Third law of Thermodynamics

• Third law of Thermodynamics

• Nernst heat theorem: In the neighborhood of absolute zero, all reactions in a liquid or solid
• in internal equilibrium take place with no change in entropy.

• Walther Nernst

• (consider e.g. a chemical reaction )

• 1

• 2

• Albert Einstein

• Max Planck

• Robert Milikan

• Max von Laue

• Motivated by considering reactions in the limit of decreasing temperature

• We know: at P, T=const. equilibrium thermodynamics determined by Gmin.

• controls reaction

• Experimental finding:

• for

• With

• and

• (see thermodynamic potentials)

• Nernst proposed as a general principle:

• for

• ,

• and

• T

• G, H

• From

• T=const.

• heat flow into bath (exotherm)
• but sometimes also
• out of the bath (endotherm)

• Planck made further hypothesis known as the third law

• Entropy of every solid or liquid substance in internal equilibrium at
• absolute zero is itself zero

• Some consequences of the third law

• Since

• finite at a given T

• From Nernst theorem

• With Maxwell relation

• (*)

• Requires quantum mechanics to
• derive it in terms of statistical mechanics

• It is impossible to reach the absolute zero temperature

• with a finite sequence of isothermal and adiabatic changes of pressure or other
• variables like the magnetic field, e.g., in the case of adiabatic demagnetization.

• S

• T

• P

• P’

• isothermal
• compression

• adiabatic
• expansion

• Gas compression refrigeration

• T+Tad

• T

• T

• T-Tad

• T

• +P

• -Q

• -P

• +Q

• P=P-P’

• According to 3rd law:

• S(T,P)=S(T,P’) for T=0

• T=0 not achievable in a finite # of
• compression and expansion steps

• 3rd law and Boltzmann’s entropy expression

• W: # of possible microstates

• S=kB ln W

• Although we don’t focus on stat. mechanics it is useful to get an idea how the third law is related to the Boltzmann formula

• -Consider system described by a Hamilton operator with a discrete spectrum of

• energy-eigenvalues having a lower bound (ground state):

• E0

• E1

• E2

• @ sufficient low T system will be in its ground state

• If there are g0 eigenstates with the same energy E0 we say ground state is degenerate

• # of microstates representing the same macro state is W=g0 and, hence

• If ground state is non degenerate

• and