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Chapter 5

Thermodynamic potentials

Thermodynamic potentials are state functions that, together with the corresponding equa- tions of state, describe the equilibrium behavior of a system as a function of so-called ”natural variables”. The natural variables are a set of appropriate variables that allow to compute other state functions by partial differentiation of the thermodynamic potentials.

5.1 Internal energyU

The basic relation of is given by the equation

m α

dU= T dS+ Fidqi + µjdNj , (5.1) i=1 j=1 � � where F, q denote the set of conjugate intensive and extensive variables that characterize a system.{ For} instance, for a

F, q P,V , { }→{− } and for a magnetic system F, q , . { }→{B M} . The number of particles in the system is a natural extensive variable for the free , we did keep it hitherto constant. The number of particle of a distinct typesj is denoted byN j in (5.1), wherej=1,...,α. The respective intensive variable,µ (and respectively theµ j), is denoted the chemical potential. The chemical potential become identical to the for a gas of (at low , as we will discuss in Sect. 13.3). Note that diffusion processes (e.g. across a membrane) may change individualN j. The number of particles needs therefore not to be constant. of a gas. Eq. (5.1) is equivalent to

dU= T dS P dV+ µdN , U=U(S,V,N) (5.2) −

47 48 CHAPTER 5. THERMODYNAMIC POTENTIALS for a gas with one species of particle, which implies that

∂U ∂U ∂U T= , P= , µ= . (5.3) ∂S − ∂V ∂N � �V,N � �S,N � �S,V

Response functions. The experimentally important response functions are obtained by second-order differentiation of the internal energy,

1 ∂2U ∂T T ∂S ∂2U − = = , C =T =T ∂S 2 ∂S C V ∂T ∂S 2 � �V,N � �V,N V � �V,N �� �V,N �

1 ∂2U ∂P 1 1 ∂2U − = = , κ = ∂V 2 − ∂V Vκ S V ∂V 2 � �S,N � �S,N S �� �S,N �

Maxwell relations.A Maxwell relation follows, as discussed already in Sect. 4.4.2, from the differentiability of thermodynamic potentials. An example of a Maxwell relation derived from the differential of the internal energyU=U(S,V,N) is

∂ ∂U ∂ ∂U ∂µ ∂P = , = , (5.4) ∂V ∂N ∂N ∂V ∂V − ∂N � � � � � �S � �V which relates the change of the chemical potentialµ with the volumeV to the (negative of the) change of the pressureP with the number of particlesN.

5.1.1 Monoatomic For the monoatomic ideal we didfind hitherto 3 3 PV= nRT, U= nRT, C = nR . 2 V 2 We now use these relations to derive an expression forU in terms of its natural variables. . We start with the entropy

3nR T V S(T,V) S(T ,V ) = log + nR log − 0 0 2 T V � 0 � � 0 � 3nR U V = log + nR log 2 U V � 0 � � 0 � of the ideal gas, as derived in Sect. 4.4.1, where we have used the ideal gas relation U=3nRT/2 to substituteT andT 0 withU andU 0 in the second step. 5.1. INTERNAL ENERGYU 49

WithC V = 3nR/2 we may write equivalently

S S U 2 V U V 2/3 − 0 = ln + ln = ln , (5.5) C U 3 V U V V � 0 � � 0 � � 0 � 0 � � which can be solved for the internal energy as

γ 1 − V0 (S S0)/C U(S,V)=U e − V ,γ=5/3. (5.6) 0 V � �

Eq. (5.6) is the fundamental equation for the ideal gas, withU(S,V ) as the thermodynamic potential.S,V are the independent natural variables.

