Statistical Mechanics Lecture on Thermodynamic Potentials

In this Statistical Mechanics lecture, the focus is on Thermodynamic Potentials such as internal energy, enthalpy, Helmholtz energy, and Gibbs' energy. The lecture covers properties of thermodynamic potentials, derivative relationships, reversible and irreversible processes, entropy, and inequalities associated with thermodynamic potentials.

• Statistical Mechanics
• Thermodynamic Potentials
• Entropy
• Reversible Processes
• Irreversible Processes

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1. PHY 770 -- Statistical Mechanics 12:30-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: http://www.wfu.edu/~natalie/s14phy770 Lecture 3-4* -- Chapter 3 Review of Thermodynamics continued 1. Properties of thermodynamic potentials 2. Response functions 3. Thermodynamic stability *Special double lecture, starting at 11 AM on 1/21/2014 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 1

2. 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 2

3. 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 3

4. 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 4

5. Summary of thermodynamic potentials Potential Variables Total Diff Fund. Eq. U S,X,Ni i i = + + U TS YX iN = + + idN dU TdS YdX i i H S,Y,Ni i = + idN dH TdS XdY = H U YX i A T,X,Ni i = A U TS = + + idN dA SdT YdX i G T,Y,Ni i = + idN dG SdT XdY = G U TS YX i T,X, i i i = U TS iN = + idN d SdT YdX i i 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 5

6. Derivative relationships of thermodynamic potentials S U , ( energy Internal N , : ) X i U U U = = = T Y i S X N , , X N S N i , , S X N i i j N Enthalpy ( , , : ) H S Y i H H H = = = T X i S Y N , , Y N S N N i Y , , S N i i j Helmholz energy free ( , , : ) A T X i A A A = = = S Y i N T X N , , X N T N i , , T X N i i j Gibb' energy free s ( , , : ) G T Y i G G G = = = S X i T Y N , , Y N T N i Y , , T N i i j 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 6

7. Properties of thermodynamic potentials Potential in the sense that energy can be stored and retrieved through thermodynamic work Equalities reversible processes Inequalities general processes d Q Entropy = + dS d S i T Entropy production due to irreversible processes Reversible entropy contribution Q d 0 di S dS T 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 7

8. Inequalities associated with thermodynamic potentials continued Entropy T d Q = + dS d S i = For a thermally isolated system : 0 d d Q = 0 dS S i At equilibrium when dS=0 , S is a maximum 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 8

9. Inequalities associated with thermodynamic potentials continued Internal energy i = Since TdS d Q dU YdX dN i i i + + dU TdS YdX dN i i N isolated an For system dU with fixed , and , S X i At equilibrium when dU=0 , U is a minimum (for fixed S, X, and {Ni}) 0 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 9

10. Inequalities associated with thermodynamic potentials continued Enthalpy i + i dH TdS XdY dN i N isolated an For system dH with fixed , and , S Y i 0 At equilibrium when dH=0 , H is a minimum (for fixed S, Y, and {Ni}) 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 10

11. Inequalities associated with thermodynamic potentials continued Helmholz free energy i + + i dA SdT YdX dN i N isolated an For system dA with fixed , and , T X i 0 At equilibrium when dA=0 , A is a minimum (for fixed T, X, and {Ni}) 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 11

12. Inequalities associated with thermodynamic potentials continued Gibbs free energy i + i dG SdT XdY dN i N isolated an For system dG with fixed , and , T Y i 0 At equilibrium when dG=0 , G is a minimum (for fixed T, Y, and {Ni}) 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 12

13. Inequalities associated with thermodynamic potentials continued Grand potential i + + d SdT YdX N d i i isolated an For system d with fixed , and , T X i 0 At equilibrium when d = , is a minimum (for fixed T, X, and { i}) 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 13

14. Example: Consider a system having a constant electrostatic potential at fixed T and P containing a fixed number of particles {Ni}fori=1, Find the change in the Gibbs free energy when dN particles, each with charge q are reversibly added to the system. = 1 = + + + dG SdT VdP dN q dN dN i i 1 i N = constant For , , : T P i ( ) + dG q dN 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 14

15. Thermodynamic response functions -- heat capacity d Q = C dT = d Q dU YdX dN i i i N = Assuming U ( , , : ) U T X i U U U = + + d Q dT Y dX dN i i T X N i , , X N T N i , , T X N i i j N For heat capacity constant at , : X i U = C , X N T i , X N i 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 15

