Lecture01 Slides

7/29/2019 Lecture01 Slides

1/18

3.205 L01 10/26/06

I r revers ib le

Thermodynamics

1


2/18

Therm odynamic s and K ine t i c s 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?

3.205 L01 10/26/06

Samuel M. Allen 2


3/18

A few m a th conc ept s Fields, scalar

Fields, vector

Fields, tensor

Variations of scalar fields

3.205 L01 10/26/06

Samuel M. Allen 3


4/18

Fields, scalar

Values of something are specified as a function of position,

, and (in kinetics) time.r

r =xi +yj+zk

Example: composition field

diffusing into a body

3.205 L01 10/26/06

4Samuel M. Allen

from a point sourcecr

r,t( )


5/18

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 fieldr

vr

r,t( )

3.205 L01 10/26/06

Samuel M. Allen 5


6/18

Fields, tensor

A second-rank tensor called the stress tensor

"

ij

r

r( ) relates forcer

Fat a point to an oriented area A

at that point

3.205 L01 10/26/06

Samuel M. Allen 6


7/18

Variations of scalar fields

Stationary field, moving point rr (a bug on a surface)c(

r

r +r

v dt) = c(r

r )+r

"c #r

v dt

dc

dt=

r

"c #r

v

Evolving field, moving pointr

r (a bug on a surface in an

earthquake)dc

dt=

r

"c #r

v +$c

$t

3.205 L01 10/26/06

Samuel M. Allen 7


8/18

Continuum limitsIs it possible to define a local value for concentration

in the limit ?"V# 0

It is, with suitable care.

3.205 L01 10/26/06

Samuel M. Allen 8

Figures removed due to copyright restrictions.See Figures 1.5 and 1.6 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Carter.

Kinetics of Materials. Hoboken, NJ: J. Wiley & Sons, 2005. ISBN: 0471246891.


9/18

Fluxes

The flux vectorr

Ji

represents mass flow of

component i at a point per unit oriented area at that

point

lim"A#0

Mi ("r

A)

"A=

r

Ji $ n

Example: swimming fish

3.205 L01 10/26/06

Samuel M. Allen 9


10/18

Accumulation

Flow field,Rate of production in dV,

r

J(r)

Accumulation of species i

"ci

"t= #$ %

r

Ji + &i

"i

For a conserved quantity like internal energy

"u

"t=#$ %

r

Ju

and for a non-conserved quantity like entropy

"S

"t= #$ %

r

JS+ &i3.205 L01 10/26/06 Samuel M. Allen

10


11/18

Syst em at Equi l ib r ium

Characterized by uniform values of various

potentials throughout the system e.g. T, P, i

Densities of conjugate extensive quantities e.g. S,

V,Ni can be inhomogeneous at equilibrium

Sis maximized in a system at constant

dS"q

T

U

For an isolated system

3.205 L01 10/26/06 Samuel M. Allen11


12/18

Syst em aw ay f rom Equi l ibr ium

Kinetics concerns the path, mechanisms, and ratesof spontaneous and driven processes.

Irreversible thermodynamics attempts to applythermodynamics principles to systems that are notin equilibrium and to suggest principles by which

they relax toward equilibrium or steady state.

3.205 L01 10/26/06 Samuel M. Allen12


13/18

Post u la t e 1

It is generally possible to use principles of thecontinuum limit to define meaningful, useful,

local values of various thermodynamicsquantities, e.g., chemical potential i.

Useful kinetic theories can be developed byassuming a functional relationship between the

rate of a process and the local departure fromequilibrium (driving force).

3.205 L01 10/26/06 Samuel M. Allen13


14/18

Post u la t e 2

For any irreversible process the rate of entropyproduction is everywhere 0.

"= #S#t

+$%r

JS& 0

More restrictive than dStot 0 for isolated system.

3.205 L01 10/26/06 Samuel M. Allen14


15/18

Post u la t e 3

Assume linear coupling between fluxesJandforcesX

J" = L"#X# ",#=Q,q,I,...,Nc

J" =$#

$"

L"#X"X#

Electrical + heat conductionJq = LqqXq +LqQXQ Directcoefficients, Lqq LQQ

JQ = LQqXq +LQQXQ Couplingcoefficients, LqQ LQq

3.205 L01 10/26/06 Samuel M. Allen15


16/18

Post u la t e 4

Onsager symmetry postulateL= L(microscopic reversibility).

L"# = L#"$%Jq

%XQ=

%JQ

%Xq

Similar to Maxwells relations in thermodynamics

3.205 L01 10/26/06 Samuel M. Allen16


17/18

Forc e/Flux Pai rs

Flux Conjugate Force Empirical law

r

JQrJqr

Ji

r

JQ="

K#Tr

Jq = "$#%r

Ji = "Mici#i

"

1

T#T

"#$

"#i

HeatCharge

Mass

FourierOhm

Fick

3.205 L01 10/26/06 Samuel M. Allen17


18/18

Mul t ip le fo rc es and f lux es

Consider simultaneous flow of heat and charge

r

JQ="L

QQX

Q"L

QqX

qrJq = "LQqXQ "LqqXq

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

3.205 L01 10/26/06 Samuel M. Allen18

Lecture01 Slides

Documents

Transcript of Lecture01 Slides