Statistical physics (2005)
.pdfThe Thermodynamical Potentials; the Legendre Transformation |
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nuclear interactions the system is no longer at zero magnetic field and becomes ordered when T tends to zero.
The present sections have put in correspondence the laws of Thermodynamics and the statistical approach of chapters 1 and 2. Their conclusion is that the description based on Statistical Physics and its postulates is indeed equivalent to the three laws of Thermodynamics.
3.5The Thermodynamical Potentials and their Di erentials ;
the Legendre Transformation
Now that we have identified the statistical entropy to the thermodynamical entropy, we can take advantage of our background in Thermodynamics to introduce the thermodynamical potential adapted to the considered physical situation, and relate it to the statistical description and to the physical parameters. The thermodynamical potential will be a function of the parameters determined by the experimental conditions and in equilibrium it will be extremum.
3.5.1Isolated System
Here the entropy S is the thermodynamical potential to consider. During a spontaneous evolution toward equilibrium, S increases and reaches its maximum in equilibrium.
In the microcanonical description, the entropy is related to the number W of microstates of energies lying in the range between E and E + dE by
S = kB ln W |
(3.23) |
dS and dU are linked through the di erential expressions of the first and second laws of Thermodynamics :
dS = |
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− |
δW |
(3.24) |
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In the case of the ideal gas where |
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δW = −P dΩ |
(3.26) |
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(3.27) |
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The di erential expression (3.27) shows that S, calculated for a fixed number N of particles, is a function of the variables U and Ω, both extensive. The pressure is deduced from the partial derivative of S versus the volume :
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∂S |
ãN,U |
= Å |
kB ∂ |
ln W (U ) |
ãN,U |
(3.28) |
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Note : we suggested in the Appendix 3 of chapter 1 that, in the case of free particles, the number of microstates associated with the energy E was of the form
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(3.29) |
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so that |
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Here we find the ideal gas law, that will be again established in the next chapter.
After having studied a given system, we now consider another similar system, with the same energy U and under the same volume Ω, which only di ers by its number of particles equal to N + dN . Its entropy is now S(U, Ω, N ) + dS, and we write
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dS = −T dN |
(3.32) |
This defines the chemical potential µ, related to the entropy variation of the system when an extra particle is added, at constant internal energy and volume. The chemical potential is an intensive parameter. For a classical ideal gas, if N increases, one expects S to increase, so that µ is negative.
Then, including the dependence in the particles number, the di erential of S for the ideal gas is written
dS = |
dU |
+ |
P dΩ |
− |
µdN |
(3.33) |
T |
T |
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The Thermodynamical Potentials; the Legendre Transformation |
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S is a function of extensive variables only; the partial derivative of S versus N is equal to
Å |
∂S |
ãΩ,U |
µ |
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We previously showed in §2.5.1 that if the two systems are exchanging particles, in equilibrium
∂S |
= −kB α |
(3.35) |
∂N |
takes the same value in both systems.
These two partial derivatives of the entropy versus the number of particles can be identified, so that
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(3.36) |
Besides, from the di erential (3.33) of S expressed for the ideal gas one deduces the expression of the di erential of its energy U :
dU = −P dΩ + T dS + µdN |
(3.37) |
The last term can be interpreted as a “chemical work.”
