Army Research Laboratory

Latent Work and Latent Heat of the

Liquid/Vapor Transformation

by Michael A. Grinfeld and Steven B. Segletes

ARL-TR-7008 August 2014

Approved for public release; distribution is unlimited.
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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-TR-7008 August 2014

Latent Work and Latent Heat of the

Liquid/Vapor Transformation

Michael A. Grinfeld and Steven B. Segletes

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
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August 2014 Final

Latent Work and Latent Heat of the Liquid/Vapor Transformation

ARL-TR-7008

Approved for public release; distribution is unlimited.

January 2014–June 2014

Michael A. Grinfeld AH80

Steven B. Segletes

U.S. Army Research Laboratory

ATTN: RDRL-WMP-C

Aberdeen Proving Ground, MD 21005-5066

Authors’ email: <michael.greenfield4.civ@mail.mil>, <steven.b.segletes.civ@mail.mil>

The key elements of the theory of the liquid/vapor phase transformations are the phase equilibrium curve pressure/temperature

and latent heat of the transformation. To the best of our knowledge, nobody has yet analyzed the latent work of phase

transformation. In this report, we try to eliminate this paradoxical gap. Through a generalized methodology, we study the latent

heat and latent work of liquid/vapor phase transformation for variously constrained thermodynamic processes.

thermodynamics, phase transformation, latent work, latent heat, vaporization

26

Steven B. Segletes

Unclassified Unclassified Unclassified 410-278-6010

UU

ii
Contents

List of Figures iv

1. Introduction 1

2. Latent Heat and Work of Thermodynamic Process 3

3. Equations of Phase Equilibrium 5

4. Vaporization/Condensation under Fixed Pressure and Prescribed Heat Supply 5

5. Vaporization/Condensation at Fixed Total Volume 9

6. Adiabatic Vaporization/Condensation 12

7. Conclusion 14

8. References 16

Appendix. A Curious Relation 17

Distribution List 19

iii
List of Figures

Fig. 1 Heterogeneous system liquid/vapor under pressure ........................................ 4

iv
1. Introduction

The theory of phase equilibrium for the “liquid/vapor” phase transformation has been under

development for centuries. In the case of one-component phases, it is presented in a very clear

form in multiple classical monographs and textbooks.1–3 The key products of the classical theory

of liquid/vapor phase co-existence are chemical potentials μ of the phases and the phase

equilibrium curve p = p (T). The phase equilibrium curve gives the pressure p at equilibrium as a

function of the absolute temperature T. This curve is universal in the sense that it does not depend

upon the mechanical and thermal loading conditions. In other words, it is one and the same for

isothermal and adiabatic systems as well as for isochoric or isobaric loading conditions. In short,

the equilibrium pressure p depends upon the absolute temperature T alone. In the work that

follows, we consider the liquid and vapor phases as co-existing without co-mingling. That is, the

two phases are in contact and may transport mass and heat across a notional divide that separates

them.

Consider a substance with the specific internal energy density e = e (ρ, η), where ρ is the mass

density and η is the specific entropy density. In order to find the function p (T), one has to use the

following three equilibrium conditions:

p1 = p2, T1 = T2, μ1 = μ2 , (1)

where 1 and 2 refer to the liquid and vapor phases, respectively.

The first of the relationships in Eq. 1 reflects the mechanical equilibrium between phases, the

second reflects the thermal equilibrium between the phases, and the third reflects the “chemical”

equilibrium. In this context, the chemical equilibrium means the equilibrium with respect to mass

exchange between the phase in the process of vaporization/condensation.

