# The Third Law of Thermodynamics

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The Third law of Thermodynamics strives to characterize the low-temperature behavior of a system. O ne way of interpreting the meaning of the third law is:

"It is impossible to reduce the temperature of a system to absolute zero (0°Kelvin) in a finite number of reversible steps"

### The Third Law

**The Principle of Thomsen and Berthelot**

Development of this characterizing principle began in 1906 with Walther Nernst, a chemist from Germany. Nernst developed the "*New Heat Theorem*" which states:
as 0°Kelvin / absolute zero is approached, the change in entropy for a transformation of a system becomes zero. Mathematically, this is,

- [math]\displaystyle{ \lim_{T \to 0} \Delta S = 0 }[/math] [1]

Max Planck proposed a different approach, which would ultimately end up becoming a stronger statement for stating the third law of thermodynamics.His statement can be described as *absolute entropy*.

Through experimental analysis, Julius Thomsen and Marcellin Berthelot developed a self named principle, stating: chemical changes, which always produce heat, follow a path that leads to maximizing the expulsion of heat / minimizing the chemical/physical systems energy.

In a system, the heat absorbed 'Q' can be described as:

- [math]\displaystyle{ Q = (E_f+P_oV_f)-(E_i+P_oV_i) }[/math]
- [math]\displaystyle{ = H_f - H_i }[/math]
- [math]\displaystyle{ = ΔH }[/math]

This tells us that Enthalpy is equivalent to the heat absorbed 'Q'. Connecting this to the principle of Thomsen and Berthelot tells us that a system will seek a state that minimizes its enthalpy.

When pressure and temperature are held constant, we know for a system to be in equilibrium, the Gibbs free energy must be minimized. Mathematically, this can be described as:

- [math]\displaystyle{ G = E+PV-TS }[/math]
- [math]\displaystyle{ dG = d[E+PV-TS] }[/math]
- [math]\displaystyle{ dG_{PT} \lt 0 }[/math]

This differs from the principle of Thomsen and Berthelot, which says enthalpy must be minimized which is something Nernst was aware of.

Enthalpy, [math]\displaystyle{ H=E+PV }[/math] can be related to the Gibbs free energy, [math]\displaystyle{ G=E+PV-TS }[/math] by recognizing

- [math]\displaystyle{ ΔH = ΔG+TΔS }[/math]

The stability criteria for constant entropy and constant pressure is that the enthalpy be minimized:

- [math]\displaystyle{ H = E+PV }[/math]
- [math]\displaystyle{ dH = d[E+PV] }[/math]
- [math]\displaystyle{ dH_{SP} \lt 0 }[/math]

Therefore, according to [math]\displaystyle{ ΔH = ΔG+TΔS }[/math], when temperature is held constant, enthalpy and gibbs free energy are equal, [math]\displaystyle{ ΔH = ΔG }[/math]. This consideration a lone limits the principle of Thomsen and Berthelot, but when we consider the change in entropy constant as the temperature is held constant, there becomes a way to apply the principle of Thomsen and Berthelot over a variety of temperatures.

**Entropy Change**

This is the first way Nernst attempted to explain the principle of Thomsen and Berthelot.

He found that as the the change in entropy is zero as the temperature goes towards absolute zero,
*The entropy change ΔS in any reversible isothermal process approaches zero as the temperature approaches zero.*

A change in entropy would mean that as temperature is held constant, certain coefficients used in a characterizing equation of a thermodynamic system will equal zero.

The 'finite entropy hypothesis' states that when entropy is a function of volume and temperature, it will remain finite as the volume is held constant and T approaches zero.

Therefore, according to this hypothesis,

[math]\displaystyle{ S=S(V,T) \to 0 }[/math] as [math]\displaystyle{ T \to 0 }[/math]

Geometrically, what this says is that the slope of graph where entropy as a function of V and T, [math]\displaystyle{ S=S(V,T) }[/math] , when V is constant, will reach a common value of entropy, [math]\displaystyle{ S_o }[/math] when absolute zero is approached.

**Unattainability**

Nernst would go on to create a statement more refined but equivalent to his original argument of *entropy change* in order to express the third law as a form of *unattainability*.

The third law in terms of *unattainability* states:

*No reversible adiabatic process starting at a non-zero temperature can take a system to zero temperature*[2]

One example of unattainability that proves Nernst's statement can be displayed through adiabatic demagnetization.

