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CHAPTER TWO
โช Develop from the first and second laws the
fundamental property relations which underlie
the mathematical structure of thermodynamics
โช Derive equations which allow calculation of
enthalpy and entropy values from PVT and heat-
capacity data.
โช Develop generalized correlations which provide
estimates of property values in the absence of
complete experimental information.
Thermodynamic Properties of Fluids
Learning Objectives
Property Relations for Homogeneous Phases
โช The 1st law of thermodynamics for a simple compressible system
that undergoes an internally reversible process of n moles is:
๐(๐ง๐)๐ซ๐๐ฏ = ๐ ๐๐ซ๐๐ฏ + ๐ ๐๐ซ๐๐ฏ ๐. ๐
โช The incremental heat interaction ๐ฟ๐๐๐๐ฃ is related directly to the
entropy change through the formal definition of entropy:(Entropy,
the measure of a systemโs thermal energy per unit temperature
that is unavailable for doing useful work. )
๐๐ =๐ ๐
๐๐ซ๐๐ฏ
๐จ๐ซ ๐ ๐๐ซ๐๐ฏ = ๐๐(๐ง๐) ๐. ๐
โช For a simple compressible system, the only reversible work mode is
compression and/or expansion, that is:
๐ ๐๐ซ๐๐ฏ = โ๐๐ ๐ง๐ ๐. ๐
Thermodynamic Properties of Fluids
โช Substituting these expressions for ฮดQrev and ฮดWrev into the 1st -
law statement yields:
๐(๐ง๐) = ๐๐(๐ง๐) โ ๐๐(๐ง๐ ) ๐. ๐(๐)
โช Rearranging Eq. (2.4a)
๐๐(๐ง๐) = ๐(๐ง๐) + ๐๐(๐ง๐ ) ๐. ๐(๐)
โช This equation contains only properties of the system.
โช Properties depend on state alone, and not on the kind of process
that leads to the state.
โช Thus Eq. (2.4) applies to any process in a system of constant
mass that results in a differential change from one equilibrium
state to another.
โช The only requirements are that the system be closed and that
the change occurs between equilibrium states.
Thermodynamic Properties of Fluids
โช All of the primary thermodynamic properties: P, V, T, U, and S are
included in Eq. (2.4).
โช Additionally two properties, also defined for convenience, are:
i. Gibbs free energy or Gibbs function, G: is a composite
property involving enthalpy and entropy and is defined as:
๐ = ๐โ ๐๐ ๐. ๐
โ The Gibbs free energy is particularly useful in defining equilibrium
conditions for reacting systems at constant P and T.
ii. Helmholtz Free Energy or Helmholtz Function, A: is also a
property, defined similarly to the Gibbs free energy, with the
internal energy replacing the enthalpy, that is,
๐ = ๐ โ ๐๐ ๐. ๐
โ The Helmholtz free energy is useful in defining equilibrium
conditions for reacting systems at constant V and T.
Thermodynamic Properties of Fluids
โช The enthalpy was defined by the equation for n moles is:
๐ง๐ = ๐ง๐ + ๐ ๐ง๐ ๐. ๐
โช Differentiating Eq. (2.7) gives:
๐ ๐ง๐ = ๐ ๐ง๐ + ๐๐ ๐ง๐ + (๐ง๐)๐๐
โช When d(nU) is replaced by Eq. (2.4(a)), this reduces to:
๐ ๐ง๐ = ๐๐ ๐ง๐ + (๐ง๐)๐๐ ๐. ๐
โช Similarly, Differentiating Helmholtz Function, Eq. (2.6):
๐ ๐ง๐ = ๐ ๐ง๐ โ ๐๐ ๐ง๐ โ (๐ง๐)๐๐
โช Eliminating d(nU) by Eq. (2.4(a)) gives:
๐ ๐ง๐ = โ๐๐ ๐ง๐ โ ๐ง๐ ๐๐ ๐. ๐
โช In the same fusion, differentiating the Gibbs function, Eq. (2.5)
๐ ๐ง๐ = ๐ ๐ง๐ โ ๐๐ ๐ง๐ โ (๐ง๐)๐๐
โช Eliminating d(nH) by Eq. (2.8) gives:
๐ ๐ง๐ = ๐ง๐ ๐๐ โ ๐ง๐ ๐๐ ๐. ๐๐
Thermodynamic Properties of Fluids
โช All the above equations are written for the entire mass of any
closed system.