Thermodynamical identities allow one to express p, T, and μ in terms of the derivatives of the

internal energy density function e (ρ, η):

p (ρ, η) ≡ ρ

2

∂e (ρ, η)

∂ρ = ρ

2

eρ T (ρ, η) ≡

∂e (ρ, η)

∂η = eη

μ ≡ e +

p

ρ

+ T η = e + ρeρ + ηeη

. (2)

Analysis of the system 1, which can be found in any thermodynamics textbook, results in the

1
following equation, known as the Clausius-Clapeyron equation:1, 2

dp(T)

dT =

η1 − η2

ρ

−1

1 − ρ

−1

2

=

L

T



ρ

−1

1 − ρ

−1

2

 . (3)

The quantity L ≡ T (η1 − η2) is called the latent heat of phase transformation and appears to be a

state function, not a process-dependent function. In other words, the latent heat L depends only

on the thermodynamic state, regardless of whether this state was achieved by way of isochoric,

isothermal, or any other particular process.

To make our point regarding L more clear, let us compare it with the similar concept of the

specific heat C. Consider any process, in which the heat supply Q and the absolute temperature T

are functions of time t : Q = Q (t), T = T (t). Then, the heat capacity C of this process is a

function of time also C = C (t). Here, the heat capacity C (t) in the process is defined as

C (t) ≡

dQ(t)/dt

dT(t)/dt . (4)

Obviously, the function C (t) essentially depends on the process under consideration. During an

adiabatic process it is equal to zero, whereas for an isothermal process it is equal to infinity, etc.

Using the First and Second Laws of Thermodynamics, we get, in addition to Eq. 4, the following

formula for the heat capacity in an isochoric process, in terms of the specific free energy function

ψ (ρ, T) = e − ηT:

Cv = T

∂

2ψ(ρ, T)

∂T2

≡ T ψT T . (5)

Although, originally, the function Cv was introduced by formula 4 applied to the isochoric

process, in fact, according to formula 5, it is a function of thermodynamic state (ρ, T) only. In

other words, it can be introduced directly with formula 5 and without mentioning of any

particular thermodynamic process.

In addition to Cv one can introduce the specific heat Cp for the isobaric or any other specified

thermodynamic process. That would lead us to other analogies of Eq. 5 and additional state

functions.

If we introduce the state function Cv directly with the help of identity 5, we cannot help asking

the question, “In what thermodynamic process will the function T ψT T be equal to the heat

capacity of the process?” In our case, the answer is almost a trivial tautology: “The function

T ψT T gives the heat capacity in the isochoric process.” But it is not always so obvious. For

instance, the quantity L ≡ T (η1 − η2) is obviously a state function—it is defined only in terms of

2
state variables, without any mention of the particular process of phase transformation. However,

the very name “the latent heat of phase transformation” says to us that it is the rate of heat supply

in a certain thermodynamic process. Unfortunately, the classical textbooks rarely specify clearly

the process for which L ≡ T (η1 − η2) is the rate of heat supply. The lacking analysis creates the

impression that L ≡ T (η1 − η2) describes the latent heat for any thermodynamic process. This is

what practitioners tacitly assume and they are surprised when somebody claims that this is a

stretched interpretation. That is why, in this report, we come back to the discussion of the latent

heat concept. We deliberately limit ourselves to the simplest classical one-component system

“liquid/vapor,” which everyone has learned in high school and college.

We believe that the general situation will become much more transparent if we introduce the

concept of “the latent heat capacity L(t) in thermodynamic process” similar to the concept of “the

specific heat capacity in thermodynamic process.” Contrary to the latent heat L ≡ T (η1 − η2),

the latent heat L(t) is not a state function. However, for more narrow classes of thermodynamic

processes, it leads to novel state functions. In addition, we introduce a novel thermodynamic

concept of “the latent work in thermodynamic process,” which is as meaningful as the concept of

“the latent heat capacity in thermodynamic process.”

2. Latent Heat and Work of Thermodynamic Process

Consider a single-component, two-phase heterogeneous system “liquid/vapor” (see Fig. 1) in a

pressurized vessel. If the piston position is fixed, then the phase transformation occurs in the

isochoric regime. Also, we assume the system can be thermally isolated (adiabatic phase

transformation) or placed within a thermostat (isothermic regime.) In what follows, we mark as 1

and 2 the parameters of the liquid and vapor phases, respectively.

