Lecture 10 Thermodynamic Potentials and Legendre Transform 19.09.2019

Lecture 10 Thermodynamic potentials and Legendre transform 19.09.2019

Fys2160, 2019 1 Free energies as thermodynamic potentials q The thermodynamic equilibrium of a system depends on how the Thermodynamic Independent system interacts with its environment. The appropriate Potentials (natural) thermodynamic potential takes into account this interaction Variables through energy, volume and particle exchanges. U (S,V,N) S, V, N H (S,P,N) S, P, N

F (T,V,N) V, T, N q Thermodynamic equilibrium is reached when the appropriate thermodynamic potential is minimized G (T,P,N) P, T, N qFor an isolated system, the thermodynamic potential which is minimized at thermodynamic equilibrium is its internal energy �

Fys2160, 2019 2 Closed system and minimum internal energy � �, �, � , �� ≤ 0 q From the 2nd law of thermodynamics, the entropy change satisfies the inequality

�d� ≥ dU + �d� − �d� �

Hence the change in the internal energy satisfies the thermodynamic inequality �, �, �

�� ≤ �� − �� + ��

Any spontaneous change in the internal energy at fixed �, �, and � will try to minimize the internal energy

��|�,�,� ≤ �

For a reversible change in an isolated system, � is conserved (and minimized): �� = 0

Fys2160, 2019 3 Isolated systems and maximum entropy S �, �, � , �� ≥ 0 q From the 2nd law of thermodynamics

�d� ≥ dU + �d� − �d� �, �, �

Any spontaneous change in the internal energy at fixed U, �, and � will try to maximize the entropy

��|�,�,� ≥ �

For a reversible change in an isolated system, � is conserved (and maximized): �� = 0

Thermodynamic equilibrium:

�� = , � = , � = − �� ,� � �� ,� � �� ,�

Fys2160, 2019 4 Entropy in the ideal gas �� , �, � = � �� , �, � = � �� + �� + �� > � �� = � = �� = → �� = �� ,� �� Pressure has an entropic origin: derived from counting the number of microstates �, �, � �, �, �

q The decrease �� balances the increase ��, i.e −�� = �� < → � < � �� q < → −�� < �� q �(�+�) > 0 → �� > 0

q A spontaneous change in volume will tend to increase the total entropy

� ��

Fys2160, 2019 5 Systems at constant � and Enthaly � �, �, � , �� ≤ 0 �, �, � �

The enthalpy � is the thermodynamic potential given by the internal energy of a system plus the work needed to keep the system at a given �, � = � + �� Thermodynamic identity: �� = �� + �� + �� q Thermodynamic inequality for any infinitesimal process (reversible «=», irreversible «>»)

�� ≤ �� − �� + �� → �� ≤ �� + �� + ��

Ø Any change in enthalpy H when �, P, and � are fixed is ��|�,�,� ≤ �. This is to lower � to its miminum value at equilibrium

Ø Reversible process at fixed �, �, and N means that � is conserved: ��|,, = 0

Ø If we are fixing P and N, but allow entropy to vary, then the change in enthalpy equals the reversible heat exchange: ��|, = �� = ��

Fys2160, 2019 6 Equilibrium at constant � and minimum � �, �, � , dF ≤ 0

The Hemholtz free energy � is the thermodynamic potential given by the internal energy of a system minus the heat exchanged with the thermal bath fixed �, � = � − �� q Thermodynamic identity for an infinitesimal reversible process � d� = −�� − �� + �d� �, �, � � q Thermodynamic inequality for any infinitesimal process (reversible «=», irreversible «>»)

�� − � �� ≤ −�� − �� + �d�

�� ≤ −�� − �� + ��

Ø Any change in � when T, V, and � are fixed must be ��|�,�,� ≤ �. This is to lower � to its miminum value at equilibrium

Ø Reversible process at fixed � , �, and N means that � is conserved: ��|,, = 0

Ø Reversible process at fixed �, and N means that : ��|, = −�� Ø Changes in � at a fixed T equals to the available mechanical work (free energy) that a system can do.

Fys2160, 2019 7 Helmholtz free energy � �, �, � in the ideal gas

q � = � − �� → � = − �� ln + ln +

� � �� (�, �, �) = −�� + �� + � � (�, � , �) � (�, � , �) � ��

q � = − → � = �� ln � → � = , ,

Ø Slopes are negative (higher volume, higher entropy, lower F) steeper

Ø − > − → gas A expands into gas B to lower its free energy �

Ø � > � the gas with higher pressure expands Δ� = −ΔV

Ø The reduction in F is larger than the increase in F

−Δ�> Δ� → Δ � + � < 0

Fys2160, 2019 8 Systems at constant � �� and Gibbs free energy G(�, �, �) q The Gibbs free energy � is the thermodynamic potential given by the internal energy of a system minus the available heat exchange plus the work done on the reservoir � = � − �� + �� q It is the energy to create something (system) out of nothing and put into an environment when there is “free” heat q An infinitesimal change in � is due to independent, infinitesimal changes in �, �, � D. Schroeder d� = dT + d� + d� , , , � Thermodynamic equilibrium:S = − , � = and � = �, �, � , , , �, � q Thermodynamic identity for an infinitesimal reversible process

�� = −�� + �� + ��

Fys2160, 2019 9 Systems at constant � �� and Gibbs free energy G(�, �, �) q Thermodynamic identity for an infinitesimal reversible process �

�, �, � d� = −�� + �� + �dN �, � q Thermodynamic inequality for any infitesimal process (reversible «=», irreversible «>»)

