exact differential in thermodynamics
TRANSCRIPT
Exact differential for the thermodynamics
Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton
Binghamton, NY 13902-6000
(date: September 05, 2018)
1. Exact differential
We work in two dimensions, with similar definitions holding in any other number of
dimensions. In two dimensions, a form of the type
dyyxBdxyxA ),(),(
is called a differential form. This form is called exact if there exists some scalar function ),( yxQ
BdyAdxdyy
Qdx
x
QdQ
xy
,
if and only if
yxx
B
y
A
or
yxxy y
Q
xx
Q
y
The vector field ),( BA is a conservative vector field, with corresponding potential Q. When a
differential Q is exact, the function Q exist,
)()( iQfQdQ
f
i
,
independent of the oath followed.
In thermodynamics, when dQ is exact, the function Q is a state function of the system. The
thermodynamics functions, E (or U) , S, H, F (or A), and G are state functions. An exact
differential is sometimes also called a total differential or a full differential.
2. Jacobian
We consider two functions u and v such that
),( yxuu , ),( yxvv
The Jacobian is defined as
y
v
x
v
y
u
x
u
yx
vu
),(
),(.
It clearly has the following properties
),(
),(
),(
),(
yx
vu
yx
uv
yx
uy
u
x
u
y
y
x
y
y
u
x
u
yx
yu
10),(
),(.
The following relations also hold:
),(
),(
1
),(
),(
vz
yxyx
yz
y
y
y
t
x
t
z
yt
yx
yt
yz
yx
yz
x
z
),(
),(
),(
),(
),(
),(
1),(
),(
),(
),(
),(
),(
xz
xy
zy
zx
yx
yz
ux
y
x
y
y
u
ux
uy
xy
xu
xu
yu
xy
xu
yx
yu
x
u
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
3. Expression of PC and VC
The heat capacity:
T
V
V
P
V
PT
VS
T
PT
VT
PT
VS
TVT
VST
T
STC
),(
),(
),(
),(
),(
),(
),(
),(
or
T
PT
P
PTTP
T
TP
TP
T
V
P
V
T
V
P
ST
T
ST
T
V
P
S
P
V
T
S
P
V
T
P
V
T
V
P
S
T
S
P
V
TC
][
Using the Maxwell’s relation
PT T
V
P
S
and the relation
T
T
V
P
TV
TPTP
TV
P
V
1
),(
),(
1
),(
),(
Then we get
TP
P
T
PPV
V
P
T
VTC
P
V
T
VT
CC
2
2
or
TP
VPV
P
T
VTCC
2
For the ideal gas we have the Mayer relation
RCC VP
This is one of the most important equations of thermodynamics, and it shows that:
(a) 0
TV
P, leading to VP CC
(b) As 0T , VP CC
(c) VP CC when 0
PT
V.
s
For example, at 4°C at which the density of water is maximum, VP CC .
Using the volume expansion and isothermal compressibility
PT
V
V
1
,
T
T
V
PVP
V
V
111
We may write the equation in the form
2TV
CC VP
4. Legendre transformation
Legendre transforms appear in two places in a standard undergraduate physics curriculum: (i)
in classical mechanics and (ii) in thermodynamics. The Legendre transform simply changes the
independent variables in a function of two variables by application of the product rule.
We start with the expression for the internal energy U as
PdVTdSdU
(a) Helmholtz free energy: F
Using the relation
PdVSdTTSddU )(
we have
PdVSdTTSUd )(
Here we introduce the Helmholtz free energy F as
STUF
Thus we have the relation
PdVSdTdF
(b) Enthalpy: H
From the relation
VdPPVdTdS
PdVTdSdU
)(
we have
VdPTdSPVUd )(
Here we introduce the enthalpy H as
PVUH
Thus we have
VdPTdSdH
(c) Gibbs Free energy:
Using the relation
VdPPVdSdT
PdVSdTdF
)(
we have
\
VdPSdTPVFd )(
We introduce the Gibbs free energy G as
PVFG
with
VdPSdTdG
5. Thermodynamic potential
Fig. Born diagram. UE (internal energy). FA (Helmholtz free energy).