Consider any process “x” in the two-phase heterogeneous system under study. Then, depending

on the pressure and thermal loading, the external heat supply Q, pressure P, and total volume Vtot

become functions of time. The same is true regarding the thermodynamic parameters of the

phases ρ1, p1, η1, e1, ψ1, etc., and ρ2, p2, η2, e2, ψ2, etc. Generally speaking, thermodynamic

processes are accompanied by the evolution of the total mass components M1, M2.

The total mass of the substance Mtot is assumed fixed,

M1 (t) + M2 (t) = Mtot , (6)

3
Fig. 1 Heterogeneous system

liquid/vapor under pressure

or in differential form

dM1(t)

dt +

dM2(t)

dt = 0 . (7)

We define the latent heat of condensation Lx (t) in the process “x” by the ratio

Lx(t) ≡

dQ(t)/dt

dM1(t)/dt . (8)

Likewise, we define the latent work of condensation Wx (t) in the process “x” by the ratio

Wx(t) ≡ P(t)

dVtot(t)/dt

dM1(t)/dt . (9)

4
3. Equations of Phase Equilibrium

Regardless of the external mechanical and thermal loading, the following phase equilibrium

conditions should be satisfied:

i) the condition of mechanical equilibrium,

p1 = p2 or ρ

2

eρ



1

=



ρ

2

eρ



2

; (10)

ii) the condition of thermal equilibrium,

T1 = T2 or eη1 = eη2 ; and (11)

iii) the condition of equilibrium with respect to mass exchange between phases,

χ1 = χ2 or (e + ρeρ − ηeη)

1 = (e + ρeρ − ηeη)2

, (12)

which is alternately expressible as

(ψ + p/ρ)1 = (ψ + p/ρ)2 . (13)

The quantity χ is known as the chemical potential of the one-component liquid (or gaseous)

phase. Therefore, condition 12 can be called the chemical equilibrium condition, although no

chemical reactions are actually occur in the transformation under consideration.

Also, the mass conservation condition should be satisfied for any external loading. All additional

conditions depend on the specifics of the external loading associated with process x.

4. Vaporization/Condensation under Fixed Pressure and Prescribed Heat

Supply

When external pressure is fixed p = p0 the vaporization/condensation process can be controlled

by the external heat supply Q(t).

Differentiating the phase equilibrium Eqs. 10–12 along the process trajectory allows us to obtain

5
governing relationships. First, Eq. 10:

 

ρ

2

eρη

dη

dt + ρ (ρe)

ρρ

dρ

dt !

1

=

 

ρ

2

eρη

dη

dt + ρ (ρe)

ρρ

dρ

dt !

2

=

dpext

dt

≡ 0 , (14)

where pext is the constant external applied pressure. Equation 14 implies

dρ1

dt = −

 

ρeρη

(ρe)

ρρ !

1

dη1

dt ,

dρ2

dt = −

 

ρeρη

(ρe)

ρρ !

2

dη2

dt . (15)

Then, differentiating the condition of thermal equilibrium, Eq. 11, we get

 

eηη

dη

dt + eηρ

dρ

dt !

1

=

 

eηη

dη

dt + eηρ

dρ

dt !

2

. (16)

Finally, differentiating the condition of chemical equilibrium, Eq. 12, we get after some algebraic

manipulations

"



(eρ)

ρρ − ηeηρ dρ

dt + (eηρρ − ηeηη)

dη

dt #

1

=

"



(eρ)

ρρ − ηeηρ dρ

dt + (eηρρ − ηeηη)

dη

dt #

2

. (17)

The system given by Eqs. 15–17 can be rewritten in the following matrix form:

[Gp]



















dη2/dt

dρ2/dt

dη1/dt

dρ1/dt

















=



















0

0

0

0



















, (18)

where Gp is the following 4 × 4 matrix:

[Gp] ≡























(eρ)

ρρ − ηeηρ

1

eηρ1

0



ρ (ρe)

ρρ

1

(eηρρ − ηeηη)

1

eηη1

0

(ρ

2

eρη)

1

−



(eρ)

