Question

What is the molar entropy of a pure crystal at \(0 \mathrm{~K} ?\) What is the significance of the answer to this question?

Short answer

The molar entropy of a pure crystal at absolute zero (\(0 \text{ K}\)) is zero joules per kelvin mole (\(0 \text{ J/K·mol}\)). The significance is that it establishes a reference point for calculating absolute entropies at other temperatures.

Step-by-step solution

Understanding the Third Law of Thermodynamics

The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero (\(0 \text{ K}\)) is exactly zero. This is because at absolute zero, a perfect crystal has only one microstate, with no disorder.

Identifying the Entropy at Absolute Zero

According to the Third Law, since the entropy, denoted as S, is a measure of the number of possible microstates, and a perfect crystal has only one microstate at absolute zero, the molar entropy is \(S = 0 \text{ J/K·mol}\) at this temperature.

Significance of the Molar Entropy at Absolute Zero

The significance of a molar entropy of zero at absolute zero is that it provides a reference point for calculating absolute entropies of substances at higher temperatures and understanding the behavior of entropy as a function of temperature.

Key concepts

Understanding Molar Entropy

Molar entropy, denoted as 'S', is a thermodynamic property that measures the disorder or randomness within a mole of substance. It's like a snapshot that tells us how much energy has dispersed or spread out at a particular temperature. When we talk about molar entropy, we're usually referring to how the atoms or molecules of a substance are scrambled up.

According to statistical thermodynamics, entropy is related to the number of ways, called microstates, that atoms or molecules can be arranged without changing the overall state of the substance. More microstates mean more disorder and, consequently, higher entropy. You can think of molar entropy as a way to count how many possible configurations one mole of particles can take on.

At temperatures above absolute zero, substances have energy to jiggle around and there can be many arrangements of the particles, leading to higher entropy. As the temperature of a substance approaches absolute zero, its molar entropy typically decreases because there is less energy available for particles to move and, therefore, fewer microstates.

Absolute Zero And Its Significance

Absolute zero is a term that seems straight out of science fiction, but it's really a fundamental concept in physics. It represents the lowest possible temperature where molecular motion comes to a complete stop. Measured in Kelvin, absolute zero is exactly 0 K or -273.15°C.

In practice, reaching absolute zero is impossible due to the Third Law of Thermodynamics. However, the concept of absolute zero is pivotal for scientists as it provides a base for the Kelvin temperature scale and helps in understanding the behavior of particles at low temperatures.

Understanding the behavior of substances at temperatures near absolute zero helps scientists to develop new materials with unique properties, such as superconductors, which have no electrical resistance. Moreover, studying the quantum mechanical properties of particles close to absolute zero can lead to advancements in quantum computing and other technologies.

Entropy Of A Perfect Crystal At Absolute Zero

When we picture a crystal, we often think of beautifully ordered structures with particles neatly arranged in a fixed, repeating pattern. A 'perfect crystal' is just a simplified way to think about a substance where all particles are identical and perfectly aligned with absolute order.

Now, the Third Law of Thermodynamics reveals a profound truth about these perfect crystals. It states that the entropy of a perfect crystal is zero when it is at absolute zero temperature. Why so? Because at this extreme, the particles within the crystal have only one way to be arranged, meaning there is only one microstate. No randomness or disorder exists when there's just one possible state, thus the entropy, which measures disorder, is zero.

This concept isn't just a quirky fact; it's the reference point for all entropy values. All entropy measurements are made in relation to this fixed point of zero for a perfect crystal at absolute zero. It helps scientists understand the theoretical minimum entropy that substances can have and assists in calculations related to changes in entropy as substances warm up, melt, or undergo chemical reactions.