Refrigeration and Cryogenics

Aleksandra Bień

Seminar

Thermodynamic relations. Maxwell equations

James Clark Maxwell

Born: 13 June 1831, Edinburgh, Scotland

Died: 5 November 1879 Cambridge, England

Author of „the second great unification in physics”* (after the first one released by Isaac Newton)

Scottish physicist and matematician

*Nahin, P.J., Spectrum, IEEE, Volume 29, Issue 3, March 1992 Page(s):45–

James Clark Maxwell - Contributions

Electromagnetism:

• the concept of the electromagnetic field in comparison to force lines that Faradaydiscovered. His famous equations, in their modern form of four partial differentialequations, first appeared in fully developed form in his textbook A Treatise onElectricity and Magnetism in 1873.

Colour analysis

• contributed to the field of optics and the study of colour vision, creating thefoundation for practical colour photography;

• Maxwell's discs were used to compare a variable mixture of three primary colourswith a sample colour by observing the spinning "colour top”.

Control theory

• Maxwell published a famous paper "On governors". This paper is quite frequentlyconsidered a classical paper of the early days of control theory. Here governors refer tothe governor or the centrifugal governor used in steam engines.

James Clark Maxwell - Contributions

Kinetic theory and thermodynamics

• investigated the kinetic theory of gases;

• Maxwell–Boltzmann kinetic theory of gases;

• the Gedankenexperiment (thought experiment), knownas Maxwell's demon;

• in 1871, he established Maxwell's thermodynamicrelations.

Maxwell's thermodynamic relations

Established in 1871. Statements of equality among the second derivatives ofthe thermodynamic potentials with respect to different thermodynamic variables.

Maxwell's relations (general)

•differentiation of an analytic function of two variables is irrelevant;

•the partial derivatives are taken with all other natural variables held constant;

•for every thermodynamic potential there are n(n − 1)/2 possible Maxwell relationswhere n is the number of natural variables for that potential.

Fundamental thermodynamic relations

Function of stateNatural variables

Formula

E Internal energy S , V E

H Enthalpy S , p H=E+pV

S Entropy U , V S=Q/T

F Free energy T , V F=E-TS

G Free enthalpy T , p G=H-TS

A thermodynamic potential is a scalar function used to represent the thermodynamic state of a system.

Thermodynamic potential – Internal energy

• total energy contained by a thermodynamic system;• energy needed to create the system;• excludes the energy to displace the system's surroundings, any energy associated with a

move as a whole, or due to external force fields;• two major components, kinetic energy and potential energy;• can be changed by heating the system or by doing work on it;• increase in internal energy is equal to the total heat added and work done;• If the system is isolated, its internal energy cannot change.

pdVTdSdEpdVTdSE −=⇒−= ∫ )(

dVV

EdS

S

EVSdE

TV

∂∂+

∂∂=),(

the total derivative of with respect to the natural variables:

pdVTdSdEpdVTdSE −=⇒−= ∫ )(

dVV

EdS

S

EVSdE

SV

∂∂+

∂∂=),(

This indicates that:

VS

ET

∂∂=

SV

Ep

∂∂=−

Differentiation of the left equation by V and the right equation by S:

SVs S

E

VV

T

∂∂

∂∂=

∂∂

VSV V

E

SS

p

∂∂

∂∂−=

∂∂

In second derivation the order of differentiation does not matter, thus:

Vs S

p

V

T

∂∂−=

∂∂

Thermodynamic potential – Free energy

• known as a Helmholtz potential;

• measures the “useful” work obtainable from a closed thermodynamic system at aconstant temperature and volume;

• change of free energy is equal to maximum work done by the system in constanttemperature;

• Very often used, because of it’s natural variables (T,V), which are easy to measure.

TSEVTF −=),(

pdVSdTSdTTdSpdVTdSSdTTdSdEVTdF −−=−−−=−−=),(

pdVSdTVTdF −−=),(

This indicates that:

VT

FS

∂∂=−

TV

Fp

∂∂=−

Differentiation of the left equation by V and the right equation by T:

TVT T

F

VV

S

∂∂

∂∂=

∂∂−

VTV V

F

TT

p

∂∂

∂∂=

∂∂−

In second derivation the order of differentiation does not matter, thus:

VT T

p

V

S

∂∂=

∂∂

pdVSdTVTdF −−=),(

dVV

FdT

T

FVTdF

TV

∂∂+

∂∂=),(

Thermodynamic potential – Enthalpy

• measure of the total energy of a thermodynamic system;• includes the internal energy and the amount of energy required to make room for it;• state function and an extensive quantity;• simplifies descriptions of energy transfer;• The total enthalpy, H, of a system cannot be measured directly;• ∆H, is a more useful quantity than its absolute value.

pVEpSH +=),(

VdpTdSVdppdVpdVTdSVdppdVdEpSdH +=++−=++=),(

VdpTdSpSdH +=),(

This indicates that:

pS

HT

∂∂=

Sp

HV

∂∂=

Differentiation of the left equation by p and the right equation by S:

SpSS

H

pp

T

∂∂

∂∂=

∂∂

pSp p

H

SS

V

∂∂

∂∂=

∂∂

In second derivation the order of differentiation does not matter, thus:

pSS

V

p

T

∂∂=

∂∂

dpp

HdS

S

HpSdH

Sp

∂∂+

∂∂=),(

VdpTdSpSdH +=),(

Thermodynamic potential – Free Enthalpy

• also known as a Gibbs Function;• a maximum work attainable in a closed system which is not related to the change of volume.

