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CHAPTER 4MATERIAL EQUILIBRIUM

ANIS ATIKAH BINTI AHMAD

anisatikah@unimap.edu.my

PHYSICAL CHEMISTRY

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SUBTOPIC

Introduction to Material Equilibrium Entropy and Equilibrium The Gibbs and Helmholtz Energies Thermodynamic Relations for a System

Equilibrium Calculation of Changes in State Function Phase Equilibrium Reaction Equilibrium

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WHAT IS MATERIAL EQUILIBRIUM?

In each phase of the closed system, the number of moles of each substances present remains constant in time

No net chemical reactions are occurring in the system

No net transfer of matter from one part of the system to another

Concentration of chemical species in the various part of the system are constant

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Material equilibrium

Reaction equilibrium

Phase equilibrium

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ENTROPY AND EQUILIBRIUM Entropy, is a measure of

the "disorder" of a system. What "disorder refers to is really the number of different microscopic states a system can be in, given that the system has a particular fixed composition, volume, energy, pressure, and temperature.

While energy strives to be minimal, entropy strives to be maximal

Entropy wants to grow. Energy wants to shrink. Together, they make a compromise.

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ENTROPY AND EQUILIBRIUM Example: In isolated system (not in material

equilibrium) The spontaneous chemical reaction or

transport of matter are irreversible process that increase the ENTROPY

The process was continued until the system’s entropy is maximized.

Once it is maximized, any further process can only decrease entropy –(violate the second law)

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• isolated systems: is one with rigid walls that has no communication (i.e., no heat, mass, or work transfer) with its surroundings. An example of an isolated system would be an insulated container, such as an insulated gas cylinder

isolated (Insulated) System:

U = constant

 Q = 0

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The system is not in material equilibrium but is in mechanical and thermal equilibrium

The surroundings are in material, mechanical and thermal equilibrium

System and surroundings can exchange energy (as heat and work) but not matter

Since system and surroundings are isolated , we have

dqsurr= -dqsyst (1)

Since, the chemical reaction or matter transport within the non equilibrium system is irreversible, dSuniv must be positive:

dSuniv = dSsyst + dSsurr > 0 (2)

Consider a system at T;

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The surroundings are in thermodynamic equilibrium throughout the process.

Therefore, the heat transfer is reversible, anddSsurr= dqsurr/T (3)

The systems is not in thermodynamic equilibrium, and the process involves an irreversible change in the system, therefore

DSsyst ≠dqsyst/T (4)

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Equation (1) to (3) give dSsyst > -dSsurr = -dqsurr/T = dqsyst/T (5)

Therefore dSsyst > dqsyst/T

dS > dqirrev/T (6) closed syst. in them. and mech. equilib.

dqsurr= -dqsyst (1)

dSsurr= dqsurr/T (3)

dSuniv = dSsyst + dSsurr >0 (2)

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When the system has reached material equilibrium, any infinitesimal process is a change from a system at equilibrium to one infinitesimally close to equilibrium and hence is a reversible process.

Thus, at material equilibrium we have, ds = dqrev/T (7)

Combining (6) and (7): ds ≥dq/T (8) material change, closed

syst. in

them & mech. Equilib

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The first law for a closed system isdq = dU – dw (9)

Eq 8 gives dq≤ TdS Hence for a closed system in mechanical and

thermal equilibrium we have dU – dw ≤ TdS Or

dU ≤ TdS + dw (10)

ds ≥ dq/T (8)

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A spontaneous process at constant-T-and-V is accompanied by a decrease in the Helmholtz energy, A.

A spontaneous process at constant-T-and-P is accompanied by a decrease in the Gibbs energy, G.

dA = 0 at equilibrium, const. T, V

dG = 0 at equilibrium, const. T, P

THE GIBSS & HELMHOLTZ ENERGIES

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dU TdS + SdT – SdT + dw

dU d(TS) – SdT + dw

d(U – TS) – SdT + dw

d(U – TS) – SdT - PdVat constant T and V, dT=0, dV=0

d(U – TS) 0

dU TdS + dw

Equality sign holds at material equilibrium

HELMHOLTZ FREE ENERGY

A U - TS

Consider material equilibrium at constant T and

V

dw = -P dV for P-V work

only

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For a closed system (T & V constant), the state function U-TS, continually decrease during the spontaneous, irreversible process of chemical reaction and matter transport until material equilibrium is reached

d(U-TS)=0 at equilibrium

HELMHOLTZ FREE ENERGY

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const. T

dwSdTTSUd )(

dwSdTdA dwdAwA

wwby

Awby const. T, closed syst.

