UPME – ME 267 – ENQ – 6/2010 1

THERMODYNAMIC RELATIONS A LITTLE MATH—PARTIAL DERIVATIVES AND ASSOCIATED RELATIONS For a function z that depends on two independent variables (x,y) such that ( , )z z x y= (1.1) its partial derivative

0 0

( , ) ( , )lim limx x

y y

z z z x x y z x yx x x∆ → ∆ →

∂ ∆ + ∆ − = = ∂ ∆ ∆

represents the variation of z with respect to one variable while holding the other variable constant. The partial derivative of z with respect to x at constant y is

(1.2)

and the partial derivative of z with respect to y at constant x is

0 0

( , ) ( , )lim limy y

x x

z z z x y y z x yy y y∆ → ∆ →

∂ ∆ + ∆ −= = ∂ ∆ ∆

(1.3)

∂ represents the partial differential change due to the variation of a

single variable

Also note that the value of the partial derivative

y

zx∂

∂

in general, is different at different y values.

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The total differential

change of a function reflects the influence of all variables. Specifically, the total differential change in z(x, y) for simultaneous changes in x and y, is given by

y x

z zdz dx dyx y

∂ ∂ = + ∂ ∂ (1.4)

This is the fundamental relation for the total differential of a dependent variable in terms of its partial derivatives with respect to the independent variables

Note:

• The changes indicated by d and ∂ are identical for independent variables

, but not for dependent variables.

For example, (∂x)y

but = dx

(∂z)y ≠ dz = (∂z)x + (∂z)y

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Partial Differential Relations If

( , )z z x y= is a continuously variable function of (x,y), then its total differential

is

dz M dx N dy= + (1.5) where

,y x

z zM Nx y

∂ ∂ = = ∂ ∂ (1.6)

Also, dz is an

exact differential if

2 2

yx

M z N zy x y x y x

∂ ∂ ∂ ∂ = = = ∂ ∂ ∂ ∂ ∂ ∂ (1.7)

Reciprocity Relation

( )1/y y

zx x z∂ = ∂ ∂ ∂

(1.8)

Cyclic Relation

1y xz

z x yx y z

∂ ∂ ∂ = − ∂ ∂ ∂ (1.9)

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Maxwell Relations These are equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible system to each other

s v

T Pv s

∂ ∂ = − ∂ ∂ (1.10)

s P

T vP s∂ ∂ = ∂ ∂ (1.11)

T v

s Pv T∂ ∂ = ∂ ∂ (1.12)

T P

s vP T∂ ∂ = − ∂ ∂ (1.13)

• These are called the Maxwell relations (Fig. 12–8). They are extremely valuable in thermodynamics because they provide a means of determining the change in entropy in terms of changes in properties P, v, and T.

• Note that the Maxwell relations given above are limited to

simple compressible systems.

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THE CLAPEYRON EQUATION • Enables us to determine the enthalpy change associated with a phase change (such

as the enthalpy of vaporization hfg

) from a knowledge of P, v, and T data alone.

Example: Vaporization Process at constant pressure and temperature The development of the enthalpy change relation starts with the 3rd

Maxwell relation

T v

s Pv T∂ ∂ = ∂ ∂

(1.12)

In the case of liquid-vapor phase change or vaporization process,

1. P = Psat is a function of T = Tsat only → Psat = f(Tsat

)

2. Then

v sat

P PT T∂ ∂ = ∂ ∂

(1.14)

which represents the slope of the saturation line

For the vaporization process (sat. liquid to sat. vapor, or f – g) at a given or constant temperature,

on a P-T diagram

.sat

P constT∂ = ∂

so that combining (1.12) and (1.14) gives

g f fg

g f fgsat

s s sPT v v v

−∂ = = ∂ −

or

fg fgsat

Ps vT∂ = ∂

(1.15)

The entropy change during vaporization is the product of the change in specific volume and the slope (∂P/∂T) of the vaporization curve at the vaporization temperature Tsat

.

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The enthalpy change for the process is obtained using the relation

dh = T ds + v dP (1.16) Since the pressure is constant during vaporization, dP = 0. Integrating eq. (1.16) noting that T = const. = Tsat

during vaporization,

fg fgsat

Ph T vT∂ = ∂

(1.17)

• The Clapeyron equation is applicable to any phase-change process that occurs at

constant temperature and pressure. It can be expressed in a general form as

12

12sat

hPT T v∂ = ∂

(1.18)

where the subscripts 1 and 2 indicate the two phases.

