EntropyProperty Relationships

Chapter 7b

The T-ds relations

dUWQ outrevrev ,intint

Consider an internally reversible process occurring in a closed system

TdSQ rev int PdVW outrev ,int

dUPdVTdS

PdvduTds Or…

T

Pdv

T

duds

To find s all you have to do is integrate!!!

First Gibbs equation – also calledFirst Tds relationship

2nd Gibbs relationship

Recall that…

Pvuh

vdPPdvdudh

Find the derivative, dh

vdPPdvdhdu

Rearrange to find du

PdvduTds vdPPdvdhdu

vdPdhTds Second Tds relationship, or

Gibbs equation

T

vdP

T

dhds

To find s all you have to do is integrate

First Tds relationship

We have two equations for ds

T

vdP

T

dhds

T

Pdv

T

duds To find s, integrate

the equation that is the easiest, or for which you have the data

First lets look at solids and liquids

T

Pdv

T

duds

T

vdP

T

dhds

Solids and liquids do not change specific volume appreciably with pressure

That means that dv=0, so the first equation is the easiest to use.

0

T

duds

For solids and liquids…

Recall that…

CdTdu For solids and liquids, so…

T

CdT

T

duds

1

2lnT

TCs

Integrate to give…

Only true for solids Only true for solids and liquids!!and liquids!!

What if the process is isentropic? What happens to s?

0ln1

2

T

TCs

The only way this expression can equal 0 is if T2 = T1

For solids and liquids, isentropic processes are also isothermal, if they are truly incompressible

Entropy change of ideal gases

RTPv

dTCdu v

dTCdh p

Some equations we know for ideal gases

Let’s use these relationships with the Gibbs equations

T

Pdv

T

duds

T

vdP

T

dhds

v

Rdv

T

dTCv

P

RdP

T

dTC p

We can integrate these equations if we assume constant Cp and constant Cv

2

1

2

1 v

Rdv

T

dTCs v

1

2

1

2 lnlnv

vR

T

TCs v

Only true for ideal gases, assuming constant heat capacities

First Gibbs equation

We can integrate these equations if we assume constant Cp and constant Cv

2

1

2

1 P

RdP

T

dTCs p

1

2

1

2 lnlnP

PR

T

TCs p

Only true for ideal gases, assuming constant heat capacities

Second Gibbs equation

Use which ever equation is easiest!!

Which should you use?

Sometimes it is more convenient to calculate the change in entropy per mole, instead of per unit mass

2

1

2

112 P

dPR

T

dTCsss up

2

1

2

112 v

dvR

T

dTCsss uv

1

2

1

212 lnln

v

vR

T

TCsss uv

1

2

1

212 lnln

P

PR

T

TCsss up

What if it’s not appropriate to assume constant specific heats?

We could substitute in the equations for Cv and Cp, and perform the integrationsCp = a + bT + cT2 + dT3

That would be time consuming and error prone

There must be a better way!!

What if it’s not appropriate to assume constant specific heats?

Someone already did the integrations and tabulated them for usThey assume absolute 0 as the starting

point

2

2 0

0 )(T

pT T

dTTCs

2

1

01

02 )(

T

T p T

dTTCss

1

1 0

0 )(T

pT T

dTTCs

See Table A-17, pg 910

1

22

1lnP

PR

T

dTCs p

1

201

02 ln

P

PRsss

1

201

02 ln

P

PRsss u

So….

These two equations are good for ideal gases, and consider variable specific heats

Remember

Entropy of an Ideal GasEntropy of an Ideal Gas6-12

The entropy of an ideal gas depends on both The entropy of an ideal gas depends on both T T and and PP. The function . The function ss° represents only the temperature-dependent part of entropy° represents only the temperature-dependent part of entropy

Isentropic Processes of Ideal GasesMany real processes can be modeled

as isentropicIsentropic processes are the standard

against which we should measure efficiency

We need to develop isentropic relationships for ideal gases, just like we developed them for solids and liquids

