (S) is a measure of disorder or randomness in a system. When a solid dissolves in a liquid, its particles become more dispersed, leading to an increase in disorder. Therefore, the dissolving process of tooth enamel will lead to an increase in entropy. Since the entropy is increasing, the change in entropy (ΔS) will be positive: $$\Delta S > 0$$

Step 3: Analyzing Gibbs free energy (ΔG)

Gibbs free energy (G) determines whether a reaction is spontaneous or not. The relationship between Gibbs free energy (ΔG), enthalpy (ΔH), and entropy (ΔS) is given by: $$\Delta G = \Delta H - T\Delta S$$ In the case of tooth enamel dissolution, as the reaction proceeds, the bonds in hydroxyapatite break, and new bonds are formed between the resulting ions and the surrounding water molecules. The process of bond breaking generally requires energy, and bond formation releases energy. Since the dissolution of tooth enamel primarily occurs due to the presence of acids, the breakdown of hydroxyapatite is likely to be an energetically favorable process (exothermic). This means that the change in enthalpy (ΔH) is likely to be negative: $$\Delta H < 0$$ Now we need to combine the signs of ΔS and ΔH to determine the sign of ΔG. If ΔH is negative and ΔS is positive, as it is in this case, the sign of ΔG depends on the temperature (T). Dissolving tooth enamel often occurs at or near room temperature (around 298 K). At this temperature, TΔS > ΔH. Therefore, we can conclude that ΔG is negative: $$\Delta G < 0$$

Step 4: Conclusion

For the dissolution of tooth enamel, the change in entropy (ΔS) is positive, and Gibbs free energy (ΔG) is negative. This means the reaction is spontaneous under the conditions of tooth enamel dissolution (presence of acids and room temperature).