Some Simplifications:

If all the following conditions exist,

1. At Low Pressures, vg >> vf so that vfg ≈ vg

,

2. and treating the vapor as an ideal gas, vg

≈ RT/P

3. and for small temperature intervals, hfg)T1 ≈ hfg)T2 ≈ hfg

≈ const. ,

then

for liquid-vapor and solid-vapor phase changes.

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GENERAL RELATIONS FOR du, dh, ds, cv, & cp

IN TERMS OF P, v, and T

• The following discussion presents the development of relations for

calculating changes in internal energy, enthalpy, and entropy

in terms of the measurable properties pressure, specific volume, temperature, and specific heats.

• Once the state of a system is defined by two independent properties (i.e., the state postulate), the other properties may be calculated using the relations developed here

• An expression relating cp and cv

will be developed so that only one of the specific heats need to be measured – the other specific heat is calculated

• Property values at specified states can be determined after the selection of an arbitrary reference state (which has “assigned” values of properties)

Internal Energy Change - du From the state postulate, let ( , )u u T v= (1.1) Its total differential is then

v T

u udu dT dvT v∂ ∂ = + ∂ ∂

(1.2)

Recall that the definition of the specific heat at constant volume is

vv

ucT∂ ≡ ∂

(1.3)

Substituting (1.3) into (1.2),

vT

udu c dT dvv∂ = + ∂

(1.4)

In this expression, we want to express T

uv∂

∂ in terms of P, v, T so that we can finally

calculate du in terms of measurable parameters. Another relationship for du in terms of P, v, T, and s which can be invoked comes from one of the Gibbs equations: du T dS P dv= − (1.5)

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We now express dS in terms of P, v, T by letting (using the state postulate) entropy be a function of the same variables (T, v) as internal energy. Thus ( , )s s T v= (1.6) and its total differential is then

v T

s sds dT dvT v∂ ∂ = + ∂ ∂

(1.7)

Substituting (1.7) into (1.5) to eliminate ds ,

v T

s sdu T dT dv P dvT v

∂ ∂ = + − ∂ ∂

Or

v T

s sdu T dT T P dvT v

∂ ∂ = + − ∂ ∂ (1.8)

Between eqs. (1.4) and (1.8), the former looks easier to express in terms of P, v, T. Note that both eqs. (1.8) and (1.4) can independently calculate du which implies that their RHS are equal. Specifically, the coefficients of dT should be equal; the same is true for the coefficients of dv . Therefore,

vv

sc TT∂ = ∂

(1.9)

and

T T

u sT Pv v∂ ∂ = − ∂ ∂

(1.10)

Eq. (1.10) is now substituted in eq.(1.4) to obtain

vT

sdu c dT T P dvv

∂ = + − ∂ (1.11)

The partial derivative of entropy in the above equation is then replaced by the Maxwell relation

T v

s Pv T∂ ∂ = ∂ ∂

(1.12)

To finally obtain du in terms of measurable properties P, v, T and cv

as

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vv

Pdu c dT T P dvT

∂ = + − ∂ (1.13)

This can be integrated between two states at (T1, v1) and (T2, v2) to determine (u2 – u1

):

2 2

1 12 1

T v

vT vv

Pu u c dT T P dvT

∂ − = + − ∂ ∫ ∫ (1.14)

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Enthalpy Change - dh From the state postulate, let ( , )h h T P= (1.15) Its total differential is then

P T

h hdh dT dPT P∂ ∂ = + ∂ ∂

(1.16)

Recall that the definition of the specific heat at constant pressure is

pP

hcT∂ ≡ ∂

(1.17)

Substituting (1.17) into (1.16),

pT

hdh c dT dPP∂ = + ∂

(1.18)

In this expression, we want to express T

hP∂

∂ in terms of P, v, T so that we can finally

calculate dh in terms of measurable parameters. Another relationship for dh in terms of P, v, T, and s which can be invoked comes from one of the Gibbs equations: dh T ds v dP= + (1.19) We now express dS in terms of P, v, T by letting (using the state postulate) entropy be a function of the same variables (T, P) as enthalpy. Thus ( , )s s T P= (1.20) and its total differential is then

