PRESSURE I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. These slides follow closely my written notes ( I went through these slides in three 90-minute lectures. 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 A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 3

A half bottle of wine 4 The reservoir of energy is a thermal system, modeled by a function S R (U R ), and by a fixed temperature T R. The weight applies a fixed force f weight. Pressure P weight = f weight /A. The entropy of the weight S weight is constant as the piston moves. The half bottle of wine is a closed system, modeled by a family of isolated systems of two independent variations, using function S(U,V). liquid f weight vapor closed system reservoir of energy, T R adiabatic diathermal 2O2O Model the wine as a family of isolated systems S(U,V) liquid vapor

1.Construct an isolated system with an internal variable, x. 2.When the internal variable is constrained at x, the isolated system has entropy S(x). 3.After the constraint is lifted, x changes to maximize S(x). 5 The basic algorithm of thermodynamics Entropy = log (number of quantum states). Entropy is additive.

Construct an isolated system with internal variables 6 Isolated system conserves space (kinematics): V = Ah Isolated system conserves energy: U composite = U + U R + f weight h = constant Entropy of the isolated system is additive: Entropy of the reservoir: liquid f weight vapor closed system S(U,V) reservoir of energy, T R adiabatic diathermal Isolated system with internal variables U and V

Entropy of the reservoir of energy 7 reservoir of energy S R, U R,T R A reservoir of energy is a thermal system, modeled by a function S R (U R ) and by a fixed temperature, T R. Clausius-Gibbs equation: Integration: Conservation of energy: Entropy of the reservoir:

Entropy of the isolated system: Calculus: Maximize S composite to reach Thermodynamic equilibrium: Temperature and pressure of the wine: Gibbs equation: Internal variables change to maximize the entropy of the isolated system 8 liquid f weight vapor reservoir of energy, T R wine reservoir weight

Count the number of quantum states of a half bottle of wine Experimental determination of S(U,V) 9 Measure V by geometry Measure U by calorimetry (e.g., pass an electric current through a resistor) Measure T by thermometry Measure P by force/area Obtain S by integrating the Gibbs equation 2O2O liquid vapor A family of isolated systems Five properties: U,V,S,P,T Two independent variables, U,V

10 liquid f weight vapor reservoir of energy, T R 2O2O liquid vapor Each member in the family is a system isolated for a long time, and is in a state of thermodynamic equilibrium. Transform one member to another by fire (heat) and weight (work). Five properties: U,V,S,P,T Name all states of thermodynamic equilibrium by two independent variables (U,V) Three functions (equations of state): S(U,V), T(U,V), P(U,V) Once S(U,V) is known, two other equations of state are determined by the Gibbs equations: isolated system U,V,S,P,T closed system Model a closed system by a family of isolated systems

Plan A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 11

Calculus: a function of two variables 12 x y z z(x,y) (x,y) x y A pair of values (x,y) corresponds to a point in the (x,y) plane. The function z(x,y) corresponds to a surface in the (x,y,z) space.

Calculus: partial derivative 13 x y z z(x,y) slope (x+dx, y)(x,y) Each partial derivative corresponds to a slope of a plane tangent to the z(x,y) surface. Partial derivatives: Increment:

14 A pair of values corresponds to a point in the (U, V) plane. S(U,V) U V A state (U,V) U V S S(U,V) (U,V) The function S(U,V) corresponds to a surface in the (U, V,S) space.

15 Given a state (U,V), draw a plane tangent to the surface S(U,V) The two slopes of the tangent plane give T(U,V) and P(U,V). T(U,V) and P(U,V) U V S (U,V) (P,T)

V Entropy of the isolated system: Internal variables change to maximize entropy: Graphic derivation of the condition of equilibrium between the wine, reservoir and weight 16 S liquid f weight vapor reservoir of energy, T R S(U,V) surface wine tangent plane plane of slopes 1/T R and P weight /T R Reservoir, weight U wine reservoir, weight An isolated system of internal variables U,V

Plan A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 17

Three phases of a pure substance 18 u v liquid solid gas intensive-intensiveextensive-intensiveextensive-extensive T liquid solid gas critical point triple point P