### Adiabatic Magnetization

Adiabatic magnetization is a process which cools down magnetic objects to very low temperatures. Adiabatic makes reference too the condition that no heat is allowed to escape our system and into the environment or vice versa. To understand this process we must understand Currie's law.

Currie's law of magnetization states the magnetization of a material is equal to a constant times the ratio of an external magnetic field divided by the temperature of the object. Mathematically, we can describe this as:

[math]\displaystyle{ M = σ(\frac{B_{ext}}{T}) }[/math] where [math]\displaystyle{ σ\gt 0 }[/math]

Now before adiabatic magnetization is to be performed, we must first use isothermal magnetization. This means the magnetic field,[math]\displaystyle{ B_{ext} }[/math], is applied while temperature is held constant. Now in order to turn on the magnetic field, we must do work. Doing work relates to a positive quantity. Now because this process is isothermal, the internal energy will change, leading us to make an argument with the first law of thermodynamics:

[math]\displaystyle{ dE=dQ+dW \to dW=-dQ }[/math]

which tells us the system is ejecting heat. When we do adiabatic demagnetization, the temperature of the system winds up having to go down due to energy being a function of temperature. This means that [math]\displaystyle{ Q \to 0 }[/math] due to this process being adiabatic. Therefore in an adibatic magnetization process, [math]\displaystyle{ E=W }[/math].

Let us hypothesize an experimental graph of entropy as a function of temperature. Adiabaticc.jpeg

The slope of this graph can be regarded as the magnetic field being applied to the magnet.

So we start at a high temperature, where [math]\displaystyle{ B=0 }[/math] and isothermally magnetize, following the orange arrow on the graph and putting us on the [math]\displaystyle{ B\gt 0 }[/math] curve. This causes the entropy to go down because heat leaves the system which can be proven by the second law of thermodynamics, [math]\displaystyle{ dQ=TdS }[/math].

Next we adiabatically demagnetize the object. Due to this being and adiabatic process, entropy remains constant. This can be displayed by the blue arrows on the graph. This causes us to end back on the [math]\displaystyle{ B=0 }[/math] curve.

Now our temperature has decreased again. We can repeat this process to get to a lower temperature.

But because the two curves converge at some non-zero value of entropy,[math]\displaystyle{ S_o }[/math], the amount of entropy expelled from the system becomes less until we are not able to expel any more entropy.

The demagnetization and magnetization process are reversible and the only way to reach absolute zero is if you let the number of steps go to infinity which is physically impossible. This experimental fact proves Nernst statement of *unattainability*.

**Absolute Entropy**

The *absolute entropy* version of the third law is a combination of the *entropy change* and *unattainability* versions of the third law. Max Planck followed Nernst's work and was able to conclude the following[3]:

The gist of the (heat) theorem is contained in the statement that, as the temperature diminishes indefinitely the entropy of a chemical homogeneous body of finite density approaches indefinitely near to a definite value, which is independent of the pressure, the state of aggregation and of the special chemical modification’

— Max Planck

Notice that when Planck makes to the temperature diminishing, he is referring to the finite entropy hypothesis, while the later half makes reference to the entropy change statement.

A more formal and modern approach to stating the third law is as follows[4]:

The entropy of a perfect crystal approaches zero when T approaches zero (but perfect crystals do not exist)

### Computational Model

Following the link provided below we can roughly observe the Third Law of Thermodynamics. You will notice that as you pump atoms into the box the temperature and entropy increase. You can keep doing this until you get around 300°Kelvin. Now when you apply the ice bucket to the box, the temperature begins to dwindle down to absolute zero. While this model does let you achieve that value we can look at something else in this model. You will notice that as you approach 0°Kelvin, it takes longer and longer to lower the temperature. In a real life application, such as adiabatic demagnetization, it would simulate this process, except it would take infinitely long, which is essentially impossible, to reach 0°Kelvin.

https://phet.colorado.edu/en/simulations/gas-properties

## References

[1].The New Heat Theorem, Nernst. W, https://archive.org/details/in.ernet.dli.2015.206086

[2]. Mere Thermodynamics, Don S. Lemons, https://www.goodreads.com/book/show/8359990-mere-thermodynamics

[3]. Treatise of Thermodynamics, Max Planck, https://www.gutenberg.org/files/50880/50880-pdf.pdf