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Thermodynamics Y Y Shan

AP3290 67

Chapter 5 Thermodynamic properties of materials

Introduction:

The thermodynamic properties of materials are intensive quantities which are

specific to a given material, such as the specific heats, ,Vc and Pc of materials , introduced in

Chapter 2. There are some other experimentally measurable properties, such coefficients of

thermal expansion and compressibility. Each is directly related to a second order differential

of a thermodynamic potentialdefined later in this chapter.

(i) Specific heat of materials ( ,Vc and Pc ): also known simply as heat capacity

( ,VV ncC = and PP ncC = ), is the measure of the heat energy required for a unit quantity of a

substance ( moln 1= ) to increase 1 degree temperature

VV

VT

ST

dT

dQC

=

=

olumeconstant vatheatSpecific

5-1a

PP

PT

ST

dT

dQC

=

=

pressureconstantatheatSpecific

5-1b

(ii) Coefficient Thermal expansion: is the tendency of matter to change in volume in

response to a change in temperature. The degree of expansion divided by the change in

temperature is called the material's coefficient of thermal expansion

PT

V

V

=

1

expansionthermaloftCoefficien

5-1c

(iii) Compressibility : is a measure of the relative volume change as a response to apressure change

TP

V

V

=

1

ilitycompressiboftCoefficien

5-1d

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These material properties, Specific heatC, thermal expansion, and Compressibility,can

be experimentally determined.

1. Thermodynamic potentials

In thermodynamics, thermodynamic potentials are parameters associated with a

thermodynamic system and have the unit of energy (Joule).

Thermodynamic potentials are very useful when calculating the properties of

materials in a chemical reaction, which are very difficult to be measured. They are

characterized bythe basic four variables: P&V(mechanical natural variable), T& S(thermal

natural variable) respectively, where P, T are called generalized forces, and V,S are called

generalized displacements introduced in Chapter 4. There are four thermodynamic

potentials: the internal energyU, the enthalpyH, the Helmholtz free energyF, and the Gibbs

free energyG.

Internalenergy: PdVTdSdU =

obtainedPandTknown,isV)(S,function UIfV.S,byzedcharacteriisU

Enthalpy: PVUH +=:asdefined

)()()( VdPPdVPdVTdSPVddUdH ++=+=

VdPTdSdH +=

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obtainedbecanVandTknown,isP)H(S,functionIfP.S,byzedcharacteriisH

Helmholtz free energy: TSUF =:asdefined

)()()( SdTTdSPdVTdSTSddUdF +==

PdVSdTdF = obtainedbecanPandSknown,isV)F(T,functionIf

Gibbs free energy: TSPVUG +=:asdefined

)()( TSdPVddUdG +=

VdPSdTdG += obtainedbecanVandSknown,isP)G(T,IfP.T,byzedcharacteriisG

These four thermodynamic potentials, U(S,V) , H(S,P) , F(T,V) , and G(T,P), introduced

above are summarized as the follows:

VdPSdTPTdG

PdVSdTVTdF

VdPTdSPSdH

PdVTdSVSdU

+=

=

+=

=

),(

),(

),(

,),(

Eqn5-2

----------------------------------------------------------------------

(*Because the mixed partial derivatives are identical:

yxZ

xyZ

yZ

xxZ

yxy

=

=

=

22

Q

Therefore thermodynamic properties of materials Specific heatC, thermal expansion, and

Compressibility , each can be directly related to a second order differential of a

thermodynamic potentialdefined in Eqn5-2.

2

2

T

FTCV

= , 2

2

T

GTCP

= , 2

21

P

G

V

= ,

TP

G

V

=

21

On the other hand, they can be experimentally determined. Therefore thermodynamic

potentials can be calculated, and vise verse)

--------------------------------------------------------------------

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AP3290 70

2. Maxwell's relations and their applications

2.1. Derivation: Maxwells relations are a set of four equations derived from the

definitions of the correspondingfourthermodynamic potentials, in Eqn 5-2.

In mathematics, for a function ),( yxZ , its differential is known to be:

Eq5-3a

i.e.

xy y

Z

Nx

Z

M

=

= , Eq5-3b

Because the mixed partial derivatives are identical:

yx

Z

xy

Z

y

Z

xx

Z

yxy

=

=

=

22

Q ,

yx

Z

x

N

xy

Z

y

M

yx

=

=

22

,

Thus:

yx x

N

y

M

=

Eq5-3c

-------------------------------------------------------------------------

* Triple product rule

For ),( yxZ : NdyMdxyxdZ +=),( ,xy y

ZN

x

ZM

=

= , ,

when Zis fixed, i.e. 0=dZ ,

NdyMdx = , 1=

Zy

x

N

M,

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AP3290 71

1=

Zxy y

x

Z

y

x

Z Eq5-3d

This is called the Triple Product Rule.