19 Equivalent statements z(x,y) is convex Each tangent plane touches the surface at a single point. Roll the tangent plane with two degrees of freedom One-to-one correspondence: (M,N)  (x,y). Calculus: convex function x y z A function of two variables: Partial derivatives: tangent plane of slopes M and N ( x,y ) z(x,y)z(x,y)

20 x y z (M,N) Calculus: an example of nonconvex function A tangent plane touches the surface at two points. Roll the tangent plane with one degrees of freedom One-to-two correspondence: (M,N)  (x’,y’) and (x’’,y’’). z(x’,y’) z(x’’,y’’) tie line

21 S(U,V) is smooth and convex. Each tangent plane touches the surface at a single point. Roll the tangent plane with two degrees of freedom. point-to-point (P,T)  (u,v). u v s Gibbs relations tangent plane of slopes 1/T and P/T (u,v) s(u,v) (P,T) A phase of a pure substance

Two co-existent phases of a pure substance 1. Each phase has its own smooth and convex s(u,v) function. 22 solid s'(u’,v’) liquid u v s s’’(u’’,v’’)

23 liquid solid isolated system of fixed u and v u = (1 - x)u’ + xu’’ v = (1 - x)v’ + xv’’ s = (1 - x)s’ + xs’’ s'(u’,v’) u v s s’’(u’’,v’’) (u’’,v’’)(u’,v’) (u,v) Two co-existent phases of a pure substance 2. Rule of mixture defines a line in the (s,u,v) space.

24 u v s (u’’,v’’)(u’,v’) (u,v) u v solid liquid solid-liquid mixture liquid solid (P,T) tie line Two co-existent phases of a pure substance 3. Isolated system conserves energy and volume, but maximizes entropy. Roll common tangent plane with one degree of freedom. One-to-many correspondence: (P,T)  (all states on the tie line)

25 u v s s(u’’,v’’) s(u’,v’) liquid gas critical point dome (P,T) liquid gas Two co-existent phases of a pure substance 4. Liquid and gas share a smooth but non-convex surface s(u,v) Roll common tangent plane with one degree of freedom. point-to-line (P,T)  (u,v)

Three co-existent phases of a pure substance Each phase has its own s(u,v) function. Common tangent plane cannot roll. point-to-triangle (P,T)  (u,v) 26 solid s'(u’,v’) u v s s’’(u’’,v’’) s’’’(u’’’,v’’’) liquid vapor Rule of mixture u = (1 – x - y)u’ + xu’’ + yu’’’ v = (1 – x - y)v’ + xv’’ + yv’’’ s = (1 – x - y)s’ + xs’’ + ys’’’ (P,T)

27 Experiment by Andrews (1869). Described by Gibbs (1873). A clay model built by Maxwell (1874) Gibbs’s thermodynamic surface S(U,V) Solid phase has its own smooth and convex s(u,v) function. Liquid and gas phases share a smooth but non-convex s(u,v) function. Use tangent planes to make S(U,V) surface convex (convexification) entropy energy volume gas liquid solid critical point liquid solid gas volume energy

28 Single phase The tangent plane touches the s(u,v) surface of a single phase. Roll the tangent plane with two degrees of freedom. T and P change independently. A single phase corresponds to a region in the T-P plane. point-to-point (P,T)  (u,v) Two coexistent phases The tangent plane touches the s(u,v) surfaces of two phases. Roll the tangent plane with one degree of freedom. T depends on P. Two coexistent phases correspond to a curve in the T-P plane. Phase boundary. point-to-line (P,T)  (u,v) Three coexistent phases The tangent plane touches the s(u,v) surfaces of three phases. The tangent plane cannot roll. T and P are fixed. Three coexistent phases correspond to a point in the T-P plane. Triple point. point-to-triangle (P,T)  (u,v) T liquid solid gas critical point triple point Gibbs’s phase rule for a pure substance Values of (P,T) give the slopes of a plane tangent to s(u,v) surface v liquid solid gas u P critical point extensive-extensive Intensive-intensive

Plan A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 29

Ideal gas When molecules are far apart, the probability of finding a molecule is independent of the location in the container, and of the presence of other molecules. Number of quantum states of the gas scales with V N Definition of entropy S = k B log  Gibbs equations: Equations of state for ideal gases: 30 U,V,S,P,T,N, 