P T

s sds dT dPT P∂ ∂ = + ∂ ∂

(1.21)

Substituting (1.21) into (1.19) to eliminate ds ,

P T

s sdh T dT v T dPT P

∂ ∂ = + + ∂ ∂ (1.22)

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Between eqs. (1.18) and (1.22), the former looks easier to express in terms of P, v, T. Note that both eqs. (1.22) and (1.18) can independently calculate dh which implies that their RHS are equal. Specifically, the coefficients of dT should be equal; the same is true for the coefficients of dP . Therefore,

pP

sc TT∂ = ∂

(1.23)

and

T T

h sv TP P∂ ∂ = + ∂ ∂

(1.24)

Eq. (1.24) is now substituted into eq.(1.18) to obtain

pT

sdh c dT v T dPP

∂ = + + ∂ (1.25)

The partial derivative of entropy in the above equation is then replaced by the 4th

Maxwell relation

T P

s vP T∂ ∂ = − ∂ ∂

(1.26)

To finally obtain dh in terms of the measurable properties P, v, T and cp

as

pP

vdh c dT v T dPT

∂ = + − ∂ (1.27)

This can be integrated between two states at (T1, P1) and (T2, P2) to determine (h2 – h1

):

2 2

1 12 1

T P

pT PP

vh h c dT v T dPT

∂ − = + − ∂ ∫ ∫ (1.28)

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Entropy Change - ds Two general relations for entropy change of a simple compressible system can be developed: one in terms of T and v, another in terms of T and P. 1st

Relation – In terms of ( T, v )

Let ( , )s s T v= (1.29) its total differential is then

v T

s sds dT dvT v∂ ∂ = + ∂ ∂

(1.30)

From the development of a general relation for du, where u = u(T,v), it has been shown that

v

v

c sT T

∂ = ∂ (1.31)

Also from the 3rd

Maxwell relation

T v

s Pv T∂ ∂ = ∂ ∂

(1.32)

Substituting (1.31) and (1.32) into (1.30) yields

v

v

c Pds dT dvT T

∂ = + ∂ (1.33)

or

2 2

1 12 1

T

T

v

v

v

v

P

T

cs dT dv

Ts

δ

∂ − =

+∫ ∫ (1.34)

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2nd

Relation – In terms of ( T, P )

Let ( , )s s T P= (1.35) its total differential is then

P T

s sds dT dPT P∂ ∂ = + ∂ ∂

(1.36)

From the development of a general relation for dh, where h = h(T,P), it has been shown that

P

P

c sT T

∂ = ∂ (1.37)

Also from the 4th

Maxwell relation

P P

s vT T∂ ∂ = − ∂ ∂

(1.38)

Substituting (1.37) and (1.38) into (1.36) yields

P

P

c vds dT dPT T

∂ = − ∂ (1.39)

or

2 2

1 12 1

T P

T P

P

P

v

T

cs dT dP

Ts

δ

∂ − =

−∫ ∫ (1.40)

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Relations for Specific Heats cv and cp

For a general pure substance the specific heats depend on pressure or specific volume as well as temperature. Relation Between Zero Pressure (or Ideal Gas) Specific Heats and Higher Pressure

Specific Heats

• Recall that ideal gas specific heats cv0 and cp0

, which are easier to determine, are valid for low pressures or high specific volume

• A relation for relating ideal gas specific heats to those at higher pressures can be developed as follows: Consider the two relations for entropy change derived previously which contain specific heats in the equations,

v

v

c Pds dT dvT T

∂ = + ∂ (1.33)

P

P

c vds dT dPT T

∂ = − ∂ (1.39)

Applying the test for exactness

yx

M Ny x

∂ ∂ = ∂ ∂

for a differential dz M dx N dy= +

on eqs. (1.33) and (1.39) give

2

2v

vT

c PTv T

∂ ∂= ∂ ∂

(1.41)

and

2

2p

PT

c vTP T

∂ ∂= − ∂ ∂

(1.42)

The deviation of cp from cp0 with increasing pressure can be determined by integrating eq.(1.42) from zero pressure to any pressure P along a constant-temperature process

. Thus

( )2

200

P

p p TP

v

Tc dPc T ∂

∂

− =

− ∫ (1.43)

The P-v-T behavior of the substance is necessary to integrate the RHS of the above equation.