5.1.2 Thermodynamic potential vs. The natural variables forU areS andV , which means that if the functionU(S,V ) is known for a given system we can obtain –all– thermodynamic properties of the system through the differentiation ofU(S,V ). The equation of state

U=U(T,V,N) (5.7)

for the internal energyU is on the contrary –not– a thermodynamic potential. This is because thefirst of ( 5.7) yield the specific heatC V and the energy equation (4.12), ∂U ∂U ∂P =C , =T P, ∂T V ∂V ∂T − � �V � �T � �V but not the dependent variablesS andP . Note the marked difference to ( 5.3).

5.1.3 vs. thermodynamics There is a certain analogy between the potentialV and internal energyU of classical mechanics and thermodynamics respectively:

Classical mechanics Thermodynamics

Potential V(x, y, z) U(S,V,N)

Independent variables x, y, z S,V,N

∂V ∂U Dependent variables F = P= x − ∂x − ∂V � �S 50 CHAPTER 5. THERMODYNAMIC POTENTIALS 5.2

A disadvantage of usingU(S,V,N) as a thermodynamic potential is that the natural variableS is difficult to control in the lab. For practical purposes, it is more convenient to deal with other thermodynamic potentials that can be defined by making use of the Legendre transformation. Legendre transformations in classical mechanics. We recall the Legendre transfor- mation connecting the Legendre functionL(q,˙q) with the Hamilton functionH(q, p), whereq is the generalized coordi- nated and,˙q andp respectively the L velocity and the , with ∂L p= , ∂˙q � �q . : p −H(p) pq at constant coordinateq, and q. . H(q, p) =p˙q L(q,˙q). (5.8) q − A Legendre transformation is hence a variable transformation,˙q p, where one of the variables, here the momentump, is defined by the slope of original→ function, viz by the slope of the Lagrange functionL(p,˙q).

5.2.1 Legendre transformations in thermodynamics As an example we consider the transformation

U(S,V,N) H(S,P,N) , → from the internal energyU(S,V,N) to the enthalpyH(S,P,V ). The form is, as discussed shortly in Sect. 3.6, ∂U H(S,P,N)=U(S,V,N)+PV,P= . (5.9) − ∂V � �S Note that the is defined as the (negative) slope of the internal energyU with respect to the volumeV , in analogy to the transformation from the velocity˙q to the momentump=∂L/∂˙q in classical mechanics. Note however the different sign conventions between (5.9) and (5.8). Differential. The Legendre transformation (5.9) allows to evaluate with

dH=d U+PV = T dS P dV+ µdN + P dV+ V dP , − the differential dH of� the � � in terms of its natural� variables� S,P and� N. The result is dH= T dS+ V dP+ µdN . (5.10) 5.2. LEGENDRE TRANSFORMATION 51

5.2.2 The four basic thermodynamic potentials A thermodynamic potential for a system with variable number of particles should depend on µ, N as well as a thermal variable and a mechanical variable, which can be (for a gas):{ }

thermal:S (extensive)T (intensive)

mechanical:V (extensive)P (intensive)

The four possible combinations of these variables,

Intensive variables Extensive variables

P EnthalpyH(S,P) S → ← ↓ ↓ Gibbs enthalpy Internal energy G(T,P) U(S,V)

↑ ↑ T Free energyF(T,V) V → ← define four basic thermodynamic potentials: U(S,V,N) – internal energy H(S,P,N) – enthalpy G(T,P,N) – Gibbs enthalpy F(T,V,N) – free energy Gibbs enthalpy. We did already discuss the differentials (5.2), dU= T dS P dV+µdN, and (5.10) of respectively the free energyU(S,V,N) and of the enthalpyH(−S,P,N). For the Gibbs enthalpy one has in analogy

G=U(S,V,N) TS+PV , dG= SdT+ V dP+ µdN . (5.11) − − Free energy. The natural variablesT,V andN of the free energyF=F(T,V,N) are straightforward to access experimentally. The free energy is hence an often considered thermodynamic potential. It obeys

