PHY304 - Statistical Mechanics
Spring Semester 2021 Dr. Anosh Joseph, IISER Mohali
LECTURE 09
Thursday, January 21, 2021 (Note: This is an online lecture due to COVID-19 interruption.)
Contents
1 The Free Enthalpy 1
2 The Grand Potential 4 2.1 Transformation of All Variables ...... 6
1 The Free Enthalpy
Consider a system with given temperature T and pressure p. We can then perform the Legendre transformation of the internal energy U(S, V, N, ··· ) with respect to two variables S and V . We have G = U − TS + pV. (1)
The function G is also a thermodynamic potential. This is called the free enthalpy. It was introduced by Gibbs in 1875. It is also known as the Gibb’s potential. Taking the total differential of G we have
dG = dU − T dS − SdT + pdV + V dp = −SdT + V dp + µdN + ··· . (2)
From this we see that G depends only on T , p, and N. PHY304 - Statistical Mechanics Spring Semester 2021
We can obtain the equations of state from G. We have
∂G −S = , (3) ∂T p,N,···
∂G V = , (4) ∂p T,N,···
∂G µ = , (5) ∂N T,p,··· . . .
Upon using the Euler’s equation
U = TS − pV + µN, (6) we have G = U − TS + pV = µN (7)
We see that G is proportional to the particle number. Thus we can say that the free enthalpy per particle is identical to the chemical potential. Note that this statement is valid only for systems containing one kind (species) of particles, which cannot exchange other forms of energy with their surroundings. To understand the meaning of the free enthalpy let us consider an isolated system consisting of the isothermal and isobaric system immersed in a heat bath. See Fig. 1. Then we have
dStot = dSsys + dSbath ≥ 0. (8)
The equality sign holds for reversible processes. The greater-than-or-equal-to sign for irreversible processes. For reversible processes we have
1 dS = −dS = − (dU + pdV − δW rev ) . (9) bath sys T sys sys other
For irreversible processes we must have
rev T dStot = T dSsys − dUsys − pdVsys + δWother ≥ 0. (10)
We can write the above two equations as
rev irr dGsys = d(U − TS + pV ) = δWother ≤ δWother. (11) sys
2 / 9 PHY304 - Statistical Mechanics Spring Semester 2021
F ⃗ = const .
T, p, N
heat bath T
Figure 1: Isothermal and isobaric system.
This tells us that the change in free enthalpy is just the work performed by the system in an isothermal, isobaric reversible process, without volume work against the constant external pressure. We see that in an isothermal, isobaric system which is left to its own, irreversible processes happen until a minimum of the free enthalpy is reached,
dG = 0, (12)
G = Gmin. (13)
Similar to enthalpy, the free enthalpy is of great importance for chemistry. This is useful in the case of many types of fuel cells or batteries. There, chemical reactions happen slowly under constant (atmospheric) pressure maintaining thermal equilibrium. We can directly calculate the electrical work obtainable from a battery as the difference of the free enthalpies in the final and initial states. As an example, the free enthalpy of the ideal gas is
G(T, p, N) = Nµ(T, p). (14)
The free enthalpy and its derivative with respect to temperature can be related to the enthalpy
3 / 9 PHY304 - Statistical Mechanics Spring Semester 2021 of the system
∂G H = G − T ∂T p
∂(G/T ) = −T 2 . (15) ∂T p
The above relation is known as the Gibbs-Helmholtz equation. If the isothermal and isobaric system consists of several chemical components (particle species) we then have
G = µiNi. (16)
The equilibrium condition takes the form
dG = µidNi = 0. (17)
Note that we can apply the above considerations to systems containing elementary particles in a hot star (or plasma), instead of a chemical reaction.
2 The Grand Potential
Let us consider systems where the chemical potential is given as a state variable, instead of the particle number N. The chemical potential can be fixed by a particle bath. The exchange of particles with a particle reservoir leads to a constant chemical potential. Let us transform the internal energy in the state variables S and N to the new variables T and µ Φ = U − TS − µN. (18)
This potential is called the grand potential. Note that this potential is of great importance for the statistical treatment of thermodynamic problems. The total differential has the form
dΦ = dU − T dS − SdT − µdN − Ndµ, = −SdT − pdV − Ndµ. (19)
4 / 9 PHY304 - Statistical Mechanics Spring Semester 2021
The equations of state are
∂Φ −S = , (20) ∂T V,µ
∂Φ −p = , (21) ∂V T,µ
∂Φ −N = . (22) ∂µ T,V
Upon using the Euler’s equation
U = TS − pV + µN we see that Φ = −pV. (23)
This potential becomes useful when we are dealing with isothermal systems with a fixed chemical potential. See Fig. 2.
particle reservoir μ
T, V, μ
heat bath T
Figure 2: Isothermal system with fixed chemical potential.