(a) The internal energy ),( VSUU
For an infinitesimal reversible process, we have
PdVTdSdU
showing that
VS
UT
, SV
UP
and
VS S
P
V
T
(Maxwell’s equation)
(b) The enthalpy ),( PSHH
The enthalpy H is defined as
PVUH
S
V
T
P
E
FG
H
For an infinitesimal reversible process,
VdPPdVdUdH
but
PdVTdSdU
Therefore we have
VdPTdSdH
showing that
PS
HT
, SP
HV
and
PS S
V
P
T
. (Maxwell’s equation)
(c) The Helmholtz free energy ),( VTFF
The Helmholtz free energy is defined as
STUF
For an infinitesimal reversible process,
PdVSdT
TdSSdTPdVTdS
TdSSdTdUdF
or
PdVSdTdF
showing that
VT
FS
, TV
FP
VT T
P
V
S
(Maxwell’s equation)
We note that
V
V T
F
TT
T
FTFSTFU ][2
(Gibbs-Helmholtz equation)
(d) The Gibbs free energy ),( PTGG
The Gibbs free energy is defined as
TSHPVFG
For an infinitesimal reversible process,
VdPSdT
TdSSdTVdPTdS
TdSSdTdHdG
or
VdPSdTdG
with
VdPTdSdH ,
showing that
PT
GS
, TP
GV
and
PT T
V
P
S
(Maxwell’s equation)
We note that
PP T
G
TT
T
GTGTSGH
2 (Gibbs-Helmholtz equation)
6. Relations between the derivatives of thermodynamic quantities
(a) First energy equation
PdVTdSdU
PV
ST
PdVTdSVV
U
T
TT
)(
Using the Maxwell’s relation VT T
P
V
S
, we get
PT
PT
V
U
VT
(First energy equation)
which is called the first energy equation. For the ideal gas ( TNkPV B ), we can make a proof of
the Joule’s law.
0
TV
U
In other words, U is independent of V: TCU V (Joule’s law for ideal gas)
(b) Second energy equation
PdVTdSdU
TT
TT
P
VP
P
ST
PdVTdSPP
U
)(
Using the Maxwell’s relation PT T
V
P
S
, we get
TPT P
VP
T
VT
P
U
(second energy equation)
which is called the second energy equation. For an ideal gas,
2( ) 0
T
U R RTT P
P P P
In other words, U is independent of P: TCU V (Joule’s law for ideal gas)
7. Generalized susceptibility
P
PT
V
V
1
; isobaric expansivity
S
ST
V
V
1
; adiabatic expansivity
T
TP
V
V
1
; isothermal compressibility
S
SP
V
V
1
; adiabatic compressibility
8. The ratio V
P
C
C
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
TV
SV
TP
SP
TP
SP
SV
TV
SV
SP
TP
TV
P
V
P
V
S
T
S
T
or
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
TV
SV
TP
SP
TP
SP
SV
TV
SV
SP
TP
TV
P
V
P
V
S
T
S
T
or
V
P
V
P
S
T
C
C
T
ST
T
ST
VT
VS
PT
PS
),(
),(
),(
),(
9. Mayer’s relation
(a) Relation
T
VVP
V
P
T
P
TCC
2
The thermodynamics first law:
PdVTdSdU
PP
PT
ST
T
QC
Transforming PC to the variable T and P,
T
VT
V
VTTV
T
TV
TV
T
P
V
P
T
P
V
S
T
S
T
P
V
S
V
P
T
S
V
P
V
P
T
P
V
S
T
S
V
P
VT
PT
VT
PS
PT
PS
T
S
][1
1
),(
),(
),(
),(
),(
),(
Here we use the Maxwell’s relation,
VT T
P
V
S
Thus we get
T
V
VP
V
P
T
P
TT
ST
T
ST
2
or
T
VVP
V
P
T
P
TCC
2
Note that
T
T
P
V
TP
TVTV
TP
V
P
1
),(
),(
1
),(
),(
T
P
V
P
V
T
V
PT
VT
PT
VP
VT
VP
T
P
),(
),(
),(
),(
),(
),(
Then we have
T
P
T
PVP VT
P
V
T
V
TCC 2
2
(b) Relation
T
PVP
P
V
T
V
TCC
2
Similarly, transforming VC to the variable T and V,
T
PT
P
PTTP
T
TP
TP
T
V
P
V
T
V
P
S
T
S
T
V
P
S
P
V
T
S
P
V
P
V
T
V
P
S
T
S
P
V
PT
VT
PT
VS
VT
VS
T
S
][1
1
),(
),(
),(
),(
),(
),(
Using the Maxwell’s relation
PT T
V
P
S
we have
T
P
PV
P
V
T
VT
T
ST
T
ST
2][
or
T
P
T
PVP VT
P
V
T
V
TCC 2
2][
10. Derivative of CV with respect to T (Reif, Blundell)
TV
TV
VTT
V
V
S
TT
V
S
TT
VT
ST
TV
ST
T
ST
VV
C
2
2
][
Using the Maxwell’s relation
VT T
P
V
S
we get
VVVT
V
T
PT
T
P
TT
V
C
2
2
11. Derivative of PC with respect to P (Reif, Blundell)
TP
TP
PTT
P
P
S
TT
P
S
TT
PT
ST
TP
ST
T
ST
PP
C
2
2
][
Using the Maxwell’s relation
PT T
V
P
S
we get
PPPT
P
T
VT
T
V
TT
P
C
2
2
12. Joule-Thomson expansion
P
T
H
T
H
P
H
TP
HP
TP
HT
HP
HT
P
T
),(
),(
),(
),(
),(
),(
Note that
VdPTdSdH
So we get
VT
VT
VP
ST
VdPTdSPP
H
T
T
TT
)(
P
P
PP
C
T
ST
VdPTdSTT
H
)(
Thus we have
][1
VT
VT
CP
T
TPH
REFERENCES
L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd edition, revised and enlarged (Pergamon,
1980).