F=U(S,V,N) TS , dF= SdT P dV+ µdN . (5.12) − − − Differential relation betwen free and internal energy. The Legendre transforma- tion (5.12) betweenU andF allow to derive a differential relation between the free and 52 CHAPTER 5. THERMODYNAMIC POTENTIALS

internal energy, ∂F ∂ F U=F+TS=F T = T 2 , − ∂T − ∂T T � �V,N � � which can be written in terms of the inverse temperatureβ=1/(k BT ) as ∂ ∂ ∂ ∂(1/T) 1 ∂ U= βF , = = − . (5.13) ∂β ∂T ∂(1/T) ∂T T 2 ∂(1/T) � � It is generally easier to obtain explicit expressions for the internal energyU than for the free energyF . Above relation will be for this reason fundamental to the formulation of , as discussed lateron in Sect. 9.2, as it allows to identifyF.

5.2.3 The differentiability of the thermodynamic potentials leads to various Maxwell relations, such as the one given by Eq. (5.4). There are furthermore four Maxwell relations relating T , respectivelyS toP respectivelyV . We drop in the following the dependence on the number of particlesN.

∂T ∂P U(S,V) dU= T dS P dV = − ∂V − ∂S � �S � �V

∂T ∂V H(S,P) dH= T dS+ V dP = ∂P ∂S � �S � �P

∂S ∂V G(T,P) dG= SdT+ V dP = − ∂P − ∂T � �T � �P

∂S ∂P F(T,V) dF= SdT P dV = − − ∂V ∂T � �T � �V

5.2.4 Entropy The differential of the entropyS=S(U, V, N) is obtained rewriting ( 5.2) as

1 P µ dS= dU+ dV dN . (5.14) T T − T 5.3. HOMOGENEITY RELATIONS 53

The conjugate (natural) intensive variables are here 1/T andP/T with respect to dU and dV respectively. It is hence not customary to perform Legendre transformations with respect to these variables.

5.3 Homogeneity relations

The thermodynamic potentials vary as extensive state properties if the and num- ber of particles vary by a factorλ, i.e., V λV → U λU. (5.15) Nj λN j ⇒ → → � Note that long-range interactions between particles, such as the unscreened Coulomb interaction, may invalidate (5.15). Surface effects will contribute in addition for the case of a mesoscopic system. Extensive and intensive variables. is intensive, but α dU= T dS P dV+ µ dN − j j j=1 � extensive. It follows likewise that extensive :U(S,V,N),H(S,P,N),G(T,P,N),F(T,V,N) :S(U, V, N),V,N intensive :T,P,µ

Scaling. Doubling the volumeV will also double the free energyF(T,V,N). All extensive functions, viz all thermodynamic potentials obey hence the scaling relations of the sort of F(T,λV,λN) =λF(T,V,N) G(T,P,λN) =λG(T,P,N) (5.16) These functional dependencies express the homogeneity of the equilibrium . Gibbs-Duhem relation. Scaling relations are generically of fundamental importance in . Here we use the extensivity ofG and differentiate ( 5.16) with respect toλ (at λ = 1),

d α ∂G λG(T,P,N)= G(T,P,λN 1,λN 2, ..) = Nj dλ λ=1 ∂(λNj) j=1 T,P,Ni=j � � � � � �λ=1 � � � � =µ j � � We obtain then with � �� �

α G(T,P,N)= µ N Gibbs-Duhem relation. (5.17) j j ⇒ j=1 � 54 CHAPTER 5. THERMODYNAMIC POTENTIALS the Gibbs-Duhem relation. Chemical potential. When there is only one class of particles (α = 1),

G(T,P,N)= µN . (5.18)

The chemical potential may hence be interpreted as Gibbs enthalpy per particle. Representation of the internal energy. The Gibbs-Duhem relation (5.18) allows to rewrite the other thermodynamic potential. We recall, e.g. the Legendre transform (5.11),

G=U TS+PV, U=TS PV+ µN . − − Note, however, that this representation is only seemingly simple.

