CHEMICAL POTENTIAL I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. I plan to go through these slides in one 90-minute lecture. Zhigang Suo, Harvard University

The play of thermodynamics 2 energy space matter charge ENTROPY temperature pressure chemical potential electrical potential heat capacity compressibility capacitance Helmholtz function enthalpy Gibbs function thermal expansion Joule-Thomson coefficient

plan Definition of chemical potential Examples of chemical potential Equilibrium of two systems Equilibrium of a chemical reaction Equilibrium of phases 3

Model an open system as a family of isolated systems 4 liquid 2O2O gasa family of isolated systems of four independent variables: U, V, N H2O, N N2 The wine contains many species of molecules (components) and two phases. The wine is an open system, exchanging energy, space, and two components with the rest of the world. Make the wine an isolated system by insulating the bottle, jam the piston, and shut the valves. A system isolated for a long time reaches a state of thermodynamic equilibrium. Define the entropy of the isolated system: S = log (number of quantum states). Isolating the wine at various values of (U,V, N H2O, N N2 ), we obtain a family of isolated systems of four independent variables. Model the family of isolated systems by function S(U,V, N H2O, N N2 ). liquid weights fire gas valve for N 2 open system valve for H 2 O H 2 O tank N 2 tank

Derivative 1. an operation in calculus 2. a thing based on something else 5 liquid 2O2O gas a family of isolated systems S(U, V, N A, N B )

Name derivatives by Gibbs equations 6 Define temperature: Define pressure: Define chemical potential: Define chemical potential Calculus:

7 These equations define the two chemical potentials. Each chemical potential is a child of entropy and a component. Chemical potential of a component is an intensive property of a system. T appears in the definition by convention. Negative sign appears in the definition by convention. Thus, an isolated system increases entropy when a component goes from a place of high chemical potential to a place of low chemical potential. Grammar: The chemical potential of a component in a system (e.g.,  A is the chemical potential of water in the wine, and  B is the chemical potential of nitrogen in the wine). Notes on chemical potentials

Why don’t we know chemical potential as well as temperature? Blame our parapets. Our parents tell us a lot about temperature, but never tell us about chemical potential. (But they do tell us about humidity, and smells of many kinds.) Blame our world. The world confuses us with many species of molecules. 8

Breed equations (Gibbs 1878) 9 Gibbs equation: Solve for dU: Calculus:

Breed more equations 10 Gibbs equation: A Legendre transform defines the Gibbs function: Combine the above two equations: Calculus:

On being extensive 11 Gibbs equation All independent variables are extensive: Increase all extensive properties proportionally: Derivative with respect to : Solve for U: Definition of the Gibbs function: Combine the above two equations:

plan Definition of chemical potential Examples of chemical potential Equilibrium of two systems Equilibrium of a chemical reaction Equilibrium of phases 12

Pure substance 13 Define Gibbs function per molecule (or per mole): Define chemical potential of a pure substance: Compare the two definitions: Recall the definition of the Gibbs function: Recall the Gibbs equation: 1.For a pure substance, we know how to measure TVPUS. 2.From TVPUS we can calculate the chemical potential . 3.Chemical potential requires the absolute entropy.

Incompressible pure substance 14 Gibbs equation: Incompressibility: Integration:

Ideal gas 15 Gibbs equation: Law of Ideal gas: Integration: Look up values: P 0 = 1 atm

Ideal-gas mixture 16 partial pressure of component A chemical potential of component A in an ideal-gas mixture chemical potential of pure ideal gas A at 1 atm The chemical potential of a component in an ideal-gas mixture is the same as the chemical potential of the component in the pure gas, provided we use the partial pressure of the component.

Chemical potential of water in moist air relates to relative humidity Model the moist air as an ideal-gas mixture 17

plan Definition of chemical potential Examples of chemical potential Equilibrium of two systems Equilibrium of a chemical reaction Equilibrium of phases 18

Two systems exchanging energy, space and molecules Isolated system conserves energy, space, and matter over time: dU’ + dU’’ = 0. dV’ + dV’’ = 0 dN A ’ + dN A ’’ = 0 dN B ’ + dN B ’’ = 0 Isolated system not in equilibrium generates entropy over time: dS’ + dS’’ > 0 Isolated system in equilibrium keeps entropy constant over time: dS’ + dS’’ = 0 19 U’, V’, N A ’, N B ’ S’(U’, V’, N A ’, N B ’) isolated system U’’, V’’, N A ’’, N B ’’ S’’(U’’, V’’, N A ’’, N B ’’) diathermal, moving, permeable to components A and B open system (‘)open system (‘’)

