THIRD LAW OF THERMODYNAMICS: FORMULAS, EQUATIONS, EXAMPLES - PHYSICAL - 2024

The third law of thermodynamics states that the entropy of a closed thermodynamic system in equilibrium tends to be minimal and constant, as its temperature approaches 0 kelvin.

Said entropy value will be independent of the system variables (pressure or applied magnetic field, among others). What happens is that as the temperature is closer to 0 K, the processes in the system stop and since entropy is a measure of internal agitation, it necessarily falls.

Figure 1. As the temperature of a system approaches absolute zero, its entropy reaches a constant minimum value. Source: Prepared by F. Zapata..

Previous concepts

To understand the scope of the third law of thermodynamics, relevant at very low temperatures, it is necessary to review the following concepts:

Thermodynamic system

Generally refers to a gas, liquid, or solid. What is not part of the system is called an environment. The most common thermodynamic system is the ideal gas, which consists of N particles (atoms) that only interact through elastic collisions.

Isolated, closed or open systems

Isolated systems are not allowed any exchange with the environment. Closed systems do not exchange matter with the environment but they do exchange heat. Finally, open systems can exchange both matter and heat with the environment.

Macrostates and microstates

The macrostate of a system is the set of values ​​that its variables have: pressure, temperature, volume, number of moles, entropy and internal energy. On the other hand, the microstate - in the case of an ideal gas - is given by the position and momentum of each of the N particles that make it up, at a given instant.

Many microstates can result in the same macrostate. In a gas at room temperature, the number of possible microstates is huge, because the number of particles that make it up, the different positions and the different energies that they can adopt is very large.

Formulas and equations

Entropy, as we said, is a thermodynamic macroscopic variable that measures the degree of molecular disorder of the system. The degree of disorder of a system is greater as the number of possible microstates is greater.

This concept is needed to formulate the third law of thermodynamics in mathematical form. Let S be the entropy of the system, then:

Entropy is a macroscopic state variable that is directly related to the number of possible microstates of a system, through the following formula:

S = k ln (W)

In the above equation: S represents the entropy, W the number of possible microstates of the system and k is the Boltzmann constant (k = 1.38 x 10 ^-23 J / K). That is, the entropy of a system is k times the natural logarithm of the number of possible microstates.

Calculation of the absolute entropy of a substance

It is possible to define the absolute entropy of a pure substance starting from the definition of the entropy variation:

δQ = n. c _p.dT

Here cp is the molar specific heat and n the number of moles. The dependence of the molar specific heat with the temperature is a data obtained experimentally and known for many pure substances.

According to the third law on pure substances:

Applications

In everyday life, the third law of thermodynamics has few applications, quite the opposite of the first and second laws. It is because it is a principle that refers to what happens in a system when it approaches absolute 0, a rare temperature range.

In fact reaching absolute 0 or −273.15 ° C is impossible (see example 1 below) However, the third law applies when studying the response of materials at very low temperatures.

Thanks to this, important advances have emerged in the Physics of condensed matter, such as:

-Superfluidity (see example 2 below)

-Superconductivity

-Laser cooling techniques

-Bose-Einstein condensate

-The superfluid gases of Fermi.

Figure 2. Superfluid liquid helium. Source: Wikimedia Commons.

At extremely low temperatures, the decrease in entropy allows interesting quantum phenomena to emerge. So let's see what happens to the entropy of a system at very low temperature.

Entropy of a system at low temperature

When you have a perfect crystalline substance, its minimum entropy is exactly zero, since it is a highly ordered system. At temperatures close to absolute 0, matter is in a condensed state (liquid or solid) and vibrations in the crystal are minimal.

Some authors consider an alternative statement of the third law of thermodynamics the following:

"If matter condenses to form a perfect crystal, when the temperature tends to absolute zero, the entropy tends to exactly zero."

Let us clarify some aspects of the previous statement:

- A perfect crystal is one in which each molecule is identical and in which the molecular structure repeats itself identically throughout.

- As the temperature approaches absolute zero, the atomic vibration decreases almost completely.

Then the crystal forms a single possible configuration or microstate, that is, W = 1, and therefore the entropy is equal to zero:

S = k ln (1) = 0

But it is not always that a material cooled near absolute zero forms a crystal, much less this crystal is perfect. This only happens if the cooling process is very slow and reversible.

Otherwise, factors such as impurities present in the glass would make the existence of other microstates possible. Therefore W> 1 and the entropy would be greater than 0.

Residual entropy

If the cooling process is abrupt, during it the system goes through a succession of non-equilibrium states, which lead to the material becoming vitrified. In such a case, an ordered crystalline structure is not produced, but an amorphous solid, whose structure is similar to that of a liquid.

In that case, the minimum entropy value in the vicinity of absolute zero is not zero, since the number of microstates is considerably greater than 1. The difference between this entropy and the null entropy of the perfect crystalline state is known as the residual entropy.

The explanation is that below a certain threshold temperature, the system has no other option but to occupy the microstates with lower energy, which, because they are quantized, constitute a fixed number.

They will take care of keeping the entropy constant, even when the temperature continues to drop towards absolute zero.

Examples

Example 1: absolute zero and Heisenberg's indeterminacy

Heisenberg's principle of indeterminacy establishes that the uncertainty in the position and momentum of a particle, for example in the atoms of a crystal lattice, are not independent of each other, but rather follow the following inequality:

Δx ⋅ Δp ≥ h

Where h is Planck's constant. That is, the uncertainty in position multiplied by the uncertainty in momentum (mass times velocity) is greater than or equal to Planck's constant, whose value is very small, but not zero: h = 6.63 x 10 -34 J s.