ρρ − ηeηρ

2

−eηρ2



ρ (ρe)

ρρ

2

0

− (eηρρ − ηeηη)

2

−eηη2

(ρ

2

eρη)

2

0























. (19)

The determinant of the matrix Gp is given by the following formula:

6
= (η1 − η2)



ρ (ρe)

ρρ eηη − (ρeρη)

2



1



ρ (ρe)

ρρ eηη − (ρeρη)

2



2

|Gp| ≡































0

eηρ1

0



ρ (ρe)

ρρ

1

0

eηη1

0

(ρ

2

eρη)

1

(ηeηρ)

2 − η1eηρ2

−eηρ2



ρ (ρe)

ρρ

2

0

(ηeηη)

2 − η1eηη2

−eηη2

(ρ

2

eρη)

2

0































. (20)

Introducing the specific volume v, instead of ρ, as v = 1/ρ, we can rewrite Eq. 20 as

|Gp| ≡ (η1 − η2) v

2

1



evveηη − e

2

vη

1

v

2

2



evveηη − e

2

vη

2

. (21)

According to the classical thermodynamic inequality, we have

evveηη − e

2

vη > 0 . (22)

Since η1 − η2 6= 0, we conclude that

|Gp| 6= 0 .

Finally, since |Gp| 6= 0, system 18 implies

dρ1

dt =

dη1

dt =

dρ2

dt =

dη2

dt = 0 . (23)

According to Eq. 23, because two independent thermodynamic parameters, (ρ1, η1) and (ρ2, η2),

within each of the phases remain unchanged in the processes with fixed external pressures, all

other thermodynamic parameters, including the absolute temperature, remain unchanged as well.

This fact is well known in classical thermodynamics.

Let us now calculate the latent heat Lp for the isobaric vaporization/condensation. The total

internal energy E and the total volume V of the system can be presented as follows:

E = M1 e(ρ1, η1) + M2 e(ρ1, η1) ≡

X

I=1,2

MI e(ρI , ηI ) , (24)

Vtot(t) = V1(t) + V2(t) = M1(t)

ρ1

+

M2(t)

ρ2

=

X

I

MI

ρI

. (25)

Note that, in light of Eq. 6, Eq. 25 may be alternately expressed for this two-phase,

7
single-component system as

Vtot(t) = M1(t)

 

1

ρ1

−

1

ρ2

!

+

Mtot

ρ2

. (26)

Since all the thermodynamic parameters of the phase remain unchanged over time, we can employ

the First Law of Thermodynamics to obtain the heat-transfer rate for this process as follows:

dQ

dt =

"

e1 − e2 + p0

 

1

ρ1

−

1

ρ2

!# dM1

dt , (27)

as it is implied by the following chain:

dQ

dt

=

"

e1 − e2 + p0

 

1

ρ1

−

1

ρ2

!# dM1

dt ,

= (e1 − e2)

dM1

dt + p0

 

1

ρ1

−

1

ρ2

!

dM1

dt

=

X

I

dMI

dt eI + p0

X

I

1

ρI

dMI

dt

=

d

dt

X

I

MI e (ρI , ηI ) + p0

d

dt

X

I

MI

ρI

=

dE

dt + p0

dVtot

dt

where we have relied upon relationships 6 and 24–26.

Comparing the general definition of the latent heat of thermodynamic process with relation 27 for

isobaric process, we arrive at the following relationship of Lp

Lp =

dQ

dM1

= e1 − e2 + p0

 

1

ρ1

−

1

ρ2

!

. (28)

Using the chemical condition of phase equilibrium, Eq. 12, we can rewrite in the following

classical form:

Lp = T0 (η1 − η2) . (29)

For the latent work Wp of vaporization/condensation at fixed external pressure we get, according

8
to definition 9

Wp(t) ≡ p0

dVtot(t)/dt

dM1(t)/dt = p0

 

1

ρ1

−

1

ρ2

!