31 Per molePer unit mass For water:For all substances:

Heat capacity of ideal gases 32

Entropy of ideal gas 33 Gibbs equation: Laws of ideal gases: Increment for ideal gas: Approximation of constant specific heat isentropic process:

Plan A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 34

Osmosis 35 membrane solution solvent In a liquid, osmosis is balanced by gravity

Osmosis 36 In a bag (or a cell), osmosis is balanced by elasticity

Theory of osmosis When solute particles are far apart, the probability of finding a particle is independent of the location in the container, and of the presence of other particles. Number of quantum states of the solution scales with V N Definition of entropy S = k B log  Gibbs equation: Osmotic pressure: 37 membrane solution solvent

Plan A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 38

39 2O2O liquid vapor Isolated system U,V,S,P,T liquid f weight vapor fire S(U,V) Model a closed system as a family of isolated systems Each member in the family is a system isolated for a long time, and is in a state of thermodynamic equilibrium. Transform one member to another by fire (heat) and weights (work). Five thermodynamic properties: P, T, V, U, S. Two independent variables, chosen to be U, V. Three equations of state: S(U,V), P(U,V), T(U,V). Determine S(U,V) by experimental measurement. Obtain P(U,V) and T(U,V) from the Gibbs equations: Eliminate U from P(U,V) and T(U,V) to obtain P(V,T). closed system

40 Calculus: Notation Function: Partial derivative: Name partial derivatives: Derivatives are functions:

U(S,V) When V is fixed, S increases with U. Invert S(U,V) to obtain U(S,V). 41 Gibbs equation: Another Gibbs equation Calculus: More Gibbs equations U V S S(U,V) U(S,V) V(U,S)

42 2O2O liquid vapor 2O2O liquid vapor Isolated system state U,V,S,P,T state U+dU, V+dV, S+dS. P+dP, T+dT liquid f weight vapor fire Going between two states of thermodynamic equilibrium via any process A special type of process to go from one state to another Slow. Quasi-equilibrium process. Reversible process TdS reversible heat through a quasi-equilibrium process PdV reversible work through a quasi-equilibrium process Gibbs equations: First law: dU = heat + work

43 Calculus: Legendre transform Turn a derivative into an independent variable A function of two variables: Partial derivatives: Increment: Define a Legendre transform: Product rule: Increment: Partial derivatives: A new function of two variables:

Enthalpy H(S,P) Define H = U + PV 44 Calculus: Gibbs equation: Combine the above: Calculus: More Gibbs equations: liquid f weight vapor reservoir of energy, T R Include the weight in the system Invert to obtain V(S,P). Convert U(S,V) to U(S,P). Obtain H(S,P)

Helmholtz function F(T,V) Define F = U - TS 45 Calculus: Gibbs equation: Combine the above: Calculus: More Gibbs equations: Invert to obtain S(T,V). Convert U(T,V) to U(T,P). Obtain F(T,V)

Gibbs function G(T,P) Define G = U - TS + PV G = H – TS = F - PV 46 Calculus: Gibbs equation: Combine the above: Calculus: More Gibbs equations:

Heat capacity under two conditions 47 liquid f weight vapor 2O2O liquid vapor fire

Name more partial derivatives 48 Coefficient of thermal expansion: Isothermal compressibility: Joule-Thomson coefficient:

Calculus: second derivatives 49

Maxwell relation 50 Gibbs equation: Gibbs relations: Maxwell relation:

Mathematical manipulation of dubious value 51 Find a reason to be unhappy with Gibbs equations: Make up a excuse to study a different function: Calculus: Clausius-Gibbs equation: Constant V: Maxwell relation: A pointless equation:

Another pointless equation 52 A pointless equation: Gibbs equation: Another pointless equation:

Breed thermodynamic properties like rabbits A single function S(U,V) produces all other functions. Inbreeding functions! Obtain T(U,V) and P(U,V) from the Gibbs equations, Eliminate U from T(U,V) and P(U,V) to obtain P(V,T). Invert S(U,V) to obtain U(S,V). Use Legendre transform of U(S,V) to define H(S,P), F(T,V), G(P,T). Invert T(U,V) to obtain U(T,V). Name lots of partial derivatives: C V (T,V), C P (T,P),  (T,P),  (T,P),  (P,H),… 53