TSHpTG −=),(

VdpSdTSdTTdSVdpTdSSdTTdSdHpTdG +−=−−+=−−=),(

VdpSdTpTdG +−=),(

This indicates that:

pT

GS

∂∂=−

Tp

GV

∂∂=

Differentiation of the left equation by p and the right equation by T:

TpTT

G

pp

S

∂∂

∂∂=

∂∂−

pTp p

G

TT

V

∂∂

∂∂=

∂∂

In second derivation the order of differentiation does not matter, thus:

pTT

V

p

S

∂∂=

∂∂−

dpp

GdT

T

GpTdG

Tp

∂∂+

∂∂=),(

VdpSdTpTdG +−=),(

Thermodynamic potentials

• When the temperature (T ) and external parameters of a closed system are held constant, theHelmholtz free energy (F ) decreases and reaches a minimum value at equilibrium.

• When the pressure (p ) and external parameters of a closed system are held constant, the enthalpy(H ) decreases and reaches a minimum value at equilibrium.

• When the temperature (T ), pressure (p ) and external parameters of a closed system are heldconstant, the Gibbs free enthalpy (G ) decreases and reaches a minimum value at equilibrium.

Thus:

• When the entropy (S ) and "external parameters" (e.g. volume) of a closed system are held constant,the internal energy (U ) decreases and reaches a minimum value at equilibrium. This follows from thefirst and second laws of thermodynamics and is called the principle of minimum energy. The followingthree statements are directly derivable from this principle.

vs

m

VS s

p

v

T

S

p

V

T

∂∂−=

∂∂→

∂∂−=

∂∂

pT

m

pTT

v

p

s

T

V

p

S

∂∂−=

∂∂→

∂∂−=

∂∂

ps

m

pSs

v

p

T

S

V

p

T

∂∂=

∂∂→

∂∂=

∂∂

vT

m

VT T

p

v

s

T

p

V

S

∂∂=

∂∂→

∂∂=

∂∂

It is much more convenient to operate on volume and entropy related to mass of thesubstance. Obtained Maxwell relations formula are recalculated for m, so that the mostcommon version of these formulas is obtained.

The four basic Maxwell relations enable transformation of any thermodynamic relationwith the aid of thermodynamic susceptibilities such as α, κ, cp.

Thermodynamic susceptibilities

Thermal capacity at v=const:V

vconstV

Vv T

S

n

TcTdSdE

T

E

nc

∂∂=⇒= →

∂∂= =1

Thermal capacity at p=const:p

pconstp

pp T

S

n

TcTdSdE

T

H

nc

∂∂=⇒= →

∂∂= =1

Thermal expansion:pT

V

V

∂∂= 1α

Compressibility at T=const:

T

T p

V

V

∂∂−= 1κ

Compressibility at S=const:

S

S p

V

V

∂∂−= 1κ

Example of application – adiabatic compressionIn this process we increase pressure at constant entropy and observe the temperature change.

dpp

TdT

S

∂∂=

11 −=

∂∂

∂∂

∂∂→−=

∂∂

∂∂

∂∂

pTS

thus

yxzT

S

S

p

p

T

x

z

z

y

y

x

T

nc

T

S p

p

=

∂∂

11 −−

∂∂=

∂∂→

∂∂=

∂∂

TT

thus

zzp

S

S

p

x

y

y

x

Maxwell relation:pT

T

V

p

S

∂∂=

∂∂

Thermodynamic susceptibilities:

pT

V

V

∂∂= 1α

VS

p

T α1−=

∂∂

⇒

dpc

vTdT

p

α=The temparature change at adiabatic compression of gas:

Mnemonic way to calculate Maxwell relationsGuggenheim’s square

Vs S

p

V

T

∂∂−=

∂∂

Mnemonic way to calculate Maxwell relations

A mnemonic used by physics students to rememberthe Maxwell relations in thermodynamics which helpsthem remember the order of the variables in the square,in clockwise direction is:

"Good Physicists Have Studied Under Very Fine Teachers”

Another mnemonic used here is (letters in the normal leftto right writing direction) :

"Valid Factsand Theoretical Understanding GenerateSolutions to Hard Problems”

Conclusions

Why use Maxwell relations?

•Certain variables in thermodynamics are hard to measure experimentally such asentropy. Maxwell relations provide a way to exchange variables;

•Relations enable derivations of formulas of thermodynamic functions such as α, κ, cp;

•Can be used to estimate the amount of energy obtained from the thermodynamicsystem under specific conditions.

Thank you for Your attention !