A number of second derivatives of the fundamental relation have clear physical significance and can be measured experimentally. For example: 1 ∂2G 1 ∂V Isothermal compressibility κ = − = − T V ∂P2 V ∂P
The property of the energy (or entropy) as being a differential function of its variables gives rise to a number of relations between the second derivatives, e. g. :
∂2U ∂ 2U = ∂S∂V ∂V∂S The Maxwell relations
∂2U ∂ 2U = ∂S∂V ∂V∂S is equivalent to: ∂P ∂T − = ∂S ∂V V ,N1 ,N2 ,... S ,N1 ,N2 ,...
Since this relation is not intuitively obvious, it may be regarded as a success of thermodynamics. The Maxwell relations
Given the fact that we can write down the fundamental relation employing various thermodynamic potentials such as F, H, G, … the number of second derivative is large. However, the Maxwell relations reduce the number of independent second derivatives.
Our goal is to learn how to obtain the relations among the second derivatives without memorizing a lot of information. For a simple one-component system described ∂2 ∂2 ∂2 U U U by the fundamental relation U(S,V,N), there are 2 ∂S ∂S∂V ∂S∂N nine second derivatives, only six of them are ∂2 ∂2 ∂2 independent from each other. U U U 2 ∂V∂S ∂V ∂V∂N In general, if a thermodynamic system has n 2 2 2 ∂ U ∂ U ∂ U independent coordinates, there are n(n+1)/2 2 independent second derivatives. ∂N∂S ∂N∂V ∂N The Maxwell relations: a single-component system
For a single-component system with conserved mole number, N, there are only three independent second derivatives: ∂2 ∂2 ∂2 G G G ∂ 2G ∂ 2G ∂2G ∂T2 ∂T∂P ∂T∂N 2 2 2 2 2 ∂ G ∂ G ∂ G ∂ ∂ ∂ ∂ 2 T P T P ∂P∂T ∂P ∂P∂N 2 2 2 ∂2U ∂ G ∂ G ∂ G Other second derivatives, such as e. g.: ∂N∂T ∂N∂P ∂N 2 ∂S∂V can all be expressed via the above three, which are trivially related to three measurable physical parameters (dg = -sdT + vdP + µdN): The Maxwell relations: a single-component system
Coefficient of thermal Molar heat capacity at Isothermal compressibility expansion constant pressure 1 ∂ 2 g 1 ∂v ∂ 2 g ∂s δQ 1 ∂ 2 g 1 ∂v α = = c = −T = T = κ = − = − v ∂T∂P v ∂T P ∂T 2 ∂T dT T v ∂P2 v ∂P
A general procedure for reducing any derivative to a combination of α, κT, and cP is given on pages 187-189 of Callen. It is tedious but straightforward, you should read it and know it exists. You will also have a chance to practice it in a couple of homework exercises. Stability of thermodynamic systems
The entropy maximum principle states that in equilibrium: dS = 0 d 2S < 0
Let us consider what restrictions these two S conditions imply for the functional form of the dependence of S on extensive parameters of a thermodynamic system. S0
Let us consider two identical systems with the following dependence of entropy on U energy for each of the systems: U0 Stability of thermodynamic systems
Consider a homogeneous Initially both sub-systems have energy U and system with the fundamental 0 entropy S0. relation shown below and divide it on two equal sub-systems Let us consider what happens to the total entropy of the system if some energy ∆U is transferred S from one sub-system to another.
In this case, the entropy of the composite system will be S(U0 + ∆U)+ S(U0 - ∆U) > 2S0. S0 Therefore the entropy of the system increases and the energy will flow from one sub-system to U the other, creating a temperature difference.
Therefore, the initial homogeneous state of the U0 ∆U system is not the equilibrium state. Phase Separation
If the homogeneous state of the system is not the equilibrium state, the system will spontaneously become inhomogeneous, or will separate into phases. Phases are different states of a system that have different macroscopic parameters (e. g. density).
Phase separation takes place in phase transitions. When a system undergoes a transition from one macroscopically different state to another it goes through a region of phase separation (e. g. water-to-vapor phase transition). Phase Separation Examples
Phase separation is ubiquitous.