It turns out that A carries a greater significance than being simply a signpost of spontaneous change:

The negative change in the Helmholtz energy is equal to the maximum work the system can do: Awby max,

Closed system, in thermal &mechanic. equilibrium

HELMHOLTZ FREE ENERGY

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dU d(TS) – SdT – d(PV) + VdP

d(H – TS) – SdT + VdP

d(U + PV – TS) – SdT + VdP

at constant T and P, dT=0, dP=0

d(H – TS) 0

Consider material equilibrium for constant T & P, into with dw = -P dV

dU T dS + dw

dU T dS + S dT – S dT - P dV + V dP – V

dP

GIBBS FREE ENERGY

G H – TS U + PV – TS

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the state function H-TS, continually decrease during material changes (constant T and P) , until material equilibrium is reached.

This is the minimisation of Gibbs free energy.

GIBBS FREE ENERGY,G=H-TS

d(H – TS) 0

G = H – TS = U + PV -

TS

GIBBS FREE ENERGY

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G H – TS U + PV – TS

dGT,P 0

Equilibrium reached

Constant T, P

Time

G

G decreases during the approach to equilibrium, reaching minimum at equilibrium

GIBBS FREE ENERGY

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As G of the system decrease at constant T & P, Suniv increases.

WHY?Consider a system in mechanical and thermal equilibrium which undergoes an irreversible chemical reaction or phase change at constant T and P.

systsurruniv SSS systsyst STH /

TGTSTH systsystsyst //)(

TGS systuniv / closed syst., const. T, V, P-V work only

GIBBS FREE ENERGY

The decrease in Gsyst as the system proceeds to equilibrium at constant T and P corresponds to a proportional increase in S univ

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G H – TS U + PV – TS

VdPPdVdwSdTdG

G U– TS + PV A + PV VdPPdVdAdG

VdPPdVdwSdTdG

PdVdwdG

const. T and P, closed syst.

If the P-V work is done in a mechanically reversible manner, then

VPnondwPdVdw

VPnondwdG VPnonbyVPnon wwG ,

const. T and P, closed syst.

Gw VPnonby ,

or

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For a reversible change

The maximum non-expansion work from a process at constant P and T is given by the value of -G

GwVPnon

max,

(const. T, P)

Gw VPnonby ,

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Thermodynamic Relations for a System in Equilibrium

6 Basic Equations:

dU = TdS - PdV

H U + PV

A U – TS

G H - TS

VV T

UC

PP T

HC

closed syst., rev. proc., P-V work only

closed syst., in equilib., P-V work only

closed syst., in equilib., P-V work only

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The rates of change of U, H, and S with respect to T can be

determined from the heat capacities CP and CV.

VV T

STC

PP T

STC

Key properties

closed syst., in equilib.

Basic Equations

Heat capacities

(CP CV )

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dG = -SdT + VdP

dA = -SdT - PdV

dH = TdS + VdP

dU = TdS - PdV

The Gibbs Equations

closed syst., rev. proc., P-V work only

How to derive dH, dA and dG?

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The Gibbs Equations

dH = d(U + PV)

dH = TdS + VdP

= dU + d(PV)= dU + PdV + VdP= (TdS - PdV) + PdV + VdP

H U + PV

dH = ?

dU = TdS - PdV

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dA = d(U - TS)

dG = d(H - TS)

dG = -SdT + VdP

dA = -SdT - PdV

= dU - d(TS)= dU - TdS - SdT= (TdS - PdV) - TdS - SdT

= dH - d(TS)= dH - TdS - SdT= (TdS + VdP) - TdS - SdT

dA = ?

dU = TdS - PdV

dH = TdS+VdP

dG = ?

A U - TS

G H - TS

dVV

UdS

S

UdU

SV

TS

U

V

PV

U

S

ST

G

P

VP

G

T

The Power of thermodynamics:

)( PdVTdSdU

Difficultly measured properties to be expressed in terms of easily measured properties.

The Gibbs equation dU= T dS – P dV implies that U is being considered a function of the variables S and V. From U= U (S,V) we have

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(dG = -SdT + VdP)

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The Euler Reciprocity

RelationsIf Z ＝ f(x ， y) ， and Z has continuous second partial derivatives, then

y

z

xx

z

y

That is

NdyMdxdz xy y

zN

x

zM

yxx

N

y

M

The Gibbs equation (4.33) for dU is

dU ＝ TdS － PdV

V

U

SS

U

V

dS ＝ 0

PV

U

S

TS

U

V

dV ＝ 0

Applying Euler Reciprocity,

SV V

T

S

P

dU = TdS - PdV

The Maxwell Relations (Application of Euler relation to Gibss equations)

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VS PS

TV

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These are the Maxwell

Relations

VT T

P

V

S

PT T

V

P

S

PS S

V

P

T

SV V

T

S

P

The first two are little used.

The last two are extremely valuable.

The equations relate the isothermal pressure and volume variations of entropy to measurable properties.

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DEPENDENCE OF STATE FUNCTIONS ON T, P, AND V

We now find the dependence of U, H, S and G on the variables of the system.

The most common independent variables are T and P.

