The Third Law of Thermodynamics

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]

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 as T \to 0 }[/math]

References

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

Mere Thermodynamics, Don S. Lemmons, https://www.goodreads.com/book/show/8359990-mere-thermodynamics