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Relation Between cp and cv

A relation between cp and cv will be advantageous since only one of the two specific heats needs to be measured (usually cp

) and the other just calculated.

The relation between cp and cv

is developed by equating the two entropy change relations derived earlier

v

v

c Pds dT dvT T

∂ = + ∂ (1.33)

P

P

c vds dT dPT T

∂ = − ∂ (1.39)

and solved for dT so that

( ) ( )/ /v P

p v p v

T P T T v TdT dv dP

c c c c∂ ∂ ∂ ∂

= +− −

(1.44)

This suggests that ( , )T T v P= (1.45) and

P v

T TdT dv dPv P

∂ ∂ = + ∂ ∂ (1.46)

Equating the coefficients of dv or dP of eqs.(1.44) and (1.46) gives

p vP v

v Pc c TT T∂ ∂ − = ∂ ∂

(1.47)

Another version of this equation which can be expressed in terms of the thermodynamic properties volume expansivity β and isothermal compressibility α is obtained using the cyclic relation applied to the P-v-T behavior of a substance:

1v P T

P T vT v P∂ ∂ ∂ = − ∂ ∂ ∂

or

v P T

P v PT T v∂ ∂ ∂ = − ∂ ∂ ∂

(1.48)

Substituting eq. (1.48) into (1.47),

2

p vP T

v Pc c TT v∂ ∂ − = − ∂ ∂

(1.49)

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Recalling the definitions

Volume Expansivity 1

P

vv T

β ∂ = ∂ (1.50)

and

Isothermal Compressibilty 1

T

vv P

α ∂ = − ∂ (1.51)

and substituting these into eq.(1.49) gives

2

p vvTc c βα

− = (1.52)

Notes:

1. Since α is positive for all substances in all phases and β2

is also always positive, then

p vc c≥ (1.53)

2. cp approaches cv

as the temperature approaches absolute zero.

3. For incompressible substances, i.e., v = const., cp ≈ cv

. The differences are usually neglected for solids and liquids.

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THE JOULE-THOMPSON COEFFICIENT The Joule-Thomson coefficient is a measure of the change in temperature with pressure during a constant-enthalpy process.

h

TP

µ ∂ = ∂ (1.54)

A careful look at its defining equation reveals that the Joule-Thomson coefficient represents the slope of h = constant lines on a T-P diagram

A throttling process proceeds along a constant-enthalpy line in the direction of decreasingpressure, that is, from right to left.

• Only somea point of zero slope or zero Joule-Thomson

of the constant-enthalpy lines have

coefficient. • The line that passes through these points is called the Inversion Line, and the temperature at a point where a constant-enthalpy line intersects the inversion line is called the inversion temperature. • During a throttling process, the temperature increases on the right-hand side of the inversion line, decreases on the left-hand side of the inversion line. • It is clear from this diagram that a cooling effect cannot be achieved by throttling unless the fluid is below its Maximum inversion temperature.

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An expression for the Joule-Thompson coefficient in terms of P-v-T can be developed from the relation for dh

pP

vdh c dT v T dPT

∂ = + − ∂ (1.55)

Noting that dh = 0 for a throttling process, this equation after re-arrangement becomes

1

JTP hv

v Tv Tc T P

µ ∂ ∂ − − = = ∂ ∂

(1.56)

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GENERAL RELATIONS FOR du, dh, & ds OF REAL GASES Real gas behavior deviates significantly from ideal gas behavior at higher pressures. The deviation from ideal gas behavior should be accounted for in the calculation of dh, du, & ds for real (non-ideal) gases. The approach taken to accomplish this is the use of the relations developed earlier for ideal gases together with the compressibility factor Z to account for the real gas (or deviation from ideal gas) behavior.

Real Gas Enthalpy Change - dh For a real gas actual process 1 – 2, the calculation of dh is carried out along a series of imaginary processes 1 – 1* - 2* - 2 starting and ending at the same states as the actual process. These processes are: 1 – 1* isothermal process from actual

initial pressure to zero pressure 1* - 2* isobaric process with change in

temperature, ideal gas behavior 2* - 2 isothermal process from zero

pressure to actual final pressure Although these series of processes is more complicated than the actual process, the calculations are greatly simplified because one property remains constant in any segment of the entire process. The general relation

2 2

1 1

2 1

T P

pPT P

vh h c dT v T dPT

∂ − = + − ∂ ∫ ∫ (1.57)

is applied to process 1 – 1* - 2* - 2 .

