A rigorous equation of state for solids, liquids, and gases
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I.
INTRODUCTION
IN the field of chemical thermodynamics, communications on equations of state that relate the pressure P, with volume V and temperature T, have been extensive. In most cases, the proposed relationships are based on well-founded but idealized models that restrict the use of the results to specific P-V-T regions of real substances. In these terms, the general relationship derived here is unrestricted. As illustrative examples, it is applied to five gases, four solids, and three liquids. II.
DERIVATION
P= V=
\Or/v
-~ T
\aT/p +
--~ r
and =y T
Equations [1] and [2] can therefore be written as follows: [31
and
r(Oq \OT/,
[4]
G.W. TOOP, Principal Research Scientist, is with Cominco Research, Cominco Ltd., Trail, BC, Canada V I R 4S4. Manuscript submitted September 13, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS B
[51
Cv =
,
[6]
Multiply Eq. [6] through by T to give
:(0q (0q
[7]
Combination of Eqs. [5] and [7] provides a simple and general equation of state that is the subject of the present work: (P + x) ( V - y) = (Cp - Cv)T
[8]
Equation [8] reduces readily to the ideal gas law for which Cp = 5R/2, Cv = 3R/2, and Cp - Cv = R. Also, for an ideal gas, E = 3RT/2 and H = 5RT/2, i.e., the energy terms depend only on temperature and hence, both x and y are zero. The ideal gas equation is therefore produced, P V = RT. The form of Eq. [8] is similar to that of the van der Waals relationship
T
(V-y)=
r40q (OVl \or/v\or/?
[21
(;):x
\~-~/~
=
The thermodynamic relationship for the difference between the heat capacities at constant pressure and constant volume, Cp - Cv, is given by
[11
where E is the molar internal energy and H is the molar enthalpy. Historically, these two expressions have been treated separately, but they can be combined to give a general equation that relates P, V, and T. To do this, let
(P + x) =
(e + x) ( V - y)
cp -
Two thermodynamic equations that relate P with T, and V with T, are well documented in the literature, tu
:/-
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