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Irreversible

Thermodynamics

Thermodynamics and Kinetics

� Thermodynamics is precise about what cannot

happen.

� How can thermodynamics be applied to systems

that are away from equilibrium?

� How are concepts from thermodynamics useful

for building kinetic models?

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A few math concepts

� Fields, scalar

� Fields, vector

� Fields, tensor

� Variations of scalar fields

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� Fields, scalar

Values of something are specified as a function of position,

� r = xi + yj + zk , and (in kinetics) time.

� r ,tExample: composition field c( ) from a point source

diffusing into a body

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� Fields, vector

Values of a vector quantity are specified as a function of position

and (in kinetics) time

Example: velocity is a vector field, e.g. in a flow field � �

r ,tv( )

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� Fields, tensor

A second-rank tensor called the stress tensor

�� r� ij ( ) relates force F at a point to an oriented area A

at that point

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� Variations of scalar fields

� • Stationary field, moving point r (a bug on a surface…)

� c� ��v dt

�)v dt

� r + = c(

�)r +c(

dc =

dt � v

� c� �

�• Evolving field, moving point r (a bug on a surface in

an earthquake…) dc

= � v +

� c� �

�c dt �t

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� Continuum limits

Is it possible to define a local value for concentration

in the limit �V � 0 ?

See Figures 1.5 and 1.6 in Balluffi, R. W., S. M . Allen & W. C. Carter. Kinetics of Materials.

New York: Wiley & Sons, forthcoming.

It is, with suitable care.

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� Fluxes�

The flux vector Ji represents mass flow ofcomponent i at a point per unit oriented area at thatpoint

�Mi (�A � )

= Ji � nlim �A�0 �A

Example: swimming fish

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� Accumulation�

Flow field, J (r) ˙Rate of production in dV, �i

� Accumulation of species i

�ci = �� � Ji + �i �t

��u For a conserved quantity like internal energy = �� � Ju

�t and for a non-conserved quantity like entropy

��S= �� � JS + �i

�t

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System at Equilibrium

� Characterized by uniform values of various “potentials” throughout the system e.g. T, P, µi

� Densities of conjugate extensive quantities e.g. S,can be inhomogeneous at equilibriumV N, i

q � S is maximized in a system at constant U

� For an isolated system dS � T

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System away from Equilibrium

� Kinetics concerns the path, mechanisms, andrates of spontaneous and driven processes.

� Irreversible thermodynamics attempts to apply thermodynamics principles to systems that are not in equilibrium and to suggest principles by which they relax toward equilibrium or steady state.

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Postulate 1

� It is generally possible to use principles of the continuum limit to define meaningful, useful, local values of various thermodynamics quantities, e.g., chemical potential µi.

� Useful kinetic theories can be developed by assuming a functional relationship between the rate of a process and the local departure from equilibrium (“driving force”).

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Postulate 2

� For any irreversible process the rate of entropy production is everywhere � 0.

��S � = + �� JS � 0 �t

� More restrictive than dStot � 0 for isolated system.

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Postulate 3

� Assume linear coupling between fluxes J and forces X

J = L X �,� = Q,q, I ,..., Nc� �� �

J = �� L X X� �� � �� �

� Electrical + heat conduction

J = L Xq + LqQXQ Direct coefficients, Lqq LQQq qq

JQ = LQqXq + LQQXQ Coupling coefficients, LqQ LQq

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Postulate 4

� Onsager symmetry postulate

� L = L�� (microscopic reversibility). ��

�Jq �JQL = L � =�� �� �XQ �Xq

Similar to Maxwell’s relations in thermodynamics

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Force/Flux Pairs

Flux Conjugate Force Empirical law

Heat

Charge

Mass

� JQ � Jq� Ji

1 �T�T

� JQ=�K�T Fourier

��� � J Ohm� �= � �q

��µi

� Ji = �Mici�µi Fick

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Multiple forces and fluxes

Consider simultaneous flow of heat and charge� JQ = �LQQXQ � LQqXq� Jq = �LQqXQ � LqqXq

Compare heat flow in an electrical insulator with heat flow in an electrical conductor

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