Third Law of Thermodynamics: Videos & Practice Problems

The Third Law of Thermodynamics establishes that the entropy (S) of a perfect crystal is zero at absolute zero (0 Kelvin), where the crystal's atoms are immobile, resulting in only one possible microstate. Entropy, a measure of disorder, is linked to the number of microstates (W) of a system—the more microstates, the higher the entropy. Ludwig Boltzmann formulated an equation,

$S=k\ln \left(W\right)$

where k is the Boltzmann constant (1.38 x 10^-23 J/K), to calculate entropy based on microstates. This principle, while theoretical since absolute zero is unattainable, helps understand that at temperatures above 0 Kelvin, particles have more freedom to move, leading to multiple microstates and increased entropy.

Entropy of Perfect Crystals

The Third Law of Thermodynamics posits that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero, defined as 0 Kelvin. A perfect crystal is characterized by a regular and ideal internal atomic arrangement, where all components are perfectly aligned. At 0 Kelvin, the crystal is effectively frozen in place, resulting in only one possible arrangement of its particles, which is referred to as having one microstate.

As the temperature rises above 0 Kelvin, the particles within the crystal gain energy and begin to vibrate and move, leading to multiple possible arrangements of the particles. This increase in movement allows for more than one microstate, which is the term used to describe the number of distinct ways in which the components of a system can be arranged energetically. Microstates are crucial in understanding the entropy of a system, as they reflect the degree of disorder or randomness present.

It is important to note that achieving absolute zero is theoretically impossible in practice, as the average temperature of the universe is around 2 Kelvin. Consequently, even solids, which may appear stable, exhibit molecular vibrations when observed under a powerful microscope due to the higher temperatures surrounding them. This concept reinforces the relationship between temperature, microstates, and entropy as outlined in the Third Law of Thermodynamics.

Third Law of Thermodynamics Example

In thermodynamics, the concept of microstates is crucial for understanding entropy and molecular motion. A microstate refers to a specific arrangement of particles in a system. The greater the number of molecular motions, the more possible microstates can exist, leading to higher entropy. However, a perfectly ordered system, such as a perfect crystal at absolute zero (0 Kelvin), has only one microstate because the particles are fixed in place and cannot move. This means that at absolute zero, the entropy is at its minimum, and the system is in a state of perfect order.

As temperature increases above 0 Kelvin, molecular motion increases, allowing for more arrangements of particles. This results in a positive change in entropy, as the system can adopt various configurations. Therefore, any system at a temperature above absolute zero will exhibit a positive change in entropy due to the increased number of possible microstates. In summary, while a perfect crystal at 0 Kelvin has only one microstate and no molecular motion, systems at higher temperatures can experience greater entropy due to increased molecular movement and the resulting variety of microstates.

The Boltzmann Equation

The Boltzmann equation, formulated by Austrian physicist Ludwig Boltzmann, establishes a fundamental relationship between entropy and the number of microstates in a system. In this context, entropy is denoted by the variable \( S \), while the number of microstates is represented by the capital letter \( W \). The equation is expressed as:

\[ S = k \cdot \ln(W) \]

Here, \( k \) is the Boltzmann constant, valued at \( 1.38 \times 10^{-23} \, \text{J/K} \). This equation illustrates that the entropy of a system, such as a chemical reaction, increases with the number of microstates. A higher number of microstates indicates greater freedom of motion, more possible arrangements, and consequently, increased disorder or chaos within the system. Thus, as \( W \) rises, so does \( S \).

For instance, if there is only one microstate (\( W = 1 \)), substituting this value into the equation yields:

\[ S = k \cdot \ln(1) = k \cdot 0 = 0 \]

This result signifies that a system with only one microstate has zero entropy, indicating a state of complete order. Understanding this relationship provides a mathematical framework for calculating the entropy of a system when the number of microstates is known, highlighting the intrinsic link between microstates and the thermodynamic concept of entropy.