โช The immediate application of these equations is to one mole (or
to a unit mass) of a homogeneous fluid of constant
composition.
โช For this case, they simplify to:
๐๐ = ๐๐๐ โ ๐๐๐ ๐. ๐๐๐๐ = ๐๐๐ + ๐๐๐ ๐. ๐๐๐๐ = โ๐๐๐ โ ๐๐๐ ๐. ๐๐๐๐ = ๐๐๐ โ ๐๐๐ ๐. ๐๐
โ These fundamental property relations; Eqs. (2.11) through
Eq. (2.14) are general equations for a homogeneous fluid
of constant composition.
Thermodynamic Properties of Fluids
โช Another set of equations follows from Eqs. (2.11) through Eq. (2.14) by
application of the criterion of exactness for a differential expression.
โช If F = F(x, y), then the total differential of F is defined as:
๐๐ =๐๐ญ
๐๐ ๐๐๐ฑ +
๐๐ญ
๐๐ ๐๐๐ฒ ๐จ๐ซ ๐๐ = ๐๐๐ฑ + ๐๐๐ฒ ๐. ๐๐
Where: ๐ =๐๐
๐๐ฑ๐ฒ
๐๐ง๐ ๐ =๐๐
๐๐ฒ๐ฑ
โช Taking the partial derivative of M with respect to y and of N with respect
to x yields:
๐๐ด
๐๐๐
=๐๐๐ญ
๐๐๐๐๐๐ง๐
๐๐ต
๐๐๐
=๐๐๐ญ
๐๐๐๐
โช The order of differentiation is immaterial for properties since they are
continuous point functions and have exact differentials.
โช Therefore, the two relations above are identical:
๐๐
๐๐ฒ๐ฑ
=๐๐
๐๐ฑ๐ฒ
๐. ๐๐
Thermodynamic Properties of Fluids
โช This is an important relation for partial derivatives, and it is used incalculus to test whether a differential dF is exact or inexact.
โช In thermodynamics, this relation forms the basis for the developmentof the Maxwell relations
โช Since U, H, A, and G are properties and thus have exact differentials.
โช Applying Eq. (2.16) to each of them, we obtain:
๐๐
๐๐๐
= โ๐๐
๐๐๐
๐. ๐๐
๐๐
๐๐๐
=๐๐
๐๐๐
๐. ๐๐
๐๐
๐๐๐
=๐๐
๐๐๐
๐. ๐๐
๐๐
๐๐๐ฉ
= โ๐๐
๐๐๐
๐. ๐๐
Thermodynamic Properties of Fluids
โช The Eqs (2.17) through Eq. (2.20) are called the Maxwell
relations.
โช They are extremely valuable in thermodynamics because they
provide a means of determining the change in entropy, which
cannot be measured directly, by simply measuring the changes
in properties P, V, and T.
โช Note that the Maxwell relations given above are limited to
simple compressible systems.
โช We develop here only a few expressions useful for evaluation of
thermodynamic properties from experimental data.
โช Their derivation requires application of Eqs. (2.11), (2.12),
(2.19), and (2.20).
Thermodynamic Properties of Fluids
Enthalpy and Entropy as Functions of T and P
โช The most useful property relations for the enthalpy and entropy
of a homogeneous phase result when these properties are
expressed as functions of T and P.
โช What we need to know is how H and S vary with temperature
and pressure; Consider first the temperature derivatives.