If we combine the heat bath and the system under consideration to a total isolated system, then we must have
dStot = dSsys + dSbath ≥ 0. (24)
In the reversible case we have
rev T dSbath = −T dSsys = − (dUsys − µdNsys − δWother) . (25)
5 / 9 PHY304 - Statistical Mechanics Spring Semester 2021
Rearranging terms
rev irr dUsys − T dSsys − µdNsys = δWother ≤ δWother. (26)
rev irr Here δWother and δWother correspond respectively, to the work performed reversibly and irre- versibly, by the system, without the explicitly considered chemical energy. At constant temperature and constant chemical potential this is equivalent to
rev irr dΦ = d(U − TS − µN) = δWother ≤ δWother. (27)
If we leave the system on its own without performing work, δW = 0, it strives for a minimum of the grand potential dΦ ≤ 0, (28) which is achieved in equilibrium
dΦ = 0, (29)
Φ = Φmin. (30)
2.1 Transformation of All Variables
Let us consider the effect of transforming all variables in U to the new variables T, p, µ. The Legendre transformation is
Ψ = U − TS + pV − µN. (31)
Upon using the Euler’s equation
U = TS − pV + µN, we get Ψ ≡ 0. (32)
That is, this potential vanishes identically. This tells us that the simultaneous transformation of all the variables has no relevance.
3 Maxwell Relations
The thermodynamic potentials U, F, H, G and Φ are all state functions. That is, they have exact differentials. From the above fact we can derive a variety of relations among themselves.
6 / 9 PHY304 - Statistical Mechanics Spring Semester 2021
We have
dU = T dS − pdV + µdN
∂U ∂U ∂U = dS + dV + dN. (33) ∂S ∂V ∂N V,N S,N S,V
Since ∂ ∂U ∂ ∂U = , (34) ∂V ∂S ∂S ∂V V,N S,N S,N V,N we have ∂T ∂p = − . (35) ∂V ∂S S,N V,N We can obtain many relations in this way, that may allow for the calculation of unknown quantities from known quantities. From Eq. (33) we have
∂T ∂p = − , (36) ∂V ∂S S,N V,N
∂T ∂µ = , (37) ∂N ∂S S,V V,N
∂p ∂µ − = . (38) ∂N ∂V S,V S,N
From free energy we can obtain a similar set of equations
dF = −SdT − pdV + µdN, (39)
∂S ∂p − = − , (40) ∂V ∂T T,N V,N
∂S ∂µ − = , (41) ∂N ∂T T,V V,N
∂p ∂µ − = . (42) ∂N ∂V T,V T,N
From enthalpy we get dH = T dS + V dp + µdN, (43)
7 / 9 PHY304 - Statistical Mechanics Spring Semester 2021
∂T ∂V = − , (44) ∂p ∂S S,N p,N
∂T ∂µ = , (45) ∂N ∂S S,p p,N
∂V ∂µ = . (46) ∂N ∂p S,p S,N
For the free enthalpy we have
dG = −SdT + V dp + µdN, (47)
∂S ∂V − = , (48) ∂p ∂T T,N p,N
∂S ∂µ − = , (49) ∂N ∂T T,p p,N
∂V ∂µ = . (50) ∂N ∂p T,p T,N
For the grand potential we have
dΦ = −SdT − pdV − Ndµ, (51)
∂S ∂p = , (52) ∂V ∂T T,µ V,µ
∂S ∂N = , (53) ∂µ ∂T T,V V,µ
∂p ∂N = . (54) ∂µ ∂V T,V T,µ
The above set of relations are called Maxwell relations. If there are even more state variables (magnetic field, magnetic dipole moment, etc.) we need to add more relations. There exists a simple device which allows for a quick overlook of the potentials and their variables, and which yields the Maxwell relations.
8 / 9 PHY304 - Statistical Mechanics Spring Semester 2021
This device is the thermodynamic rectangle. See Fig. 3.
F V T
U G
S p H
Figure 3: Thermodynamic rectangle at constant N.
The thermodynamic rectangle was conceived especially for systems with constant particle num- ber and without further state variables. The variables V, T, p and S, which are the only possible quantities at constant particle number, form the corners of this rectangle. Along the edges we have the potentials, which depend on the variables at the corresponding corners. With this way of presentation we can easily read off the partial derivatives. The derivative of a potential with respect to a variable (corner) is just given by the variable at the diagonally opposite corner. The arrows in the diagonals determine the sign. As an example, we have ∂F/∂V = −p. The minus sign occurs because the direction V → p is opposite to the direction of the flow. We also have ∂G/∂p = +V . We can also easily read off the Maxwell relations. Derivatives of variables along an edge of the rectangle (e.g., ∂V/∂S), at constant variable in the diagonally opposite corner (here p), are just equal to the corresponding derivative along the other side, i.e., in this case ∂T/∂p . S The signs have to be chosen according to the direction in which one follows the diagonals, e.g., going from V to p yields a minus sign as does the path T → S.
References
[1] W. Greiner, L. Neise, H. Stocker, and D. Rischke, Thermodynamics and Statistical Mechanics, Springer (2001).
9 / 9