3.5.2System with a Fixed Number of Particles, in Contact with a Heat Reservoir at Temperature T
We are going to show that the corresponding thermodynamical potential is the Helmholtz free energy F , which is minimum in equilibrium and is related to the N -particle canonical partition function ZN . Indeed the general definition (3.16) of S applies, with now probabilities pi equal to
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Consequently, in the canonical ensemble, |
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S = |
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U − T S = −kB T ln ZN
(3.38)
(3.39)
(3.40)
(3.41)
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Chapter 3. Thermodynamics and Statistical Physics |
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Now in Thermodynamics the free energy F is defined by |
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F = U − T S |
(3.42) |
whence we deduce the relation between the thermodynamical potential F , and the partition function ZN expressing the Statistical Physics properties :
F = −kB T ln ZN |
(3.43) |
Let us write the di erential of the thermodynamical potential F in the case where the infinitesimal work is given by δW = − Xj dxj (see eq. 3.8) :
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dF = dU − T dS − SdT
= − Xj dxj + δQ + µdN − T dS − SdT |
(3.44) |
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dF = − Xj dxj − SdT + µdN |
(3.45) |
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The interpretation of (3.45) is that, in a specific infinitesimal reversible process at constant temperature and fixed number of particles, the free energy variation is equal to the work received by the system, whence the name of Helmholtz “free energy,” i.e., available energy, for this thermodynamical potential.
For a gas, the infinitesimal work is expressed as a function of pressure, so that
dF = −P dΩ − SdT + µdN |
(3.46) |
The entropy, the pressure, and the chemical potential are obtained as partial derivatives F or of ln ZN :
S = − Å |
∂F |
ãN,Ω= |
Å |
∂(k |
B |
T ln Z |
N |
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ãN,Ω |
(3.47) |
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∂(k |
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N |
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ãN,T |
(3.48) |
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3.5.3System in Contact with Both a Heat Reservoir at Temperature T and a Particle Reservoir
It was shown in 2.5.1 that, during such an exchange of energy and particles, the parameters taking the same value in equilibrium in both the system and
The Thermodynamical Potentials; the Legendre Transformation |
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the reservoir are kB β = 1/T and kB α = µ/T : two Lagrange parameters thus come into the probabilities, which do not only depend on the energy but also on the number of particles. Thus the probability of occurrence of a macroscopic state of energy Ei and number of particles Ni can be written (see §2.5.3, eqs. (2.62) and (2.63)) :
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pi Å− |
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Consequently, |
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S = |
U |
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U − T S − αkB T N = −kB T ln ZG
(3.50)
(3.51)
(3.52)
(3.53)
(3.54)
where N is the average number of particles in the system under study.
The thermodynamical potential introduced here is the grand potential A, defined by
A = U − T S − µ N = F − µ N |
(3.55) |
It is related to the grand canonical partition function through |
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A = −kB T ln ZG |
(3.56) |
The di erential of A is deduced from its definition : |
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dA = dF − µd N − N dµ |
(3.57) |
= − Xj dxj − SdT − N dµ |
(3.58) |
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Like the other thermodynamical potentials, A is extensive. The only extensive variable in A is the one in the infinitesimal work. In the case of a gas, it is the volume so that
A = −P Ω |
(3.59) |
The entropy S and the average number of particles N are obtained by calculating the partial derivatives of A.
As a conclusion of this section on the thermodynamical potentials, remember that to a fixed physical situation, corresponding to data on specific external parameters, is associated a particular thermodynamical potential, which is extremum in equilibrium.
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Chapter 3. Thermodynamics and Statistical Physics |
3.5.4Transformation of Legendre ; Other Thermodynamical Potentials
In §3.5.2 and §3.5.3 we saw two di erent physical situations, associated with two thermodynamical potentials with similar di erentials, except for a change in one variable. The mathematical process that shifts from F (N ) to A(µ) is called “transformation of Legendre.”
The general definition of such a transformation is the following : one goes from a function Φ(x1, x2, ...) to a function
Γ = Φ(x1, x2, ...) − x1y1, with y1 = |
∂Φ |
(3.60) |
∂x1 |
the new function Γ is considered as a function of the new variable y1. Because of the change of variable and of function, the di erentials of Φ and of Γ are respectively :
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dΦ = |
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∂Φ |
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dΓ = |
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∂x1 − ∂x1 |
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The partial derivatives of Γ versus the variables x2 . . . are not modified with respect to those of Φ for the same variables.
