Would you expect the entropy of an alloy (a solution of two metals) to be zero at \(0 \mathrm{~K}\) ? Explain your answer.

The third law of thermodynamics is a fundamental principle which states that as the temperature of a crystalline solid approaches absolute zero, its entropy also approaches zero. This is based on the presumption that at 0 Kelvin (the lowest possible temperature), the particles in a perfect crystalline substance are arranged in a perfectly ordered state with no randomness or freedom of movement. Hence, entropy, which quantifies disorder, would be zero because there is only one way to arrange the particles in a state of perfect order.

However, it's important to note that this law applies to ideal cases. In reality, materials may have imperfections or be composed of different atoms, as in the case of alloys, which impacts their entropy even at absolute zero. For students tackling related exercises, understanding this principle is crucial to determine the behavior of substances in extreme cold conditions.

Entropy in crystalline substances is a measure of the degree of randomness or disorder within their atomic or molecular structure. A perfect crystal at 0 Kelvin has an entropy of zero because its atoms are arranged in a perfectly ordered lattice structure. This arrangement leaves no ambiguity in the positioning of the particles, hence no disorder.

When working with entropy concepts, notably in homework exercises, it is essential to differentiate between ideal and real systems. Textbook solutions often consider perfect crystals to illustrate the concept of entropy at absolute zero, but students should remember that actual substances might not conform to this idealized model due to imperfections or thermal vibrations that persist even at very low temperatures.

In the context of alloys, which are mixtures of two or more elements, we see a deviation from the behavior of perfect crystalline substances because of the presence of multiple types of atoms. Residual entropy is the entropy that remains in a system at absolute zero due to the mixture's inherent disorder. Unlike pure elements, alloys can have various configurations due to the different sizes and arrangements of the atoms involved.

This results in a non-zero entropy at 0 Kelvin, termed 'residual entropy'. The steps provided in the solution specifically address this concept, explaining why we would not expect the entropy of an alloy to be zero at absolute zero. When studying or teaching this aspect in exercises or homework, it's beneficial to underscore the pertinence of atomic arrangements in mixed substances like alloys that contribute to residual entropy even at the theoretical limit of zero Kelvin.