thermo & stat mech - spring 2006 class 7 1 thermodynamics and statistical mechanics tds...
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Thermo & Stat Mech - Spring 2006 Class 7
1
Thermodynamics and Statistical Mechanics
Tds Equations
Thermo & Stat Mech - Spring 2006 Class 7
2
Entropy and Available Energy
Consider two identical bodies, each of mass m, and heat capacity cP, but one at temperature T1, and the other at temperature T2. When placed in contact, heat flows from the hotter to the cooler, until they come to a common temperature,
221 TT
T f
Thermo & Stat Mech - Spring 2006 Class 7
3
Entropy Change
21
21
21
2
2121
2
21
2ln2
ln2lnln
lnln1 2
TT
TTmcS
TT
Tmc
TT
Tmc
TT
Tmc
T
T
T
Tmc
T
dT
T
dTmcS
P
fP
fP
fP
ffP
T
T
T
TP
f f
Thermo & Stat Mech - Spring 2006 Class 7
4
Entropy Increased
0)(
02
42
2
221
212
22
1
21212
22
1
2121
TT
TTTT
TTTTTT
TTTT
Thermo & Stat Mech - Spring 2006 Class 7
5
“Lost” Work
2112
2
12
2
121
1
2
2
1
1
2
2
1
1
2
2
10ln
0lnln
0
12
TTTTT
T
TT
T
T
T
T
T
T
dT
T
dT
T
dT
T
dTmc
T
Qd
T
Qd
fff
ffT
T
T
T
P
ff
Run a Carnot engine between the two bodies.
Thermo & Stat Mech - Spring 2006 Class 7
6
Amount of Work
2121
2121
21
12
12
22
2
2
)()(
TTTT
mcW
TTTTmcW
TTTmcW
TTTTmcW
QQW
P
P
fP
ffP
Thermo & Stat Mech - Spring 2006 Class 7
7
Tds Equations
dPc
dvv
cdP
P
Tcdv
v
TcTds
dPTvdTcdPT
vTdTcTds
dvT
dTcdvT
PTdTcTds
vP
vv
PP
PP
P
vv
v
Thermo & Stat Mech - Spring 2006 Class 7
8
Internal Energy
Tds = du + Pdv where u(T,v)
dvPvu
dTcTds
dvvu
dTTu
du
Tv
Tv
Thermo & Stat Mech - Spring 2006 Class 7
9
Enthalpy
dPvPh
dTcvdPdhTds
vdPTdsvdPPdvdudh
Pvuh
dPPh
dTcdPPh
dTTh
dh
PThh
TP
TP
TP
so ,
),(For
Thermo & Stat Mech - Spring 2006 Class 7
10
First Tds Equation
Pvu
Tvs
Tc
Ts
dvvs
dTTs
ds
dvPvu
TdT
Tc
ds
dvPvu
dTcTds
TT
v
v
Tv
T
v
Tv
1
1
Thermo & Stat Mech - Spring 2006 Class 7
11
First Tds Equation (cont.)
vTT
v
TT
v
v
Pvu
TTTc
v
vTs
Tvs
Pvu
Tvs
Tc
Ts
1
1
22
Thermo & Stat Mech - Spring 2006 Class 7
12
First Tds Equation (cont.)
Pvu
TP
T
Pvu
TTP
T
Pvu
TTP
vTu
TP
vu
TT
Tvu
TTu
TvTc
v
Tv
Tv
TvvT
TvT
v
2
2
2
2
11
111
11
Thermo & Stat Mech - Spring 2006 Class 7
13
First Tds Equation (cont.)
dvTP
TdTcTds
dvPvu
dTcTds
vv
Tv
Thermo & Stat Mech - Spring 2006 Class 7
14
cP – cv
vPvu
cc
vPvu
cc
dTdv
Pvu
Tu
dTqd
Tvp
Tvp
PTvp
Thermo & Stat Mech - Spring 2006 Class 7
15
cP – cv
2
1
1
Tvcc
v
Pv
v
Tv
vTvTP
Tcc
vPvu
cc
vp
T
P
vvp
Tvp
Thermo & Stat Mech - Spring 2006 Class 7
16
Ideal Gas
P
RTPRT
vPRT
PRT
PvPv
v
T
RTR
vPR
PRT
TvTv
v
TT
PP
1
11
1
11
2
Thermo & Stat Mech - Spring 2006 Class 7
17
Ideal Gas
RccTPv
PT
Tvcc
Tvcc
vp
vp
vp
2
2
1
Thermo & Stat Mech - Spring 2006 Class 7
18
Third Tds Equation
v
v
v
vvvvv
Pv
P
T
T
c
P
s
P
T
T
sT
TP
T
T
s
P
s
dvv
sdP
P
sds
vPss
1
),( Consider
Thermo & Stat Mech - Spring 2006 Class 7
19
Third Tds Equation
P
P
P
pPPPP
Pv
v
T
T
c
v
s
v
T
T
sT
Tv
T
T
s
v
s
dvv
sdP
P
sds
1
Thermo & Stat Mech - Spring 2006 Class 7
20
Third Tds Equation
dvv
TcdP
P
TcTds
dvv
T
T
cdP
P
T
T
cds
dvv
sdP
P
sds
PP
vv
P
P
v
v
Pv
Thermo & Stat Mech - Spring 2006 Class 7
21
Tds Equations
dPc
dvv
cdP
P
Tcdv
v
TcTds
dPTvdTcdPT
vTdTcTds
dvT
dTcdvT
PTdTcTds
vP
vv
PP
PP
P
vv
v
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