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6.1.2 Measurable Properties
Combining Eqs. (6.2) and (6.3) gives the following relation:
dU = TdS − PdV
(6.4) P2: KVU/KXT QC: —/— T1: IML 6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 235
In this section some thermodynamic properties that are di-
rectly measurable are defined and introduced. Heat capacity
This relation is one of the fundamental thermodynamic rela-
tions. Differentiating Eq. (6.1) and combining with Eq. (6.4) at constant pressure (C P ) and heat capacity at constant vol-
gives ume (C V ) are defined as:
δQ

(6.5) dH = TdS + VdP (6.17) C P =
dT P
Two other thermodynamic properties known as auxiliary δQ
functions are Gibbs free energy (G) and Helmholtz free energy (6.18) C V =
(A) that are defined as dT V
Molar heat capacity is a thermodynamic property that indi-
(6.6) G ≡ H − TS
cates amount of heat needed for 1 mol of a fluid to increase its
(6.7) A ≡ U − TS temperature by 1 degree and it has unit of J/mol · K (same as
◦
G and A are mainly defined for convenience and formulation J/mol · C) in the SI unit system. Since temperature units of K
◦
of useful thermodynamic properties and are not measurable or C represent the temperature difference they are both used
properties. Gibbs free energy also known as Gibbs energy is in the units of heat capacity. Similarly specific heat is defined
particularly a useful property in phase equilibrium calcula- as heat required to increase temperature of one unit mass of
◦
tions. These two parameters both have units of energy simi- fluid by 1 and in the SI unit systems has the unit of kJ/kg · K
◦
lar to units of U, H,or PV. Differentiating Eqs. (6.6) and (6.7) (or J/g · C). In all thermodynamic relations molar properties
and combining with Eqs. (6.4) and (6.5) lead to the following are used and when necessary they are converted to specific
relations: property using molecular weight and Eq. (5.3). Since heat is
a path function and not a thermodynamic property, amount
(6.8) dG = VdP − SdT of heat transferred to a system in a constant pressure process
(6.9) dA =−PdV − SdT differs from the amount of heat transferred to the same sys-
tem under constant volume process for the same amount of
Equations (6.4), (6.5), (6.8), and (6.9) are the four fundamen-
tal thermodynamic relations that will be used for property temperature increase. Combining Eq. (6.3) with (6.18) gives
calculations for a homogenous fluid of constant composition. the following relation:
In these relations either molar or total properties can be used. ∂U
Another set of equations can be obtained from mathemati- (6.19) C V = ∂T V
cal relations. If F = F(x, y) where x and y are two independent
variables, the total differential of F is defined as similarly C P can be defined in terms of enthalpy through Eqs.
(6.2), (6.5), and (6.17):
∂F ∂F

(6.10) dF = dx + dy ∂ H
∂x y ∂y x (6.20) C P =
∂T P
which may also be written as
For ideal gases since U and H are functions of only tempera-
(6.11) dF = M(x, y)dx + N(x, y)dy ture (Eqs. 5.16 and 5.17), from Eqs. (6.20) and (6.19) we have
where M(x, y) = (∂F/∂x) y and N(x, y) = (∂F/∂y) x . Consider-
2
2
ig
ing the fact that ∂ F/∂x∂y = ∂ F/∂y∂x, the following relation (6.21) dH = C dT
ig
P
exists between M and N:
ig
ig
∂M ∂N V
(6.22) dU = C dT
(6.12) =
∂y ∂x where superscript ig indicates ideal gas properties. In some
x y
references ideal gas properties are specified by superscript ◦
Applying Eq. (6.12) to Eqs. (6.4), (6.5), (6.8), and (6.9) leads to ∗ ◦ ∗
the following set of equations known as Maxwell’s equations or (i.e., C or C for ideal gas heat capacity). As will be seen
P
P
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
ig
[1, 2]: later, usually C P is correlated to absolute temperature T in
the form of polynomial of degrees 3 or 5 and the correlation
∂T ∂P

(6.13) =− coefficients are given for each compound [1–5]. Combining
∂V S ∂S V Eqs. (6.1), (5.14), (6.21), and (6.22) gives the following rela-
∂T ∂V tion between C and C through universal gas constant R:
ig ig
V
P
(6.14) =
∂P ∂S ig ig
S P (6.23) C − C = R
∂V ∂S
P V
(6.15) =− ig ig
∂T ∂P For ideal gases C and C are both functions of only temper-
P
V
P T
∂P ∂S
ature, while for a real gas C P is a function of both T and P as
(6.16) = it is clear from Eqs. (6.20) and (6.28). The ratio of C P /C V is
∂T ∂V
V T called heat capacity ratio and usually in thermodynamic texts
Maxwell’s relations are the basis of property calculations by is shown by γ and it is greater than unity. For monoatomic
relating a property to PVT relation. Before showing appli- gases (i.e., helium, argon, etc.) it can be assumed that γ = 5/3,
cation of these equations, several measurable properties are and for diatomic gases (nitrogen, oxygen, air, etc.) it is as-
defined. sumed that γ = 7/5 = 1.4.
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