, (30)

where the time-derivative of total volume has been obtained by taking the derivative of Eq. 26. In

light of Eq. 13, the formula of the latent work, Eq. 30, can be rewritten in terms of the free energy

jump:

Wp(t) ≡ p0

dVtot(t)/dt

dM1(t)/dt = ψ2 − ψ1 . (31)

Now, we can rewrite the classical Clausius-Clapeyron formula, Eq. 3, in the following transparent

and elegant form:

dp(T)

dT =

η1 − η2

ρ

−1

1 − ρ

−1

2

=

Lp

Wp

·

p0

T0

(32)

or

d ln p

d ln T

=

T0

p0

·

η1 − η2

ρ

−1

1 − ρ

−1

2

=

Lp

Wp

. (33)

5. Vaporization/Condensation at Fixed Total Volume

When heating under fixed volume, the phase transformation temperature and pressure grow. The

latent work of phase transformation Wv vanishes. The formula of the latent heat changes also.

Taking the time-derivative of Eq. 25, we obtain

dVtot

dt =

X

I

dVi

dt =

X

I

 

1

ρI

dMI

dt −

MI

ρ

2

I

dρI

dt !

= 0 , (34)

while the total-mass M conservation conditions implies

X

I

dMI

dt = 0 . (35)

For the heat supply, we get the following relationship:

dQ

dt = T (η1 − η2)

dM1

dt +

X

I

MIT

dηI

dt , (36)

9
as it is implied by the chain

dQ

dt =

dE

dt =

d

dt

X

I

MI eI

= T (η1 − η2)

dM1

dt +

X

I

MIT

dηI

dt

=

 

e1 − e2 +

p

ρ1

−

p

ρ2

!

dM1

dt +

X

I

MIT

dηI

dt

= (e1 − e2)

dM1

dt +

X

I

 

p

ρI

dMI

dt + MIT

dηI

dt !

= (e1 − e2)

dM1

dt +

X

I

MI

 

p

1

ρ

2

I

dρI

dt + T

dηI

dt !

= (e1 − e2)

dM1

dt +

X

I

MI

 

eρI

dρI

dt + eηI

dηI

dt !

.

The phase equilibrium conditions imply three conditions:

i) the mechanical equilibrium condition

dp

dt =

 



ρ

2

eρ



ρ

dρ

dt + ρ

2

eρη

dη

dt !

1

=

 



ρ

2

eρ



ρ

dρ

dt + ρ

2

eρη

dη

dt !

2

; (37)

ii) the thermal equilibrium condition

dT

dt =

 

eηρ

dρ

dt + eηη

dη

dt !

1

=

 

eηρ

dρ

dt + eηη

dη

dt !

2

; and (38)

iii) the “chemical” equilibrium condition

=

dχ

dt =

"



(ρe)

ρρ − ηeηρ dρ

dt + (ρeηρ − ηeηη)

dη

dt #

2

"



(ρe)

ρρ − ηeηρ dρ

dt + (ρeηρ − ηeηη)

dη

dt #

1

. (39)

The heat flux relation, Eq. 36, can be rewritten as

(η1 − η2)

dM1

dt +

X

I

MI

dηI

dt =

1

T

dQ

dt . (40)

10
The variable dM2/dt can be eliminated between Eqs. 34 and 35 to read

X

I

MI

ρ

2

I

dρI

dt +

 

1

ρ2

−

1

ρ1

!

dM1

dt = 0 . (41)

In matrix form, the system of equations, Eqs. 37–41, may be expressed as

[GV ]





























dM1/dt

dη2/dt

dρ2/dt

dη1/dt

dρ1/dt 



























=





























0

(1/T) · (dQ/dt)

0

0

0





























, (42)

where the matrix [GV ] is defined as

[GV ]≡





























M1/ρ2

1

0



(ρe)

ρρ − ηeηρ

1

eηρ1

ρ (ρe)

ρρ1

0

M1

(ρeηρ − ηeηη)

1

eηη1

ρ

2

eρη1

M2/ρ2

2

0

−



(ρe)

ρρ − ηeηρ

2

−eηρ2

−ρ (ρe)