-------------------------------------------------------------------------

If we replace this general function ),( yxZ by those four thermodynamic potentials

defined earlier: U(S,V), H(S,P),F(T,V),andG(T,P), applying Eq5-3cthis general result

to the four four thermodynamic potentials, the Maxwells relation is derived.

For e.g., from enthalpyH(S,P):

PS

yx

S

V

P

TVdPTdSPSdH

x

N

y

MNdyMdxyxdZ

=

+=

=

+=

),(

),(

whereSP P

HV

S

HT

=

= , . This is one of the four Maxwells relations.

Similarly, the following can be obtained:

)(-,

)(,

,

-,

PT

VT

PS

VS

VT

V

P

SVdPSdTdG

PT

P

V

SPdVSdTdF

S

V

P

TVdPTdSdH

S

P

V

TPdVTdSdU

=

=

+=

=

=

=

+=

+=

=

=

eqn5-4

The four equations on the right are known as the Maxwells relations

2.2 Some application of Maxwells relations in determining material properties

The importance of the Maxwells relations: in thermodynamics, some physical

quantities, such as P, T, V are easy to be measured, and therefore the state equation

f(P,V,T)=0 can be obtained experimentally, such as

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nRTPV = .

Also thermal expansion coefficient , Compressibility , pressure coefficient , Specific

heats Cp, Cv, can be measurable:

VTP T

PPP

VVT

VV

=

=

= 1,1,1

/=P (the Triple Product Rule, Eq5-3d is appled)

But there are some other quantities are either cannot be measured or very difficult to

measure, such as entropy. By using Maxwells relation, one can determine changes of

entropy through measurable quantities of P,V, T, and ,,,, PV CC as:

===

PdVTSP

TP

VS )(

VT

Maxwells relations are enormously useful, they provide relationships between measurable

quantities and those quantities which are not measurable or difficulty to be measured.

Each partial derivative in Maxwells relations represents a relationship between two

physical quantities in a thermodynamic process under a specific condition for e.g.,

This mathematical expression of partial derivative,

TVS

,

,when it is translated to thermodynamic language, it means the relationship between entropy

S and volume V in a constant temperature process.

The following Maxwell relation,

PS

+=

S

V

P

T,

, in thermodynamic language, means the temperature variation vs. the change of pressure in

a constant entropy process (adiabatic process) is the same as the volume variation against the

change of entropy in a isobaric process(constant pressure process)

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Example 1: To determine the change of internal energy in respect with the change of volume

for an isothermal process:

TV

U

(i) for ideal gas , (ii) for van der Waals gas.

For ),( VTUU=

PdVTdSdU =

Since mathematically: dVV

SdT

T

SVTdS

TV

),(

+

= ,

)][TV

PdVdVV

SdT

T

STPdVTdSdU

+

== i.e.

][TV

dVPVSTdT

TSTdU

+

= (a)

Also mathematically:

dVV

UdT

T

UVTdU

TV

),(

+

= (b)

Comparing (a) with (b), therefore:

VV

=

T

ST

T

U, here VC

T

U=

V

T

PV

ST

V

U

T

=

(c)

To obtainTV

U

, we need to get

T

V

S. Though

T

V

S could not be measured easily,

however by applying the Maxwells relation:

)(

VT

PdVSdTdFT

P

V

S=

=

Equation (c) becomes:

V

PT

PT

V

U

T

=

Eqn5-5

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The left side is difficult to measure, while the right side can be obtained when the state

equation ),,( TVPf is available.

(i) For ideal gas: the state equation: nRTPV =

0V

===

=

PPP

V

nRTP

T

PT

V

U

T

This is Joles law introduced in Chapter 2, and it is proved now. It tells that internal energy of

an ideal gas is not volume dependent in an isothermal expansion process. It is temperature

dependent.

(ii) For van der Waals gas, the equation of state is:

,))((2

nRTbVV

aP =+ where aand bare the constants.

)(22

V V

aP

V

aPP

bV

nRTP

T

PT

V

U

T

=+=

=

=

The change of internal energy is volume dependent.

ap3290_chapter_5_i_2009

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