Thermodynamics and Statistical Mechanics

Equations of State

Thermo & Stat Mech - Spring 2006 Class 2

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Thermodynamic quantities

Internal energy (U): the energy of atoms or molecules that does not give macroscopic motion.

Temperature (T): a measure of the internal energy of a system.

Heat (Q): a way to change internal energy, besides work. (Energy in transit.)

Thermo & Stat Mech - Spring 2006 Class 2

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Laws of Thermodynamics

First law: đQ – đW = dU

Q – W = U Energy is conserved

Thermo & Stat Mech - Spring 2006 Class 2

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Work done by a gas

f

i

V

VPdVW

PdVdW

AdsAFdW

FdsdW

Thermo & Stat Mech - Spring 2006 Class 2

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Work done by a gas

Thermo & Stat Mech - Spring 2006 Class 2

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Work done by a gas

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Configuration Work

Product of intensive variable times corresponding extensive variable:

đW = xdYGas, Liquid, Solid: PdVMagnetic Material: BdMDielectric Material: EdP

Thermo & Stat Mech - Spring 2006 Class 2

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Equation of State

For an ideal gas: PV = nRTP = pressure (N/m²)(or Pa)V = volume (m³)n = number of molesT = temperature (K)R = gas constant (8.31 J/(K·mole))

Thermo & Stat Mech - Spring 2006 Class 2

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Ideal gas law

Ideal gas law:PV = nRT

In terms of molar volume, v = V/n, this becomes:

Pv = RT, or P = RT/v

Thermo & Stat Mech - Spring 2006 Class 2

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Real Substance

Thermo & Stat Mech - Spring 2006 Class 2

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Real Substance

Thermo & Stat Mech - Spring 2006 Class 2

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van der Waals equation of state

v cannot be decreased indefinitely, so replace v by v – b. Then,

Next account for intermolecular attraction which will reduce pressure as molecules are forced closer together. This term is proportional to v-2

bvRTP

Thermo & Stat Mech - Spring 2006 Class 2

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van der Waals equation of state

RTbvvaP

va

bvRTP

2

2 or , Then,

This equation has a critical value of T which suggests a phase change. The next slide shows graphs for several values of T .

Thermo & Stat Mech - Spring 2006 Class 2

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van der Waals equation of state

Thermo & Stat Mech - Spring 2006 Class 2

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Critical Values

227

278

3

baP

RbaT

bv

C

C

C

Thermo & Stat Mech - Spring 2006 Class 2

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van der Waals equation of state

'38

31'

'3'

Then,' ,' ,'

2 Tvv

P

TTTPPPvvv CCC

This can be expressed in term of dimensionlesscoordinates, P', v', and T ' with the following Substitutions:

Thermo & Stat Mech - Spring 2006 Class 2

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van der Waals equation of state

This can also be written,

2'3

1'3'8'

vvTP

Thermo & Stat Mech - Spring 2006 Class 2

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Thermal Expansion

Expansivity or Coefficient of Volume Expansion, .

TVTTVV

PTTv

vTV

V

P

PP

),(

11

Thermo & Stat Mech - Spring 2006 Class 2

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Thermal Expansion

Usually, is positive. An exception is water in the temperature range between 0° C and 4° C.Range of is about: 10-3 for gasses. 10-5 for solids.

Thermo & Stat Mech - Spring 2006 Class 2

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Linear Expansion

Coefficient of Linear Expansion, .

TXTTXX

pTTX

X

P

P

),(

1

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Relationship Between and

3

)31()1(

)1()1()1('

)1('

3

TVTXYZ

TZTYTXVXYZV

TVTVVVVV

Thermo & Stat Mech - Spring 2006 Class 2

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Compressibility

Volume also depends on pressure.Isothermal Compressibility:

PVPPVV

PTPV

V

T

T

),( 1

Thermo & Stat Mech - Spring 2006 Class 2

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Bulk Modulus

TVPV

1 ModulusBulk

Thermo & Stat Mech - Spring 2006 Class 2

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A Little Calculus

0

0 const, If,

),( Consider,

VT

VP

TP

dPPVdT

TV

dVV

dPPVdT

TVdV

PTV

Thermo & Stat Mech - Spring 2006 Class 2

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Cyclical Relation

1

0

PVT

VTP

VTP

VT

TP

PV

TP

PV

TV

TP

PV

TV

Thermo & Stat Mech - Spring 2006 Class 2

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Application

Suppose you need:VT

P

1

PVT VT

TP

PV

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Application

T

P

T

P

PT

V

PV

V

TV

V

PVTV

VT

PVT

P1

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