And what does the uncertainty principle have to do with the third law of thermodynamics? If the position of the atoms in the crystal lattice is fixed and precise (Δx = 0) then the speed of these atoms can take any value between 0 and infinity. This is contradicted by the fact that at absolute zero, all movement of thermal agitation ceases.

Conversely, if we assume that at absolute zero temperature, all agitation ceases and the momentum of each atom in the lattice is exactly zero (Δp = 0), then the Heisenberg uncertainty principle would imply that the indeterminacy in the positions of each atom it would be infinite, that is, they can be in any position.

As a consequence of the previous statement, the number of microstates would tend to infinity and the entropy would also take an indeterminate value.

Example 2: Superfluidity and the strange case of helium-4

In superfluidity, which occurs at very low temperatures, matter loses the internal friction between its molecules, called viscosity. In such a case, the fluid could circulate without friction forever, but the problem is at those temperatures almost nothing is liquid except helium.

Helium and helium 4 (its most abundant isotope) constitute a unique case, since at atmospheric pressure and at temperatures close to absolute zero, helium remains liquid.

When helium-4 is subjected to a temperature below 2.2 K at atmospheric pressure it becomes a superfluid. This discovery occurred in 1911 in Leyden by the Dutch physicist Heike Kamerlingh Onnes (1853-1926).

Figure 3. The Dutch physicist Heike Kamerlingh Onnes (1853-1926). Source: Wikimedia Commons.

The helium-4 atom is a boson. Bosons, unlike fermions, are particles that can all occupy the same quantum state. Therefore the bosons do not fulfill the Pauli exclusion principle.

Then all helium-4 atoms at temperatures below 2.2 K occupy the same quantum state and therefore there is only one possible microstate, implying that superfluid helium-4 has S = 0.

Solved exercises

- Exercise 1

Let us consider a simple case that consists of a system made up of only three particles that have three energy levels. For this simple system:

a) Determine the number of possible microstates for three temperature ranges:

-High

-Half

-Low

b) Determine by means of Boltzmann's equation the entropy in the different temperature ranges.

c) Discuss the results and explain whether or not they contradict the third law of thermodynamics.

Solution to

On a molecular and atomic scale, the energies that a system can adopt are quantized, which means that they can only take certain discrete values. Furthermore, when temperatures are so low, the particles that make up the system only have the possibility to occupy the lowest energy levels.

High temperature

If the system has a relatively high temperature T, then the particles have enough energy to occupy any of the available levels, giving rise to 10 possible microstates, which appear in the following figure:

Figure 4. Possible states at high temperature for the resolved exercise 1. Source: Prepared by F. Zapata.

Medium temperature

In the case that the system has an intermediate temperature, then the particles that make it up do not have enough energy to occupy the highest energy level. The possible microstates are illustrated in the figure:

Figure 5. Microstates at medium temperature for the system of resolved exercise 1. Source: Prepared by F. Zapata.

Low temperature

If the temperature continues to drop in our idealized system of three particles and three energy levels, then the particles will have so little energy that they can only occupy the lowest level. In this case, only 1 possible microstate remains, as shown in Figure 6:

Figure 6. At low temperature there is a possible configuration (Own elaboration)

Solution b

Once the number of microstates in each temperature range is known, we can now use the Boltzmann equation given above to find the entropy in each case.

S = k ln (10) = 2.30 xk = 3.18 x 10 -23 J / K (High temperature)

S = k ln (4) = 1.38 xk = 1.92 x 10 -23 J / K (Average temperature)

And finally:

S = k ln (1) = 0 (Low temperature)

Solution c

First we notice that the entropy decreases as the temperature falls, as expected. But for the lowest temperature values, a threshold value is reached, from which the base state of the system is reached.

Even when the temperature is as close as possible to absolute zero, there are no lower energy states available. Then the entropy keeps its minimum value constant, which in our example is S = 0.

This exercise illustrates, at the microstate level of a system, the reason why the third law of thermodynamics holds.

- Exercise 2

Reason if the following statement is true or false:

"The entropy of a system at absolute zero temperature is exactly zero."

Justify your answer and describe some examples.

Solution

The answer is: false.

In the first place, absolute 0 of temperature cannot be reached because it would violate Heisenberg's uncertainty principle and the third law of thermodynamics.

It is very important to note that the third law does not say what happens at absolute 0, but rather when the temperature is infinitely close to absolute 0. The difference is subtle, but significant.

Nor does the third law affirm that when the temperature takes a value arbitrarily close to absolute zero, the entropy tends to zero. This would only occur in the case previously analyzed: the perfect crystal, which is an idealization.

Many systems on a microscopic scale, that is to say on a quantum scale, have their base energy level degenerate, which means the existence of various configurations at the lowest energy level.

This means that in these systems the entropy would never be exactly zero. Nor would the entropy be exactly zero in systems that vitrify when the temperature tends to absolute zero. In this case, the previously seen residual entropy remains.

It is due to the fact that their molecules were "stuck" before reaching the lowest levels of energy available, which considerably increases the number of possible microstates, making it impossible for the entropy to be exactly zero.

References

1. Cengel, Y. 2012. Thermodynamics. 7th Edition. McGraw Hill. 347.
2. Jet Propulsion Laboratory. The Coolest Spot in the Universe. Retrieved from: coldatomlab.jpl.nasa.gov.
3. González, A. Entropy and spontaneity. Recovered from: geocities.ws
4. Quora. Whats is the practical use of third law of thermodynamics ?. Recovered from: quora.com
5. General chemistry. Third principle of thermodynamics. Recovered from: corinto.pucp.edu.pe
6. Third law of thermodynamics. Recovered from: youtube.com
7. Wikipedia. Residual entropy. Recovered from: en.wikipedia.com
8. Wikipedia. Third law of thermodynamics. Recovered from: en.wikipedia.com