Equilibrium of two systems 20 U’, V’, N A ’, N B ’ S’(U’, V’, N A ’, N B ’) isolated system U’’, V’’, N A ’’, N B ’’ S’’(U’’, V’’, N A ’’, N B ’’) diathermal, moving, permeable to components A and B open system (‘)open system (‘’) Thermal equilibrium: Mechanical equilibrium: Chemical equilibrium of component A: Chemical equilibrium of component B:

Measuring chemical potential of a component in a system chemical potential of water in the wine Chemical potential affects everything. Everything measures chemical potential 21 liquid weights fire gas open system A membrane permeable to H 2 O only pure H 2 O  (T,P)

Hygrometer humidity sensors First inventor: Johann Heinrich Lambert (1755) Humidity affects everything. Everything is a hygrometer. Today’s opportunity: The Internet of things Bimaterial strip Hair-tension hygrometer Wet-bulb and dry-bulb Dew-point hygrometer Capacitor Resistor Thermal conductivity Weight

Sensors for chemical potentials of various components Humidity sensor pH sensor Oxygen sensor CO 2 sensor Electronic nose 23

24 Henry’s law (1803) N 2 dissolved in water y N2 N 2 in air P N2

25 Solubility N 2 dissolved in rubber mole/volume N 2 in air P N2

plan Definition of chemical potential Examples of chemical potential Equilibrium of two systems Equilibrium of a chemical reaction Equilibrium of phases 26

27 weights Reaction thermal reservoir, T piston Fix T,P Change U,V N A, N B, N C, N D, reaction chamber A chemical reaction conserves the number of atoms in each species ( i are stoichiometric coefficients): Increment of the number of each component (  is the degree of reaction): Conservation of energy: Entropy is additive: Isolated system (IS) Q out

Equilibrium of a reaction 28 Conservation of atoms: Conservation of energy: Entropy is additive: Condition of equilibrium: Calculus and definitions: Chemical equilibrium:

Ideal-gas reaction 29 Chemical reaction: Condition of equilibrium: Chemical potential of a component in an ideal- gas mixture: Define Condition of equilibrium: Equilibrium constant:

van’t Hoff equation 30 Equilibrium constant: Algebra: Calculus: Recall: van’t Hoff equation:

31 Simultaneous reactions Reaction 1 Reaction 2

plan Definition of chemical potential Examples of chemical potential Equilibrium of two systems Equilibrium of a chemical reaction Equilibrium of phases 32

Equilibrium of two phases 33 U’, V’, N A ’, N B ’ S’(U’, V’, N A ’, N B ’) isolated system U’’, V’’, N A ’’, N B ’’ S’’(U’’, V’’, N A ’’, N B ’’) phase (‘)phase (‘’) Thermal equilibrium: Mechanical equilibrium: Chemical equilibrium of component A: Chemical equilibrium of component B: In equilibrium, the two phases have the same temperature, the same pressure, and the same chemical potential of each component. A total of 2 + C equations. C = number of components.

34 IV = 2 + C - PH IV = number of independent variables C = number of components PH = number of phases in equilibrium The Gibbs phase rule C components: 1, 2,…, C PH phases: (‘), (‘’),… Composition of phase (‘): y’ 1, y’ 2,…y’ C-1 Composition of phase (‘’): y’ 1, y’ 2,…y’ C-1 … All phases have the same T and the same P. 2 variables Total number of number fractions: PH(C-1) Chemical potential of each component is the same in all phases: (PH-1)C equations IV = 2 + PH(C-1) –(PH-1)C

35 Gibbs rule: IV = 3 - PH IV = number of independent variables PH = number of phases in equilibrium. The Gibbs phase rule Pure substance, C = 1 Number of phases in equailibriumPHIV Single phase12 Two phases in equilibrium (two-phase boundary)21 Three phases in equilibrium (triple point)30

36 Gibbs rule: IV + PH = 4 IV = number of independent variables PH = number of phases in equilibrium. Two-component (binary) system, C = 2 Set pressure at a fixed pressure A diagram of two variables, T and y B

Three phases in equilibrium in a binary mixture: eutectic point 37

Water-salt phase diagram 38

39 Gibbs rule: IV + PH = 5 IV = number of independent variables PH = number of phases in equilibrium. Three-component (tertiary) system, C = 3 Set pressure at a fixed value Set temperature at a fixed value Gibbs triangle: Each point in the triangle represents a composition,, y B and y C Stainless steel phase diagram at 900 degrees Celsius (ASM 1-27)

Summary Chemical potential is a child of entropy and a component. Chemical potential (of a species of molecules) in a pure substance coincides with the Gibbs function per molecule. Chemical potential of a component in an ideal- gas mixture is the same as that of the pure component, provided we use partial pressure of the component. Use chemical potential to analyze equilibrium of systems, reactions, and phases. 40