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