The Boltzmann Equation Example

To determine the entropy of a system with a total of \(3 \times 10^{26}\) microstates, we can apply the Boltzmann equation, which relates entropy to the number of microstates. The equation is expressed as:

\[ S = k \ln(w) \]

In this equation, \(S\) represents the entropy, \(k\) is the Boltzmann constant, and \(w\) is the number of microstates. The Boltzmann constant \(k\) is approximately \(1.38 \times 10^{-23} \, \text{J/K}\).

Substituting the values into the equation, we have:

\[ S = 1.38 \times 10^{-23} \, \text{J/K} \times \ln(3 \times 10^{26}) \]

Calculating the natural logarithm of the number of microstates:

\[ \ln(3 \times 10^{26}) \approx 60.66 \]

Now, substituting this value back into the equation for entropy:

\[ S \approx 1.38 \times 10^{-23} \, \text{J/K} \times 60.66 \approx 8.41 \times 10^{-22} \, \text{J/K} \]

This result indicates that the change in entropy of the system is approximately \(8.41 \times 10^{-22} \, \text{J/K}\). This value reflects the degree of disorder or randomness in the system, with a higher entropy corresponding to a greater number of accessible microstates.

A brand new deck of cards which hasn't been shuffled yet, possesses only one arrangement. Another, older deck has been shuffled and possesses 8 × 10⁶⁷ arrangements. Calculate and compare entropies of each deck.

• A. deck 1 = 0, deck 2 = 2.158 × 10⁻²¹ J/K
• B. deck 1 = 0, deck 2 = 9.371 × 10⁻²² J/K
• C. deck 1 = 0, deck 2 = 9.371 × 10²⁴ J/K
• D. deck 1 = 0, deck 2 = 2.158 × 10²⁵ J/K

Answer: (A)

Here’s what students ask on this topic:

Which of the following best describes the third law of thermodynamics?

The third law of thermodynamics states that as the temperature of a system approaches absolute zero, the entropy of a perfect crystal also approaches zero. Absolute zero is the lowest limit of the thermodynamic temperature scale and is equivalent to -273.15°C or -459.67°F. At this point, a system's thermal motion would be minimal, meaning that its atoms or molecules would be arranged in the most ordered possible state, resulting in the lowest entropy. In other words, the third law suggests that it's impossible to reach absolute zero in a finite number of steps because you can't remove all thermal energy from a system. This law provides an absolute reference point for the determination of entropy, as the entropy of a perfect crystal at absolute zero is typically defined as zero.

What is the third law of thermodynamics?

The third law of thermodynamics states that as the temperature of a system approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum. This is often interpreted to mean that it is impossible to reach the absolute zero temperature (0 Kelvin or -273.15°C) through a finite number of processes. In other words, as you get closer to absolute zero, the amount of thermal energy in a system decreases to the point where the entropy, or disorder, approaches a minimum value. This minimum entropy is typically taken to be zero for a perfectly ordered crystalline solid. This law has important implications in the field of cryogenics and informs our understanding of quantum mechanics, as it implies that at absolute zero, a system is in its ground state, where it has the least amount of energy possible.

How does the third law of thermodynamics allow for the determination of absolute entropies of substances?

The third law of thermodynamics, also known as the Nernst heat theorem, states that as the temperature of a perfect crystalline substance approaches absolute zero (0 Kelvin), its entropy approaches a minimum value, which is typically taken as zero. This provides a reference point for the absolute entropy of a substance.

Because entropy is a state function, we can calculate the change in entropy for a process by considering the difference between the entropy of the final state and the initial state. With the third law as a reference, we can measure the absolute entropy of a substance at any temperature above absolute zero by adding up all the incremental changes in entropy that occur as the substance is brought from absolute zero to the temperature of interest through a reversible path.

This is done by integrating the heat capacity over the temperature and incorporating any phase changes that occur along the way, since they also involve entropy changes. The absolute entropy determined in this way is a fundamental property of the substance and is useful in various thermodynamic calculations, such as predicting the spontaneity of reactions and determining the efficiency of engines.