PRESSURE

• I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. • These slides follow closely my written notes (http://imechanica.org/node/288). • 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 capacitanceHelmholtz function enthalpy

Gibbs functionthermal expansionJoule-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 energy3

A half bottle of wine

4

• The reservoir of energy is a thermal system, modeled by a function SR(UR), and by a fixed temperature TR.

• The weight applies a fixed force fweight. Pressure Pweight = fweight/A. The

entropy of the weight Sweight 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

fweight

vaporclosed system

reservoir of energy, TR

adiabatic

diathermal 2OModel the wine asa family of isolated systemsS(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 thermodynamicsEntropy = 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: Ucomposite = U + UR + fweighth = constant

• Entropy of the isolated system is additive:

• Entropy of the reservoir:

liquid

fweight

vaporclosed systemS(U,V)

reservoir of energy, TR

adiabatic

diathermal

Isolated system with internal variables U and V

Entropy of the reservoir of energy

7

reservoir of energy SR, UR,TR

A reservoir of energy is a thermal system, modeled by a function SR(UR) and by a fixed temperature, TR.

Clausius-Gibbs equation:

Integration:

Conservation of energy:

Entropy of the reservoir:

Entropy of the isolated system:

Calculus:

Calculus:

Maximize Scomposite 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

fweight

vapor

reservoir of energy, TR

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

2O

liquid

vapor A family of isolated systemsFive properties: U,V,S,P,T Two independent variables, U,V

10

liquid

fweight

vapor

reservoir of energy, TR

2O

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 systemU,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 energy11

Calculus: a function of two variables

12

x

y

z z(x,y)

(x,y)

x

(x,y)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 toa 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 equilibriumbetween the wine, reservoir and weight

16

S

liquid

fweight

vapor

reservoir of energy, TR

S(U,V) surfacewine

tangent plane

plane of slopes1/TR and Pweight/TR

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 energy17

Three phases of a pure substance

18

u

v

liquid

solid

gas

intensive-intensive extensive-intensive extensive-extensive

T

liquid

solid

gas

criticalpoint

triplepoint

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)

20x

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 slopes1/T and P/T

(u,v)s(u,v)

(P,T)

A phase of a pure substance

Two co-existent phases of a pure substance1. 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 substance2. Rule of mixture defines a line in the (s,u,v) space.

24

u

v

s

(u’’,v’’)(u’,v’) (u,v)

u

v

solidliquid

solid-liquid mixture

liquid

solid

(P,T)

tie line

tie line

Two co-existent phases of a pure substance3. 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)

25u

v

s

s(u’’,v’’)

s(u’,v’)

liquid

gas

critical point

dome

(P,T)

liquid

gas

Two co-existent phases of a pure substance4. 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 substanceEach 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 mixtureu = (1 – x - y)u’ + xu’’ + yu’’’v = (1 – x - y)v’ + xv’’ + yv’’’s = (1 – x - y)s’ + xs’’ + ys’’’

(P,T)

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

en

erg

y

volu

me

gas

liquid

solid

critical point

liquid

solid

gas

volu

me

en

erg

y

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

criticalpoint

triplepoint

Gibbs’s phase rule for a pure substanceValues of (P,T) give the slopes of a plane tangent to s(u,v) surface

v

liquid

solid

gas

u

P

criticalpoint

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 energy29

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 VN

Definition of entropy S = kBlog

Gibbs equations:

Equations of state for ideal gases:

30

U,V,S,P,T,N,

31

Per mole Per 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 energy34

Osmosis

35

mem

bran

e

solution

solvent

In a liquid, osmosis is balanced by gravity

Osmosis

36

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

http://antranik.org/movement-of-substances-across-cell-membranes/

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 VN

Definition of entropy S = kBlog

Gibbs equation:

Osmotic pressure:

37

mem

bran

e

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 energy38

39

2O

liquid

vapor Isolated systemU,V,S,P,T

liquid

fweight

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

2O

liquid

vapor 2O

liquid

vapor

Isolated system Isolated systemstate U,V,S,P,T state U+dU, V+dV, S+dS. P+dP, T+dT

liquid

fweight

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

fweight

vapor

reservoir of energy, TR

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

fweight

vapor2O

liquid

vapor

fire 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 rabbitsA 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: CV(T,V), CP(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 energy54

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 Scomposite (U,V,Y)

•Thermal equilibrium

•Y changes to maximize 56

liquid

vapor

reservoir of energy, T

Entropy vs. Helmholtz free energyY = 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 –TSF(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 Scomposite (U,V,Y)

•Thermal equilibrium

•Mechanical equilibrium

•Y changes to maximize57

Entropy vs. Gibbs free energyY = 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 + PVG(T,P,Y)

liquid

reservoir of energy, T

fweight

vapor

wine reservoir, weight

fweight

vapor

58

Represent states on (P,V) plane

liquid

fweight

vapor

fire

Van der Waals equation (1873)

59

Critical point

• b accounts for volume occupied by molecules.• a/v2 accounts for intermolecular forces.

https://en.wikipedia.org/wiki/Van_der_Waals_equation

Maxwell construction

60

P

vl

v

vg

Psat

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:

vl

v

vgv

v

f

Non-convex f-v relation

flfg

f

Common tangent line

61Lu and Suo Large conversion of energy in dielectric elastomers by electromechanical phase transition. Acta Mechanica Sinica 28, 1106-1114 (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

criticalpoint

triplepoint

P

Claypeyron equationAll 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 variesslowly with temperature:

specific volume of the liquid is negligible:

Gas is nearly ideal:

Claypeyron equation:

Integration:

liquid

solid

gas

criticalpoint

triplepoint

P

T

65

liquid

solid

gas

criticalpoint

triplepoint

P

T

Claypeyron equation for liquid-solid mixtureWill 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:

66https://commons.wikimedia.org/wiki/File:Phase_diagram_of_water.svg

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 = NkBT.

• 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

fweight

vapor

fire