1

2

1

2 lnlnv

vR

T

TCs v For the

isentropic case, S=0

1

2

1

2 lnlnv

vR

T

TCv

Constant specific heats

vC

R

v v

v

v

v

C

R

T

T

2

1

1

2

1

2 lnlnln

vC

R

v v

v

v

v

C

R

T

T

2

1

1

2

1

2 lnlnln

1

2

1

2

1

1

2

k

C

R

v

v

v

v

T

T v

First isentropic relation for ideal gases

vp CCR v

p

C

Ck and so 1k

C

R

v

Recall that…

Similarly0lnln

1

2

1

2

P

PR

T

TCs p

pC

R

p P

P

P

P

C

R

T

T

1

2

1

2

1

2 lnlnln

k

k

P

P

T

T1

1

2

1

2

Second isentropic relationship

Only applies to ideal gases, with constant specific heats

Since…

k

k

P

P

T

T1

1

2

1

2

1

2

1

1

2

k

v

v

T

T and

k

kk

P

P

v

v1

1

2

1

2

1

Which can be simplified to…

1

2

2

1

P

P

v

vk

Third isentropic relationship

1

2

1

1

2

k

v

v

T

T

k

k

P

P

T

T1

1

2

1

2

1

2

2

1

P

P

v

vk

constant

constant

constant1

1

k

k

k

k

Pv

TP

Tv

Compact form

That works if the heat capacities can be approximated as constant, but what if that’s not a good assumption?

1

201

02 ln

P

PRsss

We need to use the exact treatment

0

1

201

02 ln

P

PRss

This equation is a good way to evaluate property changes, but it can be tedious if you know the volume ratio instead of the pressure ratio

Relative Pressure and Relative Specific Volume

1

201

02 ln

P

PRss

R

ss

P

P 01

02

1

2 exp

Rs

Rs

P

P01

02

1

2

exp

exp

s20 is a function

only of temperature!!!

1

201

02 ln

P

P

R

ss

Rs

Rs

P

P01

02

1

2

exp

exp

1

2

1

2

r

r

P

P

P

P

Rename the exponential Pr , (relative pressure) which is only a function of temperature, and is tabulated on the ideal gas tables

You can use this equation or

1

201

02 ln

P

PRss

What if you know the volume ratio?

2

22

1

11

T

vP

T

vP

Ideal gas law

2

1

1

2

1

2

P

P

T

T

v

v

2

1

1

2

r

r

P

P

T

T

1

1

2

2

T

P

P

T r

r

Rename this vr2

Rename this 1/vr1

1

2

r

r

v

v

Relative specific volumes are also tabulated in the ideal gas tables

Remember, these relationships only hold for ideal gases and isentropic processes

Summary

We developed the first and second Gibbs relationships

PdvduTds

vdPdhTds

Which can also be expressed as

T

Pdv

T

duds

T

vdP

T

dhds

SummaryFor solids and liquids

T

Pdv

T

duds

Solids and liquids do not change specific volume appreciably with pressure, so dv=0

0

T

CdT

T

duds

1

2lnT

TCs

C can be approximated as a constant in solids and liquids

SummaryFor ideal gases if we assume constant heat capacities…

T

Pdv

T

duds

T

vdP

T

dhds

v

Rdv

T

dTCv

P

RdP

T

dTC p

Which can be integrated to give

1

2

1

2 lnlnv

vR

T

TCs v

1

2

1

2 lnlnP

PR

T

TCs p

True for ideal gas with constant heat capacities

SummaryFor ideal gases with variable heat capacity

1

201

02 ln

P

PRsss

SummaryWhat if its not an ideal gas?

You’ll need to use the tables

SummaryIsentropic Processes – Ideal Gas and Constant Heat Capacity

1

2

1

1

2

k

v

v

T

T

k

k

P

P

T

T1

1

2

1

2

1

2

2

1

P

P

v

vk constant

constant

constant1

1

k

k

k

k

Pv

TP

Tv

SummaryIsentropic processes for Ideal gases – Variable Heat Capacities

1

201

02 ln

P

PRss

1

2

1

2

r

r

P

P

P

P 2 2

1 1

r

r

v v

v v

SummaryIsentropic processes if the gas is not ideal and the heat capacities are variable

Use the tables!!