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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
[email protected]@uco.edu
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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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