Electronic phase separation in perfect Two liquids. Above a certain crystals of strongly correlated materials. temperature they form a Homogeneous crystal breaks into regions homogeneous mixture. Below this of normal metal and superconductor. temperature they phase separate. Stability of thermodynamic systems
To have a stable system, the following condition has to be met:
S(U + ∆U,V , N )+ S(U − ∆U,V , N ) ≤ 2S(U,V , N ) for arbitrary ∆U
For infinitesimal dU, the above condition reduces to: ∂ 2S S(U + dU,V , N )+ S(U − dU,V , N )− 2S(U,V , N ) ≤ 0 ≤ 0 ∂U 2 Similar argument applies to spontaneous variations of any extensive variable, e. g. V: ∂ 2S S(U,V + ∆V , N )+ S(U,V − ∆V , N ) ≤ 2S(U,V , N ) ≤ 0 ∂V 2 Stability of thermodynamic systems
∂ 2S Globally stable system Locally unstable region ≥ 0 S S ∂X 2
X X
Fundamental relations of this shape are frequently obtained from statistical mechanics or purely thermodynamic models (e. g. van der Waals fluid). These fundamental relations are called the underlying fundamental relations. These relations carry information on system stability and possible phase transitions. Stability: higher dimensions
If we consider simultaneous fluctuations of energy and volume, the stability condition is that: S(U + ∆U,V + ∆V , N )+ S(U − ∆U,V − ∆V , N ) ≤ 2S(U,V , N )
for any values of ∆U and ∆V.
Local stability criteria are now a little more complex: 2 2 2 ∂2S ∂2S ∂2S ∂ S ∂ S − ≥ ≤ 0 ≤ 0 2 2 0 ∂U 2 ∂V 2 ∂V ∂U ∂U∂V
Determinant of Hessian matrix Physical consequences of stability conditions
Stability conditions result in limitations on the possible values assumed by physically measurable quantities.
1 ∂ ∂ 2S 1 ∂T 1 T = − ≤ 2 = = − 2 2 0 ∂U ∂U T ∂U V ,N T Ncv
Therefore: cv ≥ 0
The molar heat capacity at constant volume must be positive. Stability conditions in the energy representation
From the fact that the internal energy of the system should be at minimum in equilibrium, we can derive the condition for stability in the energy representation:
U (S + ∆S,V + ∆V , N )+U (S − ∆S,V − ∆V , N ) ≥ 2U (S,V , N )
The local stability criteria in the energy representation are:
2 ∂ 2U ∂T ∂ 2U ∂P ∂ 2U ∂ 2U ∂ 2U = ≥ = − ≥ − ≥ 2 0 2 0 2 2 0 ∂S ∂S ∂V ∂V ∂V ∂S ∂S∂V Stability conditions for other thermodynamic potentials
Equations used in Legendre transformations:
∂U (X ) ∂U (P) P = X = − ∂X ∂P
Here P is a generalized intensive parameter conjugate to the extensive parameter X and U(P) is the Legendre transform of U(X). Stability conditions for other thermodynamic potentials
2 ∂X ∂2U (P) ∂P ∂U (X ) = − = 2 ∂X ∂X 2 ∂P ∂P
∂P 1 ∂U 2 (X ) 1 = = − ∂X ∂X ∂X 2 ∂ 2U (P) ∂P ∂P2
Therefore, the sign of the second derivative of the Legendre transform of the internal energy is opposite to that of the internal energy. Stability conditions for other thermodynamic potentials
Exercise: derive stability ∂ 2U ∂2U Internal energy: ≥ 0 ≥ 0 conditions for the ∂S 2 ∂V 2 determinant of the Hessians matrix for ∂2 F ∂ 2 F various thermodynamic ≤ ≥ Helmholtz potential: 2 0 2 0 potentials. ∂T ∂V
2 2 ∂ H ∂ H Note the difference of Enthalpy: ≥ 0 ≤ 0 ∂S 2 ∂P2 stability criteria with respect to extensive and intensive ∂ 2G ∂ 2G Gibbs potential: ≤ 0 ≤ 0 coordinates ∂T 2 ∂P2 Physical consequences of stability conditions
Any of the just written stability conditions generates a constraint on some physical quantity, e. g.:
2 ∂ 2 F ∂ F ∂P 1 ≥ = − = ≥ 0 κT ≥ 0 2 0 2 ∂V ∂V ∂V T VκT
The isothermal compressibility of a stable thermodynamic system is non-negative. Physical consequences of stability conditions
From the Maxwell relations we can also derive:
2 Tvα κ S cv c − c = = P v κ κT T cP Tvα 2 cP − cv = cP ≥ cv ≥ 0 κT
κ S cv = κT ≥ κ S ≥ 0 κT cP