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Thus,

( ) ( ) ( )* * * *2 1 2 2 2 1 1 1h h h h h h h h− = − + − + − (1.58)

where

2 2

2 02 2

*

*2 2 0

P P

P PP PT T T T

v vh h v T dP v T dPT T

= =

∂ ∂ − = + − = − ∂ ∂ ∫ ∫ (1.59)

2 2

1 1

* *2 1 00 ( )

T T

p pT T

h h c dT c T dT− = + =∫ ∫ (1.60)

1 1

1 01 1

*

*1 1 0

P P

P PP PT T T T

v vh h v T dP v T dPT T

= =

∂ ∂ − = + − = − − ∂ ∂ ∫ ∫ (1.61)

Now define Enthalpy Departure = ( h* - h )T

= the difference in enthalpy of a real gas at ( P, T ) and ideal gas at ( P0 , T ) , where P0

≈ zero pressure

For a real gas with compressibility factor Z , /v ZRT P= The enthalpy departure at any temperature T and pressure P can be expressed as

( )* 2

0

P

TP

Z dPh h RTT P∂ − = − ∂ ∫

(1.62)

By expressing the pressure and temperature P and T in terms of the reduced properties

P = Pcr PR and T = Tcr TR

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the enthalpy departure can be expressed in non-dimensional form called Enthalpy Departure Factor

( ) ( )

*2

0

lnR

R

P

h R Ru cr R P

h h ZZ T d PR T T− ∂

= = ∂ ∫ (1.63)

• The integral on the RHS of the above equation can be evaluated graphically or

numerically using data from the compressibility charts.

• A plot of Zh = Zh ( PR , TR

) is known as the Generalized Enthalpy Departure Chart and is used to evaluate enthalpy departure of a real gas at an actual temperature T and pressure P.

The enthalpy change of a real gas from (T1, P1) to (T2, P2

) on a molar basis is then

( ) ( )2 1 2 1 2 1u cr h hidealh h h h R T Z Z− = − − − (1.64)

or on a mass basis,

( ) ( )2 1 2 1 2 1cr h hidealh h h h RT Z Z− = − − − (1.65)

where

(h2 – h1)ideal is evaluated using ideal gas behavior from T1 to T2

.

Real Gas Internal Energy Change - du The change in internal energy of a real gas is calculated from the change in enthalpy. Thus,

( ) ( )2 1 2 1 2 2 1 1uu u h h R Z T Z T− = − − − (1.66)

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Real Gas Entropy Change - ds Because of the dependence of ideal gas entropy change on both pressure and temperature, a pathway similar to enthalpy change calculation cannot be applied for entropy change because it involves the value of entropy at zero pressure, which is infinity. The pathway chosen for calculating entropy change from (P1 , T1) to (P2 , T2

) is shown in the figure.

1 – a* - 1* - 2* - b* - 2 . where * indicates ideal gas state P*

1 = P1 T*1 = T1

P

*2 = P2 T*

2 = T2 P

0

≈ zero pressure (ideal gas conditions)

1 – 1* , isothermal process 2 – 2* , isothermal process 1* - 2* , ideal gas behavior The entropy change from (P1 , T1) to (P2 , T2

) is then

The entropy difference (s – s) between the actual state at (P, T) and the corresponding ideal gas state at the same pressure and temperature (P = P, T

= T) is called the Entropy Departure at (P, T) which can be expressed as

If the specific volume for the real gas and ideal gas are respectively

ZRTv

P= and

RTvP

=

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the entropy departure can be expressed as

By expressing temperature and pressure in terms of reduced properties

T = Tcr TR and P = Pcr PR

the entropy departure can be expressed in non-dimensional form called the Entropy Departure Factor as

Values of Zs as a function of PR and TR are shown in the Generalized Entropy Departure Chart. Note that Zs

= 1 for an ideal gas.

The entropy change of a real gas from state (P1, T1) to (P2, T2

) on a molal or mass basis respectively is therefore

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