5.4 Grand canonical potential

In a system that interchanges particles with a reservoir, the chemical potential is constant and the number of particles changes accordingly. In order to describe such a system, we in- troduce the grand canonical potentialΩ(T, V, µ) as the Legendre transform ofF(T,V,N), where the natural variableN is replaced byµ, ∂F Ω(T, V, µ) =F(T,V,N) µN , µ(T,V,N)= . (5.19) − ∂N � �T,V The total differential ofΩ is

dΩ= dF µdN Ndµ, dΩ= SdT P dV Ndµ . (5.20) − − − − − Through differentiation ofΩ we can calculateS,P andN as ∂Ω ∂Ω ∂Ω S= ,P= ,N= . − ∂T − ∂V − ∂µ � �V,µ � �T,µ � �T,V Thermal equation of state. From the Gibbs Duhem relation (5.18),G= µN, it follows that Ω=F µN=G PV µN= PV,F=G PV − − − − − and hence that the thermal equation of stateP=P(T,V ), see Eq. ( 4.11), takes the from

Ω(T, V, µ) P= thermal equation of state. − V ⇒

Differential relation betwenΩ andU. With, ( 5.12),F=U TS, we may write (5.19) as − ∂Ω Ω=F µN=U TS µN, S= . − − − − ∂T � �V,µ 5.5. EQUILIBRIUM CONDITIONS 55

Reanranging the terms then leads to

∂Ω ∂ U µN=Ω+TS=Ω T , U µN= βΩ , (5.21) − − ∂T − ∂β � �V,µ � � where we have used in analogy to (5.13) that

∂ ∂ 1 ∂ ∂ ∂ ∂(1/T) 1 ∂ 1 T = T 2 =β β, = = − . − ∂T − ∂T T ∂T ∂T ∂(1/T) ∂T T 2 ∂β

5.5 Equilibrium conditions

Our discussion of thermodynamic potentials dealt hitherto mostly with the equilibrium state. The second law of thermodynamics,

T dS δQ= dU+ P dV µdN , (5.22) ≥ − allows however also to study the impact of irreversible processes, that is the of the system toward the equilibrium. We have used for (5.22) the differential form of (4.6) together with the generalizedfirst law

dU=δQ+δW+ µdN,δW= P dV , − where we have taken theδW appropriate for a gas. Equilibrium internal energy. The entropy is an exact differential dS=δQ/T at equilibrium. One can then consider situations like

dU= P dV S = const. () dU= T dS P dV − − ⇒ dU= T dS V = const. () � where selected state variables are constant. We will now generalize this discussion.

5.5.1 Principle of maximal entropy Isolated systems are defined by

δQ=0 dV=0 dU=0.  ⇒ dN=0  The second law (5.22) tells us then that 

ΔS 0 ≥ dS = 0 in equilibrium.

Principle of maximal entropy. 56 CHAPTER 5. THERMODYNAMIC POTENTIALS

In all irreversible processes, under the conditions

U = const., V = const., N = const.

the entropy increases and is maximal in the stationary equilibrium. S evolves to a maximum under conditions offixedU,V,N.

Two . Consider two isolated systems1 and2 separated by a wall, which is however movable and which allows for the exchange of energy and particles. The variablesV 1,V 2,U 1, andU 2,N 1,N 2 must obey the following constraints: U=U +U = const. dU = dU 1 2 ⇒ 1 − 2 V=V +V = const. dV = dV 1 2 ⇒ 1 − 2 N=N +N = const. dN = dN 1 2 ⇒ 1 − 2 The total entropy of1 and2 is likewise a sum of the individual :

S=S(U 1,V1,N1) +S(U 2,V2,N2) =S 1 +S 2 .

