Henry A. Bent
North Carolina Stote University Roleigh, 27607
I
I
A Note on the Notation and Terminology of Thermodynamitr
"Thermodynamics is not d i i c u ~ t , "Professor Giauque has said, "if you can just keep track of what it is you are talking about." I n chemical thermodynamics one usually needs to keep track of three things: a system, its thermal surroundings, and its mechanical surroundings. These conceptually distinct parts of the universe are shown schematically in Figure 1.
The notation in (3) and (4) is explicit but clumsy. The following abbreviations have been found useful (2, S), Figure 2. s --The system 9
UA
= The system's thermal surroundings
= The system's mechanical surroundings
...............................
Figure 1. Abrtrost, block-dimgram of the thermodynamically conceptvally important pat?$ of the "universe" of a typical rhomisd experiment.
Solid lines in Figure 1 connect mutually interacting parts of the universe. B y agreement, the system's thermal surroundings are encased in rigid, thermally conducting walls, its mechanical surroundings by moveable, non-thermally conducting walls. By convention, therefore, those two parts of the universe do not interact with each other. They cannot interact with each other; they are "orthogonal": what one can do the other cannot do, and vice-versa. They are not connected by a solid line. Most important in Figure 1 is the dashed line. It indicates that, for events considered, one may apply to the totality of interior objects Clausius' statements Ah',.,.,
0
(1)
AS,,,,, 2 0
(2)
=
How to draw, mentally, the dashed enclosure in a particular instance is, however, a skill acquired only through precept and practice. Every discipline has tacit, inarticulate features that render its practice in part an art (1). Now, by design, E and S are additive properties. By design, also, the entropy of a pure mechanical system does not change. Thus, eqns. (1) and (2) may be written 'For a specified change in a, represented by the symbol A , AE,tis determined by measurementof mechanical quantities (e.g., the altitude of s weight) and the equations of classical mechanics. The AEe corresponding to an observed changein 8 is determined in a separate, Joulean, mechanical-equivalent-ofheat type experiment involvingonly 9 and wt. AE, is then calculated by eqn. (5). 44
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Journol of Chernicol Education
..........................
i....
Figure 2.
Convenient abbreviations
(see Fig.
I).
Wt stands for a "weight", i.e., a moveable mass in (usually) the earth's gravitational field. All mechanical systems can be represented in thermodynamics by a weight. For in any friction-freethat is to say purely mechanical--system, any form of energy can be converted to any other form of (mechanical) energy at par. Without loss of generality any energy-storing and energy-releasing mechanical device-a spring, flywheel, weight-can be used to represent changes in mechanical energy. With the notation summarized in Figure 2, the algebraic expressions of the First and Second Laws, eqns. (3) and (4), become aEw.r = ASt.t,t
=
AEo
AS.
+ Ah'e + Ah'
+ ASe 2 0
aE,t=
0
(5)
(6)
Also, always, by definition Equations (5)-(7) are applicable to all conventional problems in chemical thermodynamic^.^ The First-Law equation, eqn. ( 5 ) , is usually written The algebraic meaning of AE is universally understood. Always AX = X,i.,l -X,.i,,.l. X may he any thermodynamic property (E, S, T, P, V ) of any thermodynamic system (o,0 , wt). The terms q and w require special comment. Ad& tionally, a statement is required to indicate to what part of the universe the symbol E refers.
A Short Dictiaary =Energy lost by the thermal surroundings2 w E Energy gained by the mechanical surroundings' AE E Energy-term required to conserve energy [eqn. (511
=
q
Algebraic Definitions of the Common Adjectives of Thermodynamics
- AEe
= AE,,
Adiective
( A X , = 0) - Definition
e AEu
1. Isothermal 2. Isobaric 3. Isometric 4. Isentropio 5 . Adiabatic 6. Free 7 . Free, adiabatic 8. Reversible 9. Reversible. adiabatic
AT. = 0 AP, = 0 AV. = 0 AS. = 0 AEe = 0 AE,, = 0 AE. = 0. ASt0taz = 0 AS, = 0 '
For most students eqn. (5) is more easily remembered correctly thanis eqn. (8). The u -0 -2ut notation ha? a further attractive feature. With its use, most of the common adjectives in thermodynamics can be converted immediately to compact, mathematically useful statements, identical in algebraic form (see the table). For irreversible, natural, spontaneous, real eventsthese adjectives are in thermodynamics synonymousfor such events (the only events of interest), AS.,,, > 0. For impossible, unnatural, unreal events, ASm1., < 0. The "u-0-wt-totUZ" notation makes precise, also the meaning of the expression "AS = q,,,/T." Always, by eqns. (6) and (7) AS, = - A s s
+ ASA,~.Z
=
AS,,,,
(=
-7 +
(9
-,9 + A S w d
Dictionary
Since always ASmtar2 0, a l ~ ~ a AS, y s 2 q/T. too, by eqn. (9)
=-
Clearly,
71,..,.