ρρ2

0

M2

−(ρeηρ − ηeηη)

2

−eηη2

−ρ

2

eρη2

1

ρ2

−

1

ρ1

η1 − η2

0

0

0





























. (43)

Thus, with Cramer’s Rule, we arrive at the following formula:

dM1

dt = −

|UV |

|GV |

·

1

T

dQ

dt , (44)

where the matrix [UV ] is defined as

[UV ] ≡























M1/ρ2

1



(ρe)

ρρ − ηeηρ

1

eηρ1

ρ (ρe)

ρρ1

0

(ρeηρ − ηeηη)

1

eηη1

ρ

2

eρη1

M2/ρ2

2

−



(ρe)

ρρ − ηeηρ

2

−eηρ2

−ρ (ρe)

ρρ2

0

−(ρeηρ − ηeηη)

2

−eηη2

−ρ

2

eρη2























. (45)

11
Thus, we arrive at the following formula of the latent heat at fixed volume:

LV = −T

|GV |

|UV |

. (46)

6. Adiabatic Vaporization/Condensation

In the case of adiabatic vaporization/condensation, the latent heat is, by definition, equal to zero.

Per Eq. 36, the adiabaticity condition reads (i.e., when dQ/dt ≡ 0)

(η1 − η2)

dM1

dt +

X

I

MI

dηI

dt = 0. . (47)

In order to alternately derive Eq. 47, we may first obtain

dE

dt = T (η1 − η2)

dM1

dt − p

dVtot

dt + T

X

I

MI

dηI

dt , (48)

as implied by the chain

dE

dt =

d

dt

X

I

MI e(ρI , ηI )

= T (η1 − η2)

dM1

dt − p

dVtot

dt + T

X

I

MI

dηI

dt

=

"

e1 − e2 + p

 

1

ρ1

−

1

ρ2

!# dM1

dt − p

X

I

dVI

dt + T

X

I

MI

dηI

dt

= (e1 − e2)

dM1

dt +

X

I

MI

 

p

ρ

2

I

dρI

dt + T

dηI

dt !

= (e1 − e2)

dM1

dt +

X

I

MI

 

eρ

dρI

dt + eη

dηI

dt !

=

X

I

 

dMI

dt e(ρI , ηI ) + MI

de(ρI , ηI )

dt !

where Eq. 34 has notably been used.

Now, combining the above with the First Law of Thermodynamics, we get

0 =

dQ

dt =

dE

dt + p

dVtot

dt = T (η1 − η2)

dM1

dt + T

X

I

MI

dηI

dt

12
which, when divided by T, yields Eq. 47.

For the latent adiabatic work, we get the formula

dAη

dt = −p

X

I

MI

ρ

2

I

dρI

dt − T

X

I

MI

dηI

dt −

dM1

dt (e1 − e2) (49)

by noting that dAη/dt = −dE/dt, as given in the third line of the prior chain.

The phase equilibrium conditions still imply the following:

i) the mechanical equilibrium condition

dp

dt =

 



ρ

2

eρ



ρ

dρ

dt + ρ

2

eρη

dη

dt !

1

=

 



ρ

2

eρ



ρ

dρ

dt + ρ

2

eρη

dη

dt !

2

; (50)

ii) the thermal equilibrium condition

dT

dt =

 

eηρ

dρ

dt + eηη

dη

dt !

1

=

 

eηρ

dρ

dt + eηη

dη

dt !