In this process one has shifted from the variables (x1, x2 . . . xn) to the variables (y1, x2 . . . xn). This is indeed the transformation that was performed for example, when shifting from U (S) to F (T ), the other variables Ω, N being conserved, or when shifting from F (N ) to A(µ), the variables Ω and T being unchanged.
Other thermodynamical potentials can be defined in a similar way, a specific potential corresponding to each physical situation. For example, a frequent situation in Chemistry is that of experiments performed at constant temperature and constant pressure (isothermal and isobaric condition) : the exchanges occur between the studied system A and the larger system A which acts as
The Thermodynamical Potentials; the Legendre Transformation |
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an energy and volume reservoir. We saw that the equilibrium condition, expressing the maximum of entropy of the combined system with respect to exchanges of energy and volume, corresponds to the equality of the temperatures and the pressures in both systems. The corresponding thermodynamical potential is the Gibbs free enthalpy G, in which the number of particles is fixed :
G = U − T S + P Ω
(3.65)
dG = Ω dP − SdT + µdN
This potential is defined from the free energy F
F = U − T S
(3.66)
dF = −P dΩ − SdT + µdN
using a transformation of Legendre on the set of variables (volume Ω and pressure P ), that is
G = F + P Ω |
(3.67) |
The chemical potential is interpreted here as the variation of the free enthalpy G when one particle is added to the system, at constant pressure and temperature. The free enthalpy, extensive like all thermodynamical potentials, depends of a single extensive variable, so that
G = µN |
(3.68) |
The chemical potential µ is here the free enthalpy per particle.
The free enthalpy is related to the partition function which includes all the possible values of energy and volume of the considered system (see §2.6) :
ZT,P = e−βEi − γΩi |
(3.69) |
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Note 1 : each time a transformation of Legendre is performed, an extensive variable is replaced by an intensive one. Now a thermodynamical potential has to be extensive, so that it should retain at least one extensive variable. Consequently, there is a limit to the number of such tranfomations yielding a new thermodynamical potential.
Note 2 : Appendix 4.1 presents a practical application of thermodynamical potentials to the study of chemical reactions.
Summary of Chapter 3
We have expressed the laws of Thermodynamics in the language of Statistical Physics.
– Zeroth law :
For two systems A and A in thermal equilibrium
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Å ∂E ãN,Ω = kB β = |
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– First law :
The di erential of the average energy of a system :
dU = d |
piEi = pidEi + Eidpi |
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is split into two terms.
The infinitesimal work is related to the modification of the energy states of the system due to the change of external parameters (volume, etc.)
δW = pidEi
i
The infinitesimal heat exchanged δQ is associated with variations of probabilities of occurrence of the microstates
δQ = Eidpi
i
– Second law :
We have identified the thermodynamical entropy with the statistical en-
tropy as introduced in the first chapters of this book and have shown that
T1 .
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Summary of Chapter 3 |
In Statistical Physics, an irreversible process corresponds to an increase of the number of accessible microstates for the considered system, and consequently to an increase of disorder and of entropy.
– Third law :
The entropy per particle S/N vanishes when the temperature tends to the absolute zero.
– The adapted thermodynamical potential depends on the considered physical situation, it is extremum in equilibrium. It is related to the microscopic parameters of Statistical Physics through the partition function; its partial derivatives give access to the measurable physical parameters. Using a transformation of Legendre, one gets a new thermodynamical potential, with one new variable (see the table next page).
Ensemble |
Microcanonical |
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isolated system |
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exchange of energy with a |
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Conditions |
E to δE |
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heat reservoir at fixed N |
heat reservoir and of particles with a |
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particle reservoir |
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β = |
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Description |
configurations at E |
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ZG (α, β) = |
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Thermodynamical |
S = kB ln W |
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F = −kB T ln ZN |
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Potential |
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F = U − T S |
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A = U − T S − µN |
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dS = |
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dF = −P dΩ − SdT + µdN |
dA = −P dΩ − SdT − N dµ |
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3 Chapter of Summary
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