Third Law of Thermodynamics - Online Tutor, Practice Problems & Exam Prep

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

Now the third law of thermodynamics states that the entropy of a perfect crystal is 0 at absolute 0. Now, absolute zero is 0 Kelvin. Here we're going to say a perfect crystal is just a solid with a regular and ideal internal atomic arrangement. So here we have a perfect crystal where all of its components perfectly aligned with one another. And we're going to say here this happens again at a temperature of 0 Kelvin.

It's frozen perfectly in place because the temperature is as low as it can get. There's no such thing as negative Kelvin, so this is the bottom of the temperature scale, and because it's frozen in place, it can't move around. So there's only going to be one type of arrangement. It can have the one pictured. So here we're going to say it has one micro state.

Now if the temperature happens to be above 0 Kelvin, then the particles are not frozen perfectly in place. They're going to wiggle around a little bit, move around a little bit. Here we have this one in red, could maybe move around a little bit and now it's over here. And again this happens when the temperature is above 0 Kelvin because it can arrange itself in more than one way. You say it has more than one micro state.

Now, micro states are just the number of possible energetic ways to arrange components, whether they be atoms, molecules or ions of a system which represents our chemical reaction. Now the third law, thermodynamics, is highly theoretical. We can't achieve absolute 0 here on Earth, and on average the universe has a temperature around 2 Kelvin. OK, but so this is saying that if we could obtain this absolute 0 value, this was what would happen. It'd be like freezing a structure perfectly in place.

The truth is, even solids around us don't stay perfectly still. If you have a powerful enough microscope and you look at it, you would see that the molecules of even your cell phone would be vibrating in place. That's because the temperature around us is higher than 0 Kelvin. So keep this in mind when we talk about microstates and the third law of thermodynamics.

Third Law of Thermodynamics Example

Now here in this example question, it says all the statements are correct except the greater number of molecular motion, the greater number of possible microstates. That's true. A perfectly ordered system has more than one micro state. Remember, if you're a perfectly ordered system, you are a perfect crystal. This happens at absolute 00 Kelvin. You're frozen perfectly in place, so you'd only have one microstate. So this statement here is false.

Any system at a temperature above 0 Kelvin has a positive change in entropy value. Yes, because above 0 you can arrange yourself in different ways. You can move around greater motion. Greater movement means will be greater change in the change of your entropy. So this is true.

A perfect crystal exhibits no molecular motion that is also true. A perfect crystal is frozen perfectly in place at 0 Kelvin, can't move around, can't rearrange itself, so it only has one type of arrangement, 1 microstate. So out of all the options, only option B is false.

The Boltzmann Equation

So the Boltzmann equation is credited to the Austrian physicist Ludwig Boltzmann where he related entropy to the number of microstates. Now entropy uses the variable S, microstates uses the variable capital W. Here the Boltzmann equation says that the entropy of my system, which is my chemical reaction, equals K times your natural log lane times your number of microstates.

Here K equals your Boltzmann constant and it is 1.38×10-23 joules per Kelvin. And again, capital W just represents the number of microstates. Here we would say that the greater number of the microstates, the greater the entropy. That's because more microstates means more freedom of motion, more movement, more arrangements, more chaos, more disorder. So this would mean an increase in our entropy.

Now a microstate equal to one would mean I would plug in one. Here W4 ln 1 is 0, which would mean at the end that my entropy is equal to 0 right? So keep this in mind, This is just a mathematical way to calculate the entropy of my system if the number of microstates are known.

Third Law of Thermodynamics Example

Now consider a system with a total of 3×1026 number of microstates. What is the entropy of such a system?

Alright, so here we utilize the Boltzmann equation. So we'd say that the entropy of my system, the changing entropy of my system equals K×ln(W). K represents my Boltzmann constant, which is 1.38×10-23 joules per Kelvin.

And then we'd say ln(3×1026). When we plug this in, this will give me 8.41×10-22 joules per Kelvin. This would represent the change in the entropy of my given system.

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

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.