Question

The third law of thermodynamics states that the entropy of a perfect crystal at \(0 \mathrm{K}\) is zero. In Appendix \(4, \mathrm{F}^{-}(a q), \mathrm{OH}^{-}(a q)\) and \(\mathrm{S}^{2-}(a q)\) all have negative standard entropy values. How can \(S^{\circ}\) values be less than zero?

Short answer

In summary, a negative standard entropy value for a substance (like F⁻(aq), OH⁻(aq), and S²⁻(aq)) indicates a higher degree of particle ordering in comparison to a perfect crystal. This does not violate the third law of thermodynamics; it simply reflects the relative ordering of the substance when compared to a perfect crystal at absolute zero. Negative standard entropy values are relative to a perfect crystal at 0K, and in reality, the substance still has a positive entropy value at any non-zero temperature.

Step-by-step solution

1. Understand the third law of thermodynamics

The third law of thermodynamics states that the entropy of a perfect crystal at 0K (absolute zero) is zero. Entropy is a measure of the disorder in a system, and at absolute zero, a perfect crystal has all its particles in a perfectly ordered structure, contributing to an entropy of zero.

2. Define standard entropy (S°)

Standard entropy (S°) is a measure of the entropy of one mole of a substance in its standard state at a specific temperature (usually 298 K) and pressure (1 bar). It reflects the inherent randomness or disorder of particles in the substance as well as the contribution from the configurational entropy (number of possible arrangements of particles). Higher S° values indicate greater randomness, while lower S° values indicate more ordered states.

3. Explain how standard entropy is measured

To measure the standard entropy of a substance, we typically perform a series of experiments to determine the change in entropy in response to temperature changes. The substance is heated (or cooled) from absolute zero (0K) to the desired temperature (often 298 K) under constant pressure. The experimental values of heat capacity at various temperatures are used to calculate the total change in entropy from 0K to the desired temperature.

4. Explain why some substances have negative standard entropy values

The fact that some substances have negative standard entropy values seems to contradict the third law of thermodynamics. However, it's because these substances are not perfect crystals. The standard entropy values (at 298K, for example) for these substances are derived from comparing them to a perfect crystal at absolute zero (0 K). In this context of comparing with a perfect crystal, these substances in their standard state show a higher degree of ordering compared to a perfect crystal. Hence, their calculated S° values can be less than zero. It's important to understand that the negative S° values are relative to a perfect crystal. In reality, the substance still has a positive entropy value at any non-zero temperature. The negative value is an artifact of the comparison with the idealized perfect crystal. In summary, a negative standard entropy value for a substance (like F⁻(aq), OH⁻(aq), and S²⁻(aq)) indicates a higher degree of particle ordering in comparison to a perfect crystal. This does not violate the third law of thermodynamics; it simply reflects the relative ordering of the substance when compared to a perfect crystal at absolute zero.

Key concepts

Entropy

Imagine a classroom at the end of the day: books scattered, papers strewn about, and chairs askew. This is a state of disorder, or in thermodynamics, we call it high entropy. Entropy is a key concept in understanding not just thermodynamics, but the nature of order, disorder, and energy distribution within a system. Entropy is a measure of the randomness or disorder within a system. The greater the disorder, the higher the entropy.

In any natural thermodynamic process, entropy tends to increase, leading to what's called the 'arrow of time'. This increase occurs because energy naturally spreads out and disperses unless work is done to maintain or create order. In closed systems, this leads to the second law of thermodynamics, which states that entropy never decreases over time.

Standard Entropy (S°)

Think of standard entropy, denoted as S°, as the quantum of how much disorder exists in one mole of a substance under standard conditions (usually 1 bar of pressure and 298 K temperature). It tells us how much energy per temperature unit is needed to change the disorder of a substance when we compare it to an absolute point: the perfect crystal at absolute zero.

The concept of S° allows chemists to describe and predict how changes in temperature and other conditions might affect a system's disorder. This is critical in reactions where entropy plays a role in the spontaneity of a process. Measuring standard entropy involves indirectly assessing the energy distributed among all the possible microstates of a substance.

Perfect Crystal

A perfect crystal is a hypothetical ideal where a substance’s atoms or molecules are in a perfectly ordered state, with no random motion. At absolute zero, the third law of thermodynamics tells us such a crystal has an entropy of zero, because its internal structure has only one way to exist—there's zero randomness, and thus, zero entropy.

Understanding this baseline is crucial because it establishes a reference point from which all other entropy values are compared. Entropy is relative, and a perfect crystal at absolute zero represents the ultimate ordered state. In reality, perfect crystals don't exist, but they are an ideal that helps scientists ground their calculations.

Absolute Zero

The term absolute zero refers to 0 Kelvin (K), or -273.15 degrees Celsius—the temperature at which particles theoretically stop moving. This is the coldest temperature possible, and it serves as the limit for how cold things can get.

At absolute zero, motion ceases and randomness (entropy) is at its lowest because atoms and molecules would be perfectly stationary, occupying the lowest possible energy state. Understandably, reaching absolute zero is a practical impossibility because it would require removing every last bit of heat energy from a substance. However, striving to reach or approach this limit provides invaluable insight into material behaviors at extremely low temperatures.