�� − � �� + � �� ≤ −�� + �� + �d�

�� ≤ −�� + �� + ��

q Any change in the Gibbs free energy � when T, P, and � are fixed must be ��|�,�,� ≤ � q Reversible process at fixed � , �, and N means that � is conserved: ��|,, = 0 q Reversible process at fixed �, and P means that : ��|, = ��. Chemical potential is defined as the Gibs free energy per particle. q Changes in � at a fixed T and P equals to the available chemical work (free energy) to increase or decrease the number of particles, or any other work which is not mechanical

Fys2160, 2019 10 Gibbs free energy G(�, �, �) and chemical potential � q Gibbs free energy G(�, �, �) is an extensive thermodynamic potential. Since, � �� are intensive variable, the only extensive variable that it depends on is �, therefore the � extensivity property means that the Gibbs free energy of N particles equals the sum of Gibbs free energy of each particle � �, �, � = �� , �, 1 = �� (�, �) �, �, � Combining this with the definition of the chemical potential � = ��

,, � = = � �, �, 1 = ,

,, q � �, � = , � �, �, � = ��(�, �) Chemical potential is the Gibbs free energy per unit particle at fixed pressure P and temperature T

Ø This means that when we add a particle, the system’s Gibbs energy increases by an amount equal to � Ø By adding more particles at fixed P and T, we don’t change the value of �: each particles comes with the same amount of energy independent of the particle density in the system, when we keep the pressure and temperature constant!

Fys2160, 2019 11 Ideal gas: G(�, �, �) and �

q Gibbs free energy G(�, �, �) � � = � − �� + �� = � + �� , �, � = −�� + � + �� , �, � � � = �� , �, � = −�� q Chemical potential �(�, �) � � �� , � = = −�� q Change in chemical potential Δ�(�) at fixed �

� �� = � � − �(��) = ��

Fys2160, 2019 12 Extensive thermodynamic potentials:

Thermodynamic potentials are extensive functions: invariant under dilation

• U S, V, N = N U S, V , S = S/N and V = V/N • H S, P, N = N H S, P , S = S/N

• F T, V, N = N F T, V , V = V/N • G T, P, N = N G(T, P)

Fys2160, 2019 13 Thermodynamic potentials are related by Legendre transforms

• The extensive variables (�, �, �) and intensive variables (�, �, �) are conjugate variables When one is an independent (control) variable fixed by the surroundings, its conjugate variable is a derivative:

�� When S is � �� ℎ�� = �� , • Variables like � are hard to control experimentally, hence it is better to transform the internal energy � into another thermodynamic potential that has instead � as natural variable • Legendre transform does precisely this: it is a transformation from one thermodynamic potential to another by a change between the conjugate variables Example: the transformation from internal energy �(�, �, �) to entalphy �(�, �, �)

� �, �, � → �(�, �, �)

�� , �, � = � �, �, � + ��, �ℎ�� = − �� ,

Fys2160, 2019 14 Legendre transform: graphic intepretation

Given a function of x with the slope u �

�� , �� = ��

�� If � = � � − ��, then = − � = � → �(�) �� (�) } �� (�) The Legendre transform is � � � = � � − ��

Fys2160, 2019 15 Legendre transform Suppose we have a function F �, � where z = then the function � �, � is related to F �, � by the Legendre transform �(�, �) = �(�, �) − ��

Proof: �� (� − zx) = �� + �� − �� − ��

�� − zx = �� − ��, �� ℎ�� z = �� The y-dependence is not changed thus = . Also �(�, �) = � �, � − �� → � = −

�� − zx = �� − �� → � � − zx = �� + �� = ��

� � − �� = �� → � = �

Fys2160, 2019 16 Legendre transforms: Thermodynamic identities

U �, �, � dU = �� − �� + ��

�� dF = −�� − �� + �� H �, �, � = � �, �, � + ��, � = − �� , d� = �� + �� + �� d� = −�� + �� + �� , �, � = � �, �, � − ��, � = �� ,

G �, �, � = �(�, �, �) − �� + ��

Fys2160, 2019 17 Thermodynamic square

Extensive variables Intensive variables

V �(�, �) T

� �, � = � �, � + �� (�, �) Mechanical �(�, �) � �, � = � �, � + ��

S �(�, �) P

Fys2160, 2019 18 Thermodynamic square Extensive variables Intensive variables

V �(�, �) T

�(�, �) �(�, �)

S �(�, �) P

q Derivative of the thermodynamic potential with respect to one of its arguments while keeping the other constant is determined by going along a diagonal line either with(+) or against(-) the direction of the arrow

�� = − = − ��

�� = = ��

Fys2160, 2019 19 Thermodynamic square

Extensive variables Intensive variables

V �(�, �) T � �, � = � �, � − ��

� �, � = � �, � − �� (�, �) �(�, �) Thermal

S �(�, �) P

Fys2160, 2019 20 Thermodynamic square Extensive variables Intensive variables

V �(�, �) T

�(�, �) �(�, �)

S �(�, �) P

�� = − = − ��

�� = = ��

Fys2160, 2019 21 Maxwell’s relations �� = �� − ��

�� = �� + ��

q Thermodynamic potential � is a state variable, which implies that =

� �� = → = − �� q Other Maxwell relaxations follow from the other thermodynamic potentials q Used to compute relations between response functions: heat capacities, thermal expansion coefficients

Fys2160, 2019 22 Maxwell’s relations �� = −�� − ��

�� = �� + �� q Find the Maxwell’s relation corresponding to �(�, �)

Fys2160, 2019 23