Wwilson@uco.edu Thermodynamics of a Single Particle System W. J. Wilson Department of Engineering and Physics University of Central Oklahoma Edmond, OK.

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Thermodynamics of a Single Particle System

W. J. Wilson

Department of Engineering and PhysicsUniversity of Central Oklahoma

Edmond, OK 73034

2011 Oklahoma Academy of Sciences AnnualTechnical MeetingSoutheastern Oklahoma State University

Durant, OKNovember 11, 2011

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Thermodynamics of a Particle?Single particle thermodynamics is a bit of an uncharted subject.

Some, steeped in statistical basis of thermodynamics, conclude that the laws of thermodynamics fall apart at the single particle level.

“The law of entropy increasing is only a statistical law; it is not ‘fundamental’ because it cannot describe the behavior of an individual atom or molecule; it deals with the average number of them. Entropy is not a concept that can be meaningfully applied to a single particle, or even to a small number of particles.” – John Wheeler

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Thermodynamics of a Single Particle

đQ

21v

2dU m

đW d F r

v

F

System

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First Law – Sign ConventionThe positive work done by system sign convention (used in physics and engineering) is

đQ dU đW

Energy Internal Work Done

toSystem Ener

Added In

gy of System the Syste

cr

m

ease By

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The positive work on system sign convention (used in chemistry and some upper-level physics treatments) is

đQ dU đW

Energy Internal Work Done

toSystem Ener

Added In

gy of System the Syste

cr

m

ease On

Energy Work Done Internal

toSystem the System Energy of Syst

Added On Incre

em

ase

đQ đW dU

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For a single particle, U = KE+PE = U(x,v) and F = F(x,v) are functions of position and velocity only so if we use the work on system positive sign convention

which becomes for a free(non-relativistic) particle,

đQ dU đW đQ dU Fdx

2 21 1v v v v

2 2U m dU d m m d

v vđQ m d Fdx

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Now if the change dx and dv occur in time dt

So for a single particle system,

v v

vv

đQ m d Fdx

d dxm dt F dt

dt dt

vv v

dđQ m F dt

dt

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So for a single particle system,

And for a closed system (no energy allowed to enter or escape)

(i.e., an “Adiabatic” Process)We find

vv

dđQ m F dt

dt

0đQ

v0 v

dm F dt

dt

vd

F mdt

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So for a single particle system, using the work done on the system is positive sign convention,

1.Yields Newton’s 2nd Law in the standard form2.Particle motions governed by Newton’s 2nd Law correspond to “adiabatic” processes

đQ dU Fdx

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Integrating FactorFor a single particle, U = KE+PE = U(x,v) and F = F(x,v) are functions of position and velocity only so

vv

vv

đQ dU Fdx

U Udx d Fdx

x

U UF dx d

x

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For “adiabatic” processes

or

The right-hand side is a function of x and v only, so we are guaranteed a unique solution through a given initial state (x0,v0)

v 0v

U UđQ F dx d

x

v

v xd U F

dx U

v v( ) or ( .v)x x C

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For “adiabatic” processes

v

v0 xd U F

đQdx U

( .v)x

( .v)x v

x12

( .v)x

(v, )x x or

v v( , )x

0 0( .v )x

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For “adiabatic” processes

and

So we must have a factor λ(x,v) such that

or

v 0v

U UđQ F dx d

x

( .v) v 0v

x d dx dx

v vv v

U UF dx d dx d

x x

and vv v

U UF

x x

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Now since

These yield an exact differential

As

So

and vv v

U UF

x x

vv

U UđQ dU Fdx F dx d

x

v vv v

đQ dx d dx dx x

đQ d d đQ

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Summary / Conclusions

15

• The natural sign convention for physics

• There is an exact differential (“entropy”) for the single particle

ReferencesE.C.G. Stueckelberg and P. B. Scheurer, Thermocinéque Phénoménologique Galiléenne, (Birkhauser, Stuttgart, 1974)

P.E. Williams, On a Possible Formulation of Particle Dynamics in Terms of Thermodynamics Conceptualizations and the Role of Entropy in It (M.S. Thesis, Naval Postgraduate School, Monterey, CA, 1976).

đQ dU đW dU Fdx

where ( , v) (v, )đQ

d x or