F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).
S. Blundel and K. Blundel, Concepts of Thermal Physics (Oxford, 2015).
APPENDIX
Stokes theorem and conservative force
1 Conservative force
1.1 Path integral
The work done by a conservative force on a particle moving between any two points is
independent of the path taken by the particle.
321 Pathc
Pathc
Pathc ddd rFrFrF
for any path connecting two points A and B.
1.2 Path integral along the closed path
The work done by a conservative force on a particle moving through any closed path is zero.
(A closed path is one for which the beginning point and the endpoint are identical).
0 rF dc , for any closed path
2 Potential energy U
The quantity rF dc can be expressed in the form of a perfect differential
rF dWdU cc
where the function U(r) depends only on the position vector r and does not depend explicitly on
the velocity and time. A force Fc is conservative and U is known as the potential energy.
)()( BUAUdUd
B
A
B
A
c rF
which does not depend on the path of integration but only on the initial and final positions. It is
clear that the integral over a closed path is zero
0 rF dc (1)
which is a different way of saying that the force field is conservative
Using Stoke’s theorem;
For any vector A,
aArA dd )( .
Since
0)( aFrF dd cc
we have
0 cF , (2)
where is a differential operator called del or nabla. The operator can be written in n terms of
the Cartesian components x, y, z in the form
zyx
kji ˆˆˆ
In this case, Fc can be expressed by
Uc F
or
z
U
y
U
x
Uc ,,F
which leads to the relation
z
F
x
F
y
F
z
F
x
F
y
F
cxcz
czcy
cycx
.
This relation can be used to decide whether a force is conservative or not on physical grounds.
We note that
y
F
x
F
z
F
x
F
z
F
y
F
FFyx
FFzx
FFzy
FFFzyx
cxcycxczcycz
cycxczcxczcy
czcycx
c
kji
kji
kji
F
ˆˆˆ
ˆˆˆ
ˆˆˆ
3 Stoke’s theorem
Stokes' theorem (or Stokes's theorem) in differential geometry is a statement about the
integration of differential forms which generalizes several theorems from vector calculus. It is
named after Sir George Gabriel Stokes (1819–1903), although the first known statement of the
theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes. The
theorem acquired its name from Stokes' habit of including it in the Cambridge prize
examinations. In 1854, he asked his students to prove the theorem on an examination. It is
unknown if anyone was able to do so.
Let’s start off with the following surface with the indicated orientation.
Around the edge of this surface we have a curve C. This curve is called the boundary curve.
The orientation of the surface S will induce the positive orientation of C. To get the positive
orientation of C think of yourself as walking along the curve. While you are walking along the
curve if your head is pointing in the same direction as the unit normal vectors while the surface is
on the left then you are walking in the positive direction on C.
Now that we have this curve definition out of the way we can give Stokes’ Theorem.
Stokes’ Theorem
Let S be an oriented smooth surface that is bounded by a simple, closed, smooth boundary
curve C with positive orientation. Also let F be a vector field, then
dadd nFaFrF )()(
In this theorem note that the surface S can actually be any surface so long as its boundary curve
is given by C. This is something that can be used to our advantage to simplify the surface
integral on occasion.
4. Path integral and conservative force
((Example)) Serway Problem 7-15
A force acting on a particle moving in the x y plane is given by
Njxiy )ˆˆ2( 2F ,
where x and y are in meters. The particle moves from the origin to a final position having
coordinates x = 5.00 m and y = 5.00 m as in Fig. Calculate the work done by F along (a) OAC,
(b) OBC, (c) OC. (d) Is F conservative or nonconservative? Explain.
((Solution))
dyxydxdyFdxFd yx
22 rF
Path OAC
On the path OA; x = 0 – 5, y = 0; jx ˆ2F and dy = 0.
0. A
O
drF
On the path AC; x = 5, y = 0 - 5; jj ˆ25ˆ52 F
5
0
12525. dyd
C
A
rF
Then we have
Jd
C
O
125. rF for the path OAC
Path OBC
On the path OB; x = 0, y = 0 - 5;
ydxd 2 rF and dx = 0
or
0B
O
drF
On the path BC; x = 0 - 5, y = 5; dxydxd 102 rF
5
0
5010dxd
C
B
rF
Then we have
50C
O
drF J for the path OBC
Path OC
x = t, y = t for t = 0 – 5.
dtttdtttdtdyxydx )2(22 222
Jdtttd
C
O3
200
2
52
3
125)2(
25
0
2 rF
for the path OC line. Then the force is non-conservative.
((Note))
Nxy )ˆˆ2( 2 jiF