16. Thermodynamic response functions -- heat capacity -- continued Q d C = dT = d Q dU YdX dN i i i N = Assuming U ( , , : ) U T X i U U U = + + d Q dT Y dX dN i i T X N i , , X N T N i , , T X N i i j N For heat capacity constant at Y , : i X X X = + + Note that : dX dT dY dN i T Y N i , , Y N T N i Y , , T N i i j X N = constant At Y , : dX dT i T , Y N i U U X d = + Q dT Y dT T X T , , , X N T N Y N i i i U X = + C C Y , , Y N X N X T i i PHY 770 Spring 2014 -- Lecture 3 & 4 , , T N Y N i i 1/21/2014 16

17. N i = of ideal gas : i U kT Example: Heat capacity 1 i N i = PV kT i N For heat capacity constant at , X V i U = C , V N T i , V N N i = i C k , V N at 1 i i i N For heat capacity constant Y , : P i U V = + + C C P , , P N V N V T i i , , T N P N i i N N i i i = + = i i i 1 k N k k i 1 i i 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 17

18. Thermodynamic response functions -- heat capacity -- continued = Alternate analysis = in terms of : d S Q TdS N Assuming ( , , : ) S S T X i S S U = + + d Q T dT dX dN i T X N i , , X N T N i , , T X N i i j N For heat capacity constant at , : X i S d = Q T dT T , X N i 2 S A 2 = = C T T , X N T T i , X N , X N i i N For heat capacity constant at Y , : i ( After some algebra - -) 2 S G 2 = = C T T , Y N T T i , Y N , Y N i i 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 18

19. Thermodynamic response functions -- mechanical response Results stated here without derivation: 2 X G 2 = = Isothermal susceptibl ity : , T N Y Y i , T N , T N i i 2 X H = = Adiabatic susceptibl ity : , S N 2 Y Y i , S N , S N i i X = Thermal expansivit y : , Y N T i , Y N i C , , T N Y N = Relationsh between ips response ) ) T = functions ) )2 i N : i i C , , S N X N ( ( ( ( i i 2 = C C T , , , , T N Y N X N Y N i i i i C , , , , Y N T N T N Y i i i 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 19

20. Thermodynamic response functions -- mechanical response Typical parameters: Isothermal susceptiblity isothermal compressibility V V , Adiabatic susceptiblity adiabatic compressibility 2 1 1 G 2 = = , T N P V P i T N , T N i i 2 1 1 V H 2 = = , S N V P V P i , S N , S N i i Thermal expansivity Thermal expansivity 1 V = , P N V T i , P N i 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 20

21. Example: mechanical responses for an ideal gas N = For ideal gas : U kT 1 = PV NkT 1 1 V = = Isothermal compressib ility : T V P P , T N i 1 V = Adiabatic compressib ility : S V P S 1 0 = = Note that : PV P V 0 S P 1 1 V = = Thermal expansivit y : P V T T P 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 21

22. Stability analysis of the equilibrium state = U U , , , P V T , , , P V T A B B N B B A A A , , = V V N A B iB iB iA iA = N N iA iB Assume that the total system is isolated, but that there can be exchange of variables A and B. 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 22

23. Stability analysis of the equilibrium state , , , P V T , , , P V T B N B B A A A If we can assume : , , N iB iB = iA iA U U A B = V V A B = N N iA iB Analysis of entropy fluctuatio ns : S S S = A i 1 = + + S U V N total i U V N , B i 1 P P i = + + iA iB A B U V N A A iA T T T T T T A B A B A B = = = equilibriu At Note that this result does not hold for non-porous partition. m : , , T T P P A B A B iA iB 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 23

24. Stability analysis of the equilibrium state continued Note that the equilibrium condition can be a stable or unstable equilibrium. f f x x Stable function (convex) Unstable function (concave) 2 d f 2 d f 0 0 2 dx 2 dx 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 24

25. Stability analysis of the equilibrium state -- continued Extending analysis the of entropy fluctuatio multi a in ns - partitione system d : analyzing leading second order term s 1 S S S i = + + S U V N i total 2 U V N i 2 2 2 S S S S i = + + 2 U V N i U U V U N U i relation law first the Using and other identities : i = + T S U P V N i i second the order expansion becomes : 1 i = + S T S P V N i i total 2 T 1/21/2014 PHY 770 Spring 2014 -- Lecture 3 & 4 25