๐๐ฏ
๐๐ป๐
= ๐ช๐ท ๐. ๐๐
โช Equation (2.21) defines the heat capacity at constant pressure:
โช Another expression for this quantity is obtained by division of
Eq. (2.12) by dT and restriction of the result to constant P:
๐๐ฏ
๐๐ป ๐= ๐ป
๐๐บ
๐๐ป ๐๐. ๐๐
Thermodynamic Properties of Fluids
โช Combination of Eq. (2.22) with Eq. (2.21) gives:
๐๐บ
๐๐ป๐
=๐ช๐ท๐ป
๐. ๐๐
โช The pressure derivative of the entropy results directly from Eq.
(2.20):
๐๐บ
๐๐ท๐ป
= โ๐๐ฝ
๐๐ป๐
๐. ๐๐
โช The corresponding derivative for the enthalpy is found by division
of Eq. (2.12) by dP and restriction to constant T:
๐๐ฏ
๐๐ท๐ป
= ๐ป๐๐บ
๐๐ท๐ป
+ ๐
โช As a result of Eq. (2.24) to the above equation becomes:
๐๐ฏ
๐๐ท๐ป
= ๐ฝ โ ๐ป๐๐ฝ
๐๐ป๐
๐. ๐๐
Thermodynamic Properties of Fluids
โช The functional relations chosen here for H and S are:
๐ = ๐ ๐,๐ท ๐๐ง๐ ๐ = ๐(๐, ๐)
โช Whence;
๐๐ =๐๐ฏ
๐๐ป๐ท
๐๐ +๐๐ฏ
๐๐ท๐ป
๐ ๐ท ๐๐๐ ๐ ๐บ =๐๐บ
๐๐ป๐ท
๐๐ +๐๐บ
๐๐ท๐ป
๐ ๐ท
โช The partial derivatives in these two equations are given by the
following eqauations and (2.23) through (2.25):
๐๐ = ๐ช๐ท๐๐ + ๐ฝ โ ๐ป๐๐ฝ
๐๐ป๐
๐๐ ๐. ๐๐
๐๐ = ๐๐๐๐
๐โ
๐๐
๐๐๐ฉ
๐๐ ๐. ๐๐
โช These are general equations relating the properties of homogeneous
fluids of constant composition to temperature and pressure
Thermodynamic Properties of Fluids
Internal Energy as a Function of P
โช The pressure dependence of the internal energy is obtained by
differentiation of the equation:
U = H โ PV
๐๐
๐๐๐
=๐๐
๐๐๐
โ ๐๐๐
๐๐๐
โ ๐
โช Then by Eq. (2.25):
โช๐๐ฏ
๐๐ท ๐ป= ๐ฝ โ ๐ป
๐๐ฝ
๐๐ป ๐๐. ๐๐
โช Substitute Eq. (2.25) to the above equations:
๐๐ผ
๐๐ท๐ป
= โ๐๐๐ฝ
๐๐ป๐ท
โ ๐๐๐ฝ
๐๐ท๐ป
๐. ๐๐
Thermodynamic Properties of Fluids
The Ideal-Gas State
โช The coefficients of dT and dP in Eqs. (2.26) and (2.27) are
evaluated from heat-capacity and PVT data.
โช The ideal-gas state provides an example of PVT behavior:
๐๐๐ข๐ = ๐๐๐๐๐ข๐
๐๐๐
=๐
๐
โช Where: superscript "ig" denotes an ideal-gas value.
๐๐๐ข๐ = ๐๐๐ข๐ ๐๐ ๐. ๐๐
๐๐๐ข๐ = ๐๐๐ข๐ ๐๐
๐โ ๐
๐๐
๐๐. ๐๐
Thermodynamic Properties of Fluids
Alternative Forms for Liquids
โช Eq (2.23) through (2.26) are expressed in an alternative form by
elimination of๐V
๐T Pin favor of the volume expansivity by Eq. (1.3)
and of๐V
๐P Tin favor of the isothermal compressibility ฮบ by Eq. (1.4):
๐๐
๐๐๐
= โ๐ ๐. ๐๐
๐๐
๐๐๐
= (๐ โ ๐๐)๐ ๐. ๐๐
๐๐ผ
๐๐ท๐ป
= (๐ฟ๐ท โ ๐๐)๐ฝ ๐. ๐๐
โช The above general equations, incorporating ฮฒ and ฮบ are usuallyapplied only to liquids.