We can relate the temperature and pressure variations of H, S, and G to the measurable Cp,α, and κ

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Volume dependence of UThe Gibbs equation gives dU ＝ TdS － PdV

PV

ST

V

U

T

T

T

T d

d

d

d

From Maxwell Relations

PV

ST

V

U

TT

VT T

P

V

S

Divided above equation by dVT, the infinitesimal volume change at constant T, to give

PT

PT

PT

V

U

VT

For an isothermal process dUT ＝ TdST － PdVT

T subscripts indicate that the infinitesimal changes dU, dS, and dV are for a constant-T process

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from Gibbs equations, dH ＝ TdS ＋ VdP

Pressure dependence of H

Temperature dependence of UTemperature dependence of H

PT T

V

P

S

VP

ST

P

H

TT

VTVVT

VT

P

H

PT

PP

CT

H

VV

CT

U

From Basic Equations

From Maxwell Relations

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Temperature and Pressure dependence of G

Temperature dependence of S

Pressure dependence of S

T

C

T

S P

P

From Basic Equations

The Gibbs equation (4.36) for dG is

dG ＝ -SdT + VdP

dT ＝ 0

VP

G

T

ST

G

P

dP ＝ 0

VT

V

P

S

PT

From Maxwell Relations

The equations of this section apply to a closed system of fixed composition and also to a closed system where the composition changes reversibly

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Joule-Thomson Coefficient(easily measured quantities)

HJT P

T

PT

JT CP

H/

from Chapter 2

)1(])[1(

T

C

VVTVC

PPJT

VTVVT

VT

P

H

PT

From pressure

dependence of H

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Heat-Capacity Difference(easily measured quantities)

PTVP T

VP

V

UCC

PT

PT

PT

V

U

VT

PVP T

VTCC

2TV

CC VP V

T

V

T

VV

P

P

1

From volume

dependence of U

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2TV

CC VP

1. As T 0, CP CV

Heat-Capacity Difference

2. CP CV (since > 0)

3. CP = CV (if = 0)

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EXAMPLE 1

2TV

CC VP 2

,,m

mVmP

TVCC

÷ n

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TV

U

Ideal gases 0

TV

U

Solids 300 J/cm3 (25 oC, 1 atm)

Internal Pressure

Liquids 300 J/cm3 (25 oC, 1 atm)

PT

PT

PT

V

U

VT

Strong intermolecular forces in solids and liquids.

Solids, Liquids, & Non-ideal Gases

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CALCULATION OF CHANGES IN STATE FUNCTION

1. Calculation of ΔS Suppose a closed system of constant composition

goes from state (P1,T1) to state (P2,T2), the system’s entropy is a function of T and P

dPP

SdT

T

SdS

TP

VdPdTT

CdS P

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Integration gives:

Since S is a state function, ΔS is independent of the path used to connect states 1 and 2. A convenient path (Figure 4.3) is first to hold P constant at P1 and change T from T1 to T2. Then T is held constant at T2, and P is changed from P1 to P2.

For step (a), dP=0 and gives

For step (b), dT=0 and gives

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1

PPconstdTT

CS

T

T

Pa

22

1

TTconstdPVSP

Pb

dPVdTT

CSSS P

2

1

2

112

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EXAMPLE 2

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ΔU can be easily found from ΔH using :

ΔU = ΔH – Δ (PV) Alternatively we can write down the equation for ΔU similar to:

dPTVVdTCH P 2

1

2

1

2. Calculation of ΔH

dPP

HdT

T

HdH

TP

dPVTVdTCP )(

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3. Calculation of ΔG

For isothermal process:

Alternatively, ΔG for an isothermal process that does not involve an irreversible composition change can be found as:

A special case:

TconstSTHG

TconstVdPG

P

P

2

1

PandTconstatprocessrevG 0[Since ]TqSqH /,

VP

G

T

from slide 28

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Phase Equilibrium

A phase equilibrium involves the same chemical species present in different phase. [ eg:C6H12O6(s) C6H12O6(g) ]

-

-

Phase equilib, in closed syst, P-V work only

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For the spontaneous flow of moles of j from phase to phase

- Closed syst that has not yet reached phase equilibrium

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Suppose that substance j is initially absent in phase .

If initially > , then j flows from phase β to phase δ until the equilibrium is reached.

However, if

Then, j cannot flow out of δ (since it is absent from δ ). The system will therefore unchanged with time and hence in equilibrium. So, when a substance is absent from a phase, the equilibrium condition becomes:

Phase equilib, j absent from

Reaction Equilibrium

A reaction equilibrium involves different chemical species present in the same phase.

During a chemical reaction, the change Δn in the no. of moles of each substance is proportional to its stoichometric coefficient v, where the proportionality constant is the same for all species.This proportionality constant is called the extent of reaction For general chemical reaction undergoing a definite amount of reaction, the change in moles of species i, , , equals multiplied by the proportionality constant :

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The condition for chemical-reaction equilibrium in a closed system is

Reaction equilib, in closed system., P-V work only

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EXAMPLE 3