Predict the sign of \(\Delta S\) for the following. (a) ice cream melting (b) boiling water (c) dissolving instant coffee in hot water (d) sugar, \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\), decomposing to carbon and steam

Understanding phase transitions is crucial as these transformations have significant impacts on the entropy of a system. Simply put, phase transitions are the changes from one state of matter to another, such as solid to liquid, liquid to gas, or the reverse. During phase transitions, the organization of particles is altered, directly influencing the system's entropy.

Consider the melting of ice cream or boiling water, as mentioned in the original exercise. These processes are classic examples of phase transitions, where solid becomes liquid, and liquid turns into gas, respectively. In each case, the change involves the absorption of energy, which disrupts the orderly arrangement of particles, allowing them more freedom of motion. This increase in particle liberty and randomness is directly associated with an increase in entropy, hence predicting a positive sign for \(\Delta S\) is correct.

Entropy is fundamentally a measure of the disorder or randomness within a system. It's a concept deeply rooted in the second law of thermodynamics, which implies that in an isolated system, entropy can never decrease over time. When dealing with exercises such as dissolving coffee in water or the melting of ice cream, the commonality is the change in the system's structure towards more randomness.

For instance, as the ice cream melts, the tightly packed structure of the ice crystals becomes a more chaotic arrangement of liquid molecules. The increased number of microstates available to these molecules translates to higher disorder. A similar case occurs with the dissolution of coffee; the uniform solid coffee particles become irregularly spread in the solvent, showcasing a more disordered state. Entropy is essentially a numeric tally of this disorder, so when the randomness increases, so does the entropy.

Chemical decomposition is a process where a single compound breaks down into two or more simpler substances, which is often associated with an increase in entropy. In the sugar decomposition from the original exercise, the sugar molecule \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\) breaks down into separate carbon and steam molecules. The initial crystalline structure is highly ordered, and upon decomposition, the products are spread out and disordered.

The decomposition process is a chemical reaction that directly converts a structured substance into less ordered products. It's a prime example of how chemical changes can trigger significant changes in entropy. The predictably positive \(\Delta S\) value arises because the system moves from a state of low entropy (well-organized sugar molecules) to a state of high entropy (dispersed carbon atoms and steam), which matches with our observation that systems tend to shift towards higher levels of disorder— a principle that is at the heart of understanding entropy in chemical processes.