โช However, for liquids not near the critical point, the volume itself issmall, as are ฮฒ and ฮบ.
Thermodynamic Properties of Fluids
Alternative Forms for Liquids
๐๐บ
๐๐ท๐ป
= โ๐๐ฝ
๐๐ป๐
๐. ๐๐
๐๐ฏ
๐๐ท๐ป
= ๐ฝ โ ๐ป๐๐ฝ
๐๐ป๐
๐. ๐๐
๐๐ผ
๐๐ท๐ป
= โ๐๐๐ฝ
๐๐ป๐ท
โ ๐๐๐ฝ
๐๐ท๐ป
๐. ๐๐
Thermodynamic Properties of Fluids
โช Thus at most conditions pressure has little effect on the
properties of liquids;
โ Which is the special case of an incompressible fluid.
โช When๐๐
๐๐ ๐is replaced in Eqs. (2.26) and (2.27) in favor of the
volume expansivity, they become:
๐๐ = ๐ช๐ท๐๐ + (๐ โ ๐๐)๐๐๐ ๐. ๐๐
๐๐ = ๐๐๐๐
๐โ ๐๐๐๐ ๐. ๐๐
โช Since ฮฒ and ฮบ are weak functions of pressure for liquids, they
are usually assumed constant at appropriate average values for
integration of the final terms of Eqs. (2.34) and (2.35).
Thermodynamic Properties of Fluids
Internal Energy and Entropy as Functions of T and V
โช Temperature and volume often serve as more convenient
independent variables than do temperature and pressure.
โช The most useful property relations are then for internal energy and
entropy.
โช Required here are the derivatives ฮค๐U๐T V, ฮค๐U
๐V T, ฮค๐S๐T V and
ฮค๐S๐V T.
โช The first two of these is directly from Eq. (2.11): (๐๐ = ๐๐๐ โ ๐๐๐)
๐๐ผ
๐๐ป๐ฝ
= ๐๐๐บ
๐๐ป๐ฝ
๐๐ผ
๐๐ฝ๐ป
= ๐๐๐บ
๐๐ฝ๐ป
โ ๐
โช Eq. (2.36) defines the heat capacity at constant volume:
๐๐ผ
๐๐ป๐ฝ
= ๐ช๐ฝ ๐. ๐๐
Thermodynamic Properties of Fluids
โช With Eq. (2.36); the left side of the above equation becomes:
๐๐บ
๐๐ป๐ฝ
=๐ช๐ฝ๐ป
๐. ๐๐
โช With Eq. (2.19); the right side of the above eq becomes:
๐๐ผ
๐๐ฝ๐ป
= ๐๐๐ท
๐๐ป๐ฝ
โ ๐ ๐. ๐๐
โช The chosen functional relations here are:
๐ = ๐ ๐, ๐ฝ ๐ = ๐(๐, ๐)
โช Whence;
๐๐ =๐๐ผ
๐๐ป๐ฝ
๐๐ +๐๐ผ
๐๐ฝ๐ป
๐๐ ๐๐ง๐ ๐๐ =๐๐บ
๐๐ป๐ฝ
๐๐ +๐๐บ
๐๐ฝ๐ป
๐๐
โช The partial derivatives in these two equations are given by Eqs.
(2.36), (2.37), (2.38), and (2.19):
Thermodynamic Properties of Fluids
โช The following equations are therefore, the general equations relatingthe internal energy and entropy of homogeneous fluids of constantcomposition to temperature and volume.