Equilibrium conditions. Both systems will interchange particles and energy until equi- librium is attained. In the equilibrium state

0 = dS= dS 1 + dS2

∂S1 ∂S2 = dU1 ∂U1 − ∂U2 �� �V1,N1 � �V2,N2 �

∂S1 ∂S2 + dV1 ∂V1 − ∂V2 �� �U1,N1 � �U2,N2 �

∂S1 ∂S2 + dN1 ∂N1 − ∂N2 �� �U1,V1 � �U2,V2 � 1 1 P P µ µ = dU + 1 2 dV + 1 + 2 dN . T − T 1 T − T 1 −T T 1 � 1 2 � � 1 2 � � 1 2 � where we have used the steady-state version of the second law (5.22), dS= dU+ P dV µdN. −

Homogeneity condition.U 1,V 1, andN 1 are independent variables. The prefactors of dU1, dV1, and dN1 need hence to vanish individually:

T1 =T 2 =T;P 1 =P 2 =P;µ 1 =µ 2 =µ. Instead of two systems we can subdivide a given system in an arbitrary number of ways. The system needs consequently to be homogeneous: An isolated system in equilibrium has everywhere the same temperature, pressure and chemical potential. 5.5. EQUILIBRIUM CONDITIONS 57

5.5.2 Principle of minimal free energy Closed systems in a reservoir with heat, but without work interchange, are defined by

closed N j = const dN j = 0, in a reservoir⇒T= const dT⇒ = 0, without work interchange⇒V= const⇒ dV = 0. ⇒ ⇒

The second law of thermodynamics (5.22) reduces then to

T dS dU+ P dV µdN= dU, dV=0= dN . ≥ −

Principle of minimal free energy. For constant temperature one has T dS=d(TS) and consequently that

0 T dS dU=d(TS U), dF=d(U TS) 0 , ≤ − − − ≤ where we have used (5.12), namely thatF=U TS. − The free energy decreases and reaches a minimum at equilib- rium, dF 0 dF ≤= 0 in equilibrium

for all irreversible processes characterized by a constant tem- peratureT , volumeV and particle numberN.

Composite system. Consider a composite system in a heat reservoir at temperatureT, whose particles can pass through the wall. It obeys the conditions

V=V +V = const. dV = dV 1 2 ⇒ 1 − 2 N=N +N = const. dN = dN 1 2 ⇒ 1 − 2 F=F(T,V 1,N1) +F(T,V 2,N2) =F 1 +F 2

In equilibrium,

0 = dF= dF 1 + dF2

∂F1 ∂F2 = dV1 ∂V1 − ∂V2 �� �N1,T � �N2,T �

∂F1 ∂F2 + dN1 ∂N1 − ∂N2 �� �V1,T � �V2,T � = P +P dV + µ µ dN . {− 1 2} 1 { 1 − 2} 1 58 CHAPTER 5. THERMODYNAMIC POTENTIALS

WithV 1 andN 1 being independent variables if follows that

P1 =P 2 =P, µ 1 =µ 2 =µ.

A with constant volumeV which is in contact with a heat bath at constant temperatureT reaches an equilibrium state characterized by a uniform pressureP and chemical potentialµ.

Principle of minimal Gibbs energy. Generalizing above considerations we consider closed system in a heat reservoir, but at constant pressureP (previouslyV was constant). One can easily show that

dT=0, dN j = 0, dP=0

implies that

dG 0 ≤ dG = 0 in equilibrium.

The Gibbs energy decreases in an for a system with constant pressureP in contact with a heat bath at constant temperatureT.G(T,P,N) reaches its minimum at equilibrium, which is characterized by a uniform chemical potentialµ.