Tsble
-"en*
Dictionary
Further, t,he c-0-tot not,ation renders explicit a simple physical interpretation of the "PAV" term. A useful representation of isobarice vents (see table) is shown schematically in Figure 3. Let IM represent the mechanical effect (mass) of the atmosphere resting on a piston of cross-sectional area A. Let h be mass M's altitude, g the local acceleration of gravity. Then always, for any problem for which Figure 3 is a suitable abstract representation4 Sat, = AEns = MgAh = ( M g / A ) ( A A h )
PocmAVo
See definitions 5 and 6 and eqn. ( 5 ) . Wee definitions 5 and 8 and eqns. (7) and ( 6 ) .
L
............................. 3
Figure 3. Abstract, block-diogrm of a system s held a t conrtont prerrure and conrtont temperature and coupled to a mechoniral system wt and capable, therefore, of doing or consuming "useful work' LsAE,tl.
Finally, the u-0-tut notation allows one to use more widely than before the precise notation of partial differentiation in the formal exposition of thermodynamics, thereby helping to insure that the assumptions and special conditions introduced in the derivation of derived relations are stated clearly and explicitly. Two examples will be given: the free expansion of a gas and the entropy change of a gas in an isothermal expansion. Free expansion of a gas. Joule discovered that, for a universe c 0, u a dilute gas
+
(11)
a ( E , d d - Eo Ce
(12)
avo
-E
+
+ AE,,, + AE,,
-AEe = AE, = (AE,
+ P.,,AVV) +
aE,t
By the 1st Law BY eqn. (12)
1
T,
The "heat of reaction" for a change within the 0 atm wt (Fig. 3) is given by the universe a following expressions
+ +
d
=
To
Definition of Cs Additivity of E First Law EUc = constant
Since Cs # 0, Joule's discovery implies that
Diotionary
(= q) - -
The added phrase "to the system" is superfluous for 2 Period. the universes of Figures (1) and (Z), and often counter-productive. I t directs attention away from B to the wrong object (n), as does the common textbook definition of o: "Heat absorbed bv the svs= tem"; this statement leads to such barbarisms as q.,
- R"..~""di".l.
Period. The added phrase "from the system" is generally superfluous and counter-productive. I t directs attention away from wt t o the wrong object (a),as does the usual texthook-definition of w: "Work done by the system"; this statement leads to din., and confusion, in even simthe barba&n we,,,, = -w ple problems (4). ' P o t , (--Mg/A) is written, not P., since "P," has not been defined in the previous discussion. a
.,.,.,
Now, if E.
= function (V.,
T.)
Since T , = To,dT, = dTs. Using the latter relation in eqn. (15), dividing through by dV,, and imposing the condition E,, = constant, one finds that
Volume 49, Number I, January 7972
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The last equality follows from eqns. (13) and (14). Entropy change of a dilute gas in an isothermal expansion (or compression). The quantity sought is (bS,/ ~ V J T ~ .
So does not depend on S8..z =
[h(Starz - Se)] avc T..,s,o~s,
Additivit.y of S
Slor.t constant Eqn. (7)
Te
aVr
Ew'))Tr,St.roc Additivity of E
1
=
bE,,
6( ~ , ) T ~ . S r n u ~
Eqns. (5) and (16)
1 1)efinition of Pr Te (Po) For an ideal gas, P , / T e = n R / V , . I n summary, the "a-8-wt-total" notat,ion enables one to express the First and Second Laws of thermodynamics in symmetrical, analogous, easily remembered forms, while at the same time emphasizing explicitly (in the subscripts 9 and wt) that to apply the = -
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equations of t,hermodynamicsrequires afamiliarity wit,h the primit,ivc conoept,s thermal and mechanical interactions (5) (embraced, iu part, in t,hepropert,ies assigned to diathermic and adiabatic walls). The notattion brings c1arit.y t,o the definit.ions of "q" and "w" and offers one the opt,ion of purging from one's thermodynamic vocabulary (3) t,he oft.en trouhlesome terms "heatl" and "work," by suhstit,ut,ing, respect,ively, t,hese operationally more expressive, if grammat,ically less succint, phrases: "energy lost by the thermal surroundings"; "energy gained by t,he mechanical surroundings." Finally, the notatlion highlighk a simple, physical interp~etat~ion of t,he P A V term; permit,s one t,o give precise, algebraic definitions for the word "reversible" and other characteristic adject,ives of t,hermodynamics; and allows one to extend in t,hermodynamics the range of application of t,he techniques of part,ial differentiation. Literature Cited (1) P o ~ m n .M.. "Personal Knowledge." University of Chicago Pleas. Chioago, Ill., 1968, Ch. 4. (2) l3s.wr H. A. J. CHEW. Ennc.. 39, 491 (1962): "The Second Lkw," Oxfbrd it^ Press. New York. 1965. (3) C n k ~ oN. . C.. J . C I I ~ MRUUC. . 47 342 (1970). (4) see.r.CII.M. nnrrc.. 41, 102.874 iis641. ( 5 ) I$RIVOMIN, P. \V. "The Nature of Therrnodynarnios," Harper Torch I~ooks,Harper &hdRow, New York. 1961, Chap. 1.