๐๐ = ๐ช๐ฝ๐๐ + ๐๐๐ท
๐๐ป๐ฝ
โ ๐ท ๐๐ ๐. ๐๐
๐๐ = ๐ช๐ฝ๐ ๐ป
๐ป+
๐๐ท
๐๐ป๐ฝ
๐๐ ๐. ๐๐
โช Equation (1.5) applied to a change of state at constant volumebecomes: and the alternative forms of eq (2.39 & 2.40) are:
โ๐๐ท
๐๐ป๐ฝ
=๐ท
๐ฟ
๐๐ = ๐๐๐๐ +๐
๐๐ โ ๐ ๐๐ ๐. ๐๐
๐๐ = ๐๐๐๐
๐+๐ท
๐ฟ๐๐ ๐. ๐๐
Thermodynamic Properties of Fluids
The Gibbs Energy as a Generating Function
โช The fundamental property relations for homogenous fluids ofconstant composition given by Eqs. (2.11) through (2.14) showthat each of the thermodynamic properties U, H, A, and G isfunctionally related to a special pair of variables.
โช In particular:
๐๐ = ๐ฝ๐ ๐ท โ ๐๐๐ ๐. ๐๐
โช Expresses the functional relation: ๐ = ๐(๐, ๐)
โช Thus the special or canonical (variables confirm to a generalrule that is both simple and clear) variables for the Gibbsenergy are temperature and pressure.
โช Since these variables can be directly measured andcontrolled, the Gibbs energy is a thermodynamic property ofgreat potential utility.
Thermodynamic Properties of Fluids
The Gibbs Energy as a Generating Function
โช An alternative form of Eq. (2.14), a fundamental propertyrelation, follows from the mathematical identity:
๐๐ฎ
๐น๐ป=
๐
๐น๐ป๐ ๐ฎ โ
๐ฎ
๐๐ป๐๐๐
โช Substitution for dG by Eq. (2.14) and for G=H-TS gives, afteralgebraic reduction:
๐๐ฎ
๐น๐ป=
๐ฝ
๐น๐ป๐ ๐ท โ
๐ฏ
๐๐ป๐๐๐ ๐. ๐๐
โช The advantage of this equation is that:
โ All terms are dimensionless;
โ Moreover, in contrast to Eq. (2.14), the enthalpy rather thanthe entropy appears on the right side.
Thermodynamic Properties of Fluids
โช Eqs such as Eqs. (2.14) and (2.43) are too general for direct
practical application, but they are readily applied in restricted
form.
โช Thus, from Eq. (2.43),
๐
๐๐=
๐ ฮค๐ ๐๐
๐๐๐
๐. ๐๐
๐
๐๐= โ๐
๐ ฮค๐ ๐๐
๐๐๐
๐. ๐๐
โช When G/RT is known as a function of T and P, V/RT and H/RT
follow by simple differentiation.
Thermodynamic Properties of Fluids
โช The remaining properties are given by defining equations.
โช In particular,
๐
๐=
๐
๐๐โ
๐
๐๐๐๐ง๐
๐
๐๐=
๐
๐๐โ๐๐
๐๐
โช Thus, when we know how G/RT (or G) is related to its canonical
variables, T and P, i.e., when we are given G/RT = g(T, P), we
can evaluate all other thermodynamic properties by simple
mathematical operations.
โ The Gibbs energy when given as a function of T and P
therefore serves as a generating function for the other
thermodynamic properties, and implicitly represents
complete property information.
Thermodynamic Properties of Fluids
โช The partial derivatives in these two equations are given by Eqs.
(2.36), (2.37), (2.38), and (2.19):
๐๐ = ๐๐๐๐ +๐
๐๐ โ ๐ ๐๐ ๐. ๐๐
๐๐ =๐๐๐๐๐ +
๐
๐๐๐ ๐. ๐๐
Example
โช Determine the enthalpy and enropy changes of liquid water for a
change of state from 1bar and 298.15K to 1000bar and 323.15K.
The following data for water are available.
Thermodynamic Properties of Fluids