5.5.3 Stationarity principles in From the microscopic point of view the equilibrium thermodynamic state is stationary in the sense that the positions and the velocities of the constituent particle change contin- uously. The ongoing reconfiguration of the microscopic state leaves however the macro- scopic properties of the state intact. Stationary distribution functions. A functions p(y) quantifies the probability that the statey (an event) occurs, with

p(y)dy=1, p(y) 0. ≥ � It is the task of statistical mechanics tofind the expression forp(y) (see Chap. 9) Statistical mechanics dealing hence with probability distribution functions which are stationary with respect to the reconfiguration of microscopic states. Entropy and disorder. In statistical mechanics the entropy has, as we will discuss in Chap. 12, the form

S[p] = k ln(p) = k p(y) ln(p(y)) dy . − B � � − B � It measures the amount of disorder in the sense that aflat (maximally disordered) dis- tribution functionp(y) has the largest entropyS[p]. Here we have writtenS=S[p] as a 5.5. EQUILIBRIUM CONDITIONS 59

functional ofp=p(y). This connection allows us to reinterpret the principle of maximal entropy derived in Sect. 5.5.1.

Irreversible processes increase the amount of disorder in iso- lated systems. The entropy of the universe can only increase.

Entropy and information. Information theory deals with information processing as de- scribed by probability distribution functions. The central quantity of information theory is the Shannon entropy I[p] = p(y) log(p(y)) dy , (5.23) − � which is identical, apart from an overall factor,k B, to the thermodynamic entropy. The Shannon entropy is however used as a measure of the . The context is here information transmitted via communication channels.

bit sequence type information content

..01010101.. ordered zero zero (predictable)

..00110101.. disordered high high You do not learn anything new if you can predict what another person says. Shannon’s source coding theorem. A precise connections between the information- theoretical entropyI[p] and the information content of a communication channel is pro- vided by the Shannon coding theorem, which states theI[p] provides a lower bound for lossless compression. Information is inevitably lost when compressing more. Life as a stationary state. Life is by definition a stationary state of which retains its functionality despite the continuously ongoing turnover of its constituent elements. The proteins, and cells of our body have lifetimes ranging typically from hours to months. Equilibrium principles in the neurosciences. The stationary equilibrium of the brain with both the environment and the body is dependent on a continuousflux of information. Information theoretical principles are consequently of widespread use in the neurosciences. An example is the free energy principle, as proposed by Carl Friston. It suggests that the brain may be minimizing the information-theoretical analog of the thermodynamic free energy (compare Sect. 5.5.2). 60 CHAPTER 5. THERMODYNAMIC POTENTIALS 5.6 Summary of thermodynamic potentials

Potential Natural Conjugated Maxwell relations independent dependent and others variables variables

∂U Internal energy T= ∂S � �V,N ∂U ∂T ∂P U S,V,N P= = − ∂V ∂V − ∂S � �S,N � �S � �V ∂U µ= ∂N � �S,V 1 ∂S Entropy = T ∂U � �V,N P ∂S S U,V,N = T ∂V � �U,N µ ∂S = −T ∂N � �U,N ∂F ∂S ∂P Free energy S= = − ∂T ∂V ∂T � �V,N � �T � �V ∂F ∂ F F=U TS T,V,N P= U= T 2 − − ∂V − ∂T T � �T,N � � ∂F µ= ∂N � �T,V ∂H Enthalpy T= ∂S � �P,N ∂H ∂T ∂V H=U+PV S,P,N V= = ∂P ∂P ∂S � �S,N � �S � �P ∂H µ= ∂N � �S,P ∂G ∂S ∂V Gibbs enthalpy S= = − ∂T ∂P − ∂T � �P,N � �T � �P ∂G ∂ G G=H TS=Nµ T,P,N V= H= T 2 − ∂P − ∂T T � �T,N � � ∂G µ= ∂N � �T,P 5.6. SUMMARY OF THERMODYNAMIC POTENTIALS 61

Potential Natural Conjugated Maxwell relations independent dependent and others variables variables

∂Ω Grand canonical S= − ∂T � �V,µ ∂Ω potential T,V,µ P= − ∂V � �T,µ ∂Ω Ω=F µN= PV N= − − − ∂µ � �T,V S ∂µ Chemical potential T s= = N − ∂T P G V ∂�µ � µ= P v= = N N ∂P � �T 62 CHAPTER 5. THERMODYNAMIC POTENTIALS