Plan A system with variable energy and volume Graphic representations Theory of co-existent phases Theory of ideal gases Theory of osmosis Breed properties and equations of state Basic algorithm of thermodynamics in terms of free energy 54

S(U,V,Y) Fix U and V, but let Y change. The system is an isolated system with an internal variable Y. Y changes to maximize S(U,V,Y). 55 liquid vapor Y = number of molecules in the vapor

Thermal system with constant T One internal variable Y. Thermal equilibrium determines U(T,V,Y) Y changes to minimize U - TS. Isolated system Two internal variables U and Y. U and Y change to maximize S composite (U,V,Y) Thermal equilibrium Y changes to maximize 56 liquid vapor reservoir of energy, T Entropy vs. Helmholtz free energy Y = number of molecules in the vapor. Fix T and V, but let Y change. Heat between wine and reservoir. wine reservoir liquid vapor reservoir of energy, T Define F = U –TS F(T,V,Y)

liquid Closed system with constant T and P One internal variable Y. Thermal equilibrium Mechanical equilibrium U(T,P,Y), V(T,P,Y), S(T,P,Y) Y changes to minimize U-TS +PV. Isolated system Three internal variables U, V and Y. U, V and Y change to maximize S composite (U,V,Y) Thermal equilibrium Mechanical equilibrium Y changes to maximize 57 Entropy vs. Gibbs free energy Y = number of molecules in the vapor. Fix P and T, but let Y change. Energy between wine, weight, reservoir. reservoir of energy, T Define G = U - TS + PV G(T,P,Y) liquid reservoir of energy, T f weight vapor wine reservoir, weight f weight vapor

58 Represent states on (P,V) plane liquid f weight vapor fire

Van der Waals equation (1873) 59 Critical point b accounts for volume occupied by molecules. a/v 2 accounts for intermolecular forces.

Maxwell construction 60 P vlvl v vgvg P sat T = constant Gibbs equation: Slope: Non-monotonic P-v relation: Rule of mixture: Fix T and v, minimize f. Coexistent phases: Integrate Gibbs equation: Maxwell construction: vlvl v vgvg v v f Non-convex f-v relation flfl fgfg f Common tangent line

61 Lu and Suo Large conversion of energy in dielectric elastomers by electromechanical phase transition.Large conversion of energy in dielectric elastomers by electromechanical phase transition. Acta Mechanica Sinica 28, (2012).

Two co-existent phases of a pure substance 62 Know the Gibbs function of each phase: g’(T,P), g’’(T,P). Rule of mixture: g(T,P,x) = (1-x)g’(T,P) + xg’’(T,P) The quality x is the internal variable. For fixed (T,P), x changes to minimize g(T,P,x). The condition for the two phases to equilibrate: g’(T,P) = g’’(T,P). This condition determines the phase boundary as a curve in the (T,P) plane. T liquid solid gas critical point triple point P

Claypeyron equation All quantities are for two coexistent phases of a pure substance 63 T P liquid gas Phase boundary: Definition of the Gibbs function: Increment along the phase boundary: Gibbs equation: Claypeyron equation:

Claypeyron equation for liquid-gas mixture 64 Claypeyron equation: Latent heat varies slowly with temperature: specific volume of the liquid is negligible: Gas is nearly ideal: Claypeyron equation: Integration: liquid solid gas critical point triple point P T

65 liquid solid gas critical point triple point P T Claypeyron equation for liquid-solid mixture Will pressure under a sharp blade cause ice to melt? Claypeyron equation: Latent heat varies slowly with temperature: specific volume of water: Specific volume of ice: Melting point varies slowly with pressure: Slope of phase boundary: A person stands on a sharp blade:

66

Summary Model a closed system as a family of isolated systems. Gibbs’s thermodynamic surface, S(U,V). Tangent plane of S(U,V). Gibbs equations Theory of phases (rule of mixture. convexification. single phases, two coexistent phases, three coexistent phases, critical point). Ideal gas has two basic equations of state: U(T), PV = Nk B T. For a closed system, S(U,V) generates all thermodynamic properties and equations of state (Alternative independent variables, Partial derivatives, Legendre transforms, Maxwell relations). Basic algorithm of thermodynamics in terms of Gibbs function, G = U - TS - PV. For a closed system of fixed P and T, the internal variable Y changes to minimize G(P,T,Y). 67 liquid f weight vapor fire