2

; and (51)

iii) the “chemical” equilibrium condition

=

dχ

dt =

"



(ρe)

ρρ − ηeηρ dρ

dt + (ρeηρ − ηeηη)

dη

dt #

2

"



(ρe)

ρρ − ηeηρ dρ

dt + (ρeηρ − ηeηη)

dη

dt #

1

. (52)

The system of equations given by Eqs. 47 and 49–52 can be rewritten in the following matrix

form:

[Gη]





























dM1/dt

dη2/dt

dρ2/dt

dη1/dt

dρ1/dt 



























=





























−dAη/dt

0

0

0

0





























, (53)

13
where [Gη] is the matrix

[Gη]≡





























p M1/ρ2

1

0



(ρe)

ρρ − ηeηρ

1

eηρ1

ρ (ρe)

ρρ1

M1T

M1

(ρeηρ − ηeηη)

1

eηη1

ρ

2

eρη1

p M2/ρ2

2

0

−



(ρe)

ρρ − ηeηρ

2

−eηρ2

−ρ (ρe)

ρρ2

M2T

M2

−(ρeηρ − ηeηη)

2

−eηη2

−ρ

2

eρη2

e1 − e2

η1 − η2

0

0

0





























. (54)

Thus, we arrive at the following formula:

dM1

dt = −

dAη

dt

|Uη|

|Gη|

, (55)

where Uη is the matrix

[Uη] ≡























0



(ρe)

ρρ − ηeηρ

1

eηρ1

ρ (ρe)

ρρ1

M1

(ρeηρ − ηeηη)

1

eηη1

ρ

2

eρη1

0

−



(ρe)

ρρ − ηeηρ

2

−eηρ2

−ρ (ρe)

ρρ2

M2

−(ρeηρ − ηeηη)

2

−eηη2

−ρ

2

eρη2























. (56)

Thus, the specific latent work Wη of vaporization/condensation for an adiabatic process is given

by the formula:

Wη = −

|Gη|

|Uη|

(57)

7. Conclusion

We introduce several novel notions relating to the first-order phase transformation in liquid/vapor

one-component systems. In addition to the standard state function L, called the latent heat, we

introduce the concept of the latent-heat-for-the-process function Lx. Contrary to the L-function,

the Lx-function is not a state function but a function that changes depending on the particular

process under study. The function Lx coincides with L only for the phase transformation under

fixed pressure and temperature.

14
We introduce the concept of the latent work W of the phase transformation. Here again, we

distinguish between the latent work W as the state function and the latent-work-for-the-process

function Wx.

Importantly, this report lays out a generalized methodology for obtaining the latent heat and latent

work for variously constrained thermodynamic processes. Our methodology, though lengthier

than the textbook derivation, is technically straightforward and can be used in different

circumstances.

We calculate the Lx and Wx function for variously constrained thermodynamic processes

accompanied by phase transformations. For the classical vaporization process under fixed

pressure and temperature, we give a new elegant form, Eq. 33, of the celebrated

Clausius-Clapeyron relationship.

15
8. References

1. Sears FW. An Introduction to Thermodynamics: The Kinetic Theory of Gases, and Statistical

Mechanics. Reading: Addison-Wesley; 1953.

2. Callen HB. Thermodynamics. New York: Wiley & Sons; 1960.

3. Holman JP. Thermodynamics. New York: McGraw-Hill; 1974.

16
Appendix. A Curious Relation

Curiously, Holman3 derives the relationship for a one-component single-phase liquid as

∂p

∂T =

∂η

∂v (A-1)

which, to some extent, is reminiscent of the Clausius-Clapeyron equation. He begins with the

Helmholtz free energy ψ = e − T η, from which an exact differential may be taken:

dψ = de − T dη − η dT (A-2)

The increment of internal energy de may be eliminated by way of the heat/work energy balance,

de = T dη − p dv, where v is the specific volume, equal to 1/ρ. Substitution of this, in turn,

eliminates the term T dη, leaving the expression

dψ = −p dv − η dT (A-3)

The implication of this relationship is that

−p =

∂ψ(T, v)

∂v , −η =

∂ψ(T, v)

∂T (A-4)

Furthermore, one may equate the mixed partial derivatives as follows:

−

∂

2ψ

∂T ∂v =

∂p(T, v)

∂T =

∂η(T, v)

∂v (A-5)

The similarity of Eq. A-5 to the Clausius-Clapeyron relation of Eq. 3 is obvious, upon realizing

that (η1 − η2)/(ρ

−1

1 − ρ

−1

2

) is simply ∆η/∆v.

17
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