The First Law
Henry A. Bent North Carolina State University Raleigh, 27607
For scientists, citizens, poets and philosophers
"The laws of thermodynamics," we say, "are based on experience." "But whose experience?" students often wonder. "Not ours" ( 1 ) . Yet, nearly every playground graduate has sufficient experience with seesaws, falling balls, thrown balls, bouncing balls, friction, combustion, and arithmetic to he able to appreciate intuitively, even analytically, as I shall try to show, the immediately practical, the longer term scientific, and perhaps also the philosophical issues surrounding, and summarized by, the First Law of Thermodynamics. The Nearly Balanced Seesaw Nearly everyone knows that a light person sitting far out on a seesaw can balance a more ponderous person sitting closer in. Two such objects, Figure 1, are in equilihrium, states Archimedes in his treatise "De Aequiponderantibus," when their weights W I and WZ are inversely proportional to their respective distances L1 and L2 from the. point of support.
Figure 1 A seesaw. A lever. The mechanical contrivance is balanced (in internal equiiibrium),asserted Archimedes, when W 2 = W , ( L , / L d
Figure 2 . Changes in altitude of ponderable objects on a nearly balanced seesaw.
Expression ( 1 ) is the famous "Law of Levers," usually written that, a t equilibrium, W l L 1 = WzL2. I t expresses analytically an action Aristotle deemed most wonderful. For suppose the first object's weight slightly exceeds the equilibrium value W 2 ( L 2 / L ~ )Then . "though it appears contrary to reason," remarked Aristotle in a tract on "Mechanical Problems," "the larger weight will he set in motion upwards by the smaller one," Figure 2. The heavier object's height, hz, increases: Ahz = [ ( h z ) ~ i ,-, ~(~h d r n i is positive; Ah1 = [ ( h d f i n a i ( h ~ ) , , ~ , ~is, ,negative. ] Here, for convenience, we have defined, as in mathematics Ax
XM
-
Xinltirl
(2)
X is the numerical measure of anv numericallv measurable quantity. From Fieure 2. hv the Law of Andes. one sees that. in the A-notation, - h h l l ~ ,= A ~ , / L ~ , -Ah L 2= 1 Ah, L,
For a nearly balanced seesaw, therefore, the object's vertical displacements are, by eqns. (1) and (31, inversely proportional to their weights. -Ah, Ah,
W?
-q
A(W,h, W,h,
+ W,hJ
=
0
+ W?h, = constant
(6)
(7)
Other Machines "For the comprehension of machines," ohserves Mach, "it is not necessary to search after a new principle beyond that of the lever" (2). In any mechanical contrivancewedge, screw, pulley system, or whateverPc'it is possible to replace every force," continues Mach, "by the pull of a heavy body on a string" over an appropriately placed pulley. Thus, notes Newton, a "like account is to he given of all machines . . [Bly diminishing the velocity [or, equivalently, for a rigid, mechanical contrivance, the displacement; e.g., by making [Ah21 < I A h ~ l l ,we may augment the force [make W z > W J 1 (3). T h ~ result s may be generalized. Let h , represent the altitude of a Mach-replacement weight W , . Then, as indicated by Galileo, more fully by Torricelli, and, finally, most generally hy John Bernoulli in the Principle of Virtual Displacements, in the play a t equilibrium of perfect, frictionless machines [cf. eqns. (5) and (V1
.
ZW,Ahj = 0
( 8)
ZIY,h, = constant
(9)
or
Expression (4) may he written in several algebraically equivalent ways.
Remarks Mach, "There is no saving work [ Z W , A h i ] with machines" (2).
Prepared for an NSF-AAAS Chautauqua-type short course far college teachers, 1972-73, on Thermodynamics, Chance and Necessity, Abstract Art, Language, and the Entropy-Ethic.
Free Fall With machines motion is constrained. The natural motion of weight 1, Figure 1, is opposed by the natural motion of weight 2. Remove the constraints (the pivot; the beam) and, as everyone knows, both weights would fall. Volume 50, Number 5, May 1973 / 323
Further. as evervone knows. as an obiect falls. its motion quickens. Its veiocity increases. The tarther you fall, the harder vou hit. How much harder? ~ a l i i e obelieved a falling ohject's velocity increased in the simplest way. And "no increase of velocity is more simple," stated Galileo, "than that which is always added in the same manner" 14). Galileo had difficulty, however, in deciding how best to define "in the same manner." The velocity might increase equally for equal increments in the distance fallen; or for equal increments in the squareroot of the fallen distance; or, perhaps, for equal increments in the elapsed time.
Galileo tried first the first assumption: u = ks. (Distance is simpler to "see" than time.) That assumption leads, however, to a logical contradiction (though not for the reasons stated by Galileo). For suppose that initially v = s = 0.By assumption, u can exceed zero only if s > 0. By definition, s can exceed zero only if u > 0.In short, the object can't get started. [By calculus (not available to Galileo), v = dsldt = ks implies that s = Aek'. The initial conditions, s = t = 0, imply, however, that A = 0. Thus s = 0 for all time.] Five years later Galileo tried the third assumption: u = gt. Had he tried the second assumption, as would Kepler, believes Mach (2), the entire development of mechanics [and thermodynamics] would probably have been different. We'd probably he accustomed to generating forcefunctions from an energy-function [and, e.g., indices of thermal behavior, T, from an irreversibility-function, Sl by differentiation [as in recent treatments of thermodynamics 15-71] rather than, conversely, generating, as is more usual (and usually mathematically more difficult) an energy-function from a force-function [and S from TI by integration. Terminal Velocities a n d Elapsed Distances if V = gt
t
S
U
Figure 3. Galilea's experiments with a string suspended pendulum. ReIeaSed at E , the pendulum, swinging unimpeded, rises la (nearly) its starting altitude, along path EeE'. With a nail at P" or P"' the pendulum bob's path is EeE" or EeE"'. The motions are reversible: released at E', E", or E " ' , the bob rises to E . The bob's speed at e evidently depends an its change in altitude, not its precise path of descent.
The connection between columns 1 and 2 of the table is given hy eqns. (11) and (12); that between columns 1and 3 by eqn. (13); that between columns 2 and 3 by (14)-(16). Equation (15) is Mach's "Kepler Hypothesis." Other Descents
C-alileo boldly assumed that eqns. (14)-(16) hold for all frictionless descents, free or constrained (e.g., motion down an inclined plane), prouided the distance s is interpreted as the distance of uertical descent, the change of altitude. A falling object's terminal speed, Galileo asserted, depends only on its change in altitude, not on its precise path of descent. An object that has fallen can, like people, be "saved: it can ascend by various paths to Inearlv) its initial height. - , as do amusement- ark rollercoasters (approximately) and pendulums, Figure 3. Additionallv. in .. Galileo demonstrated.. a~oroximatelv, .. the Tower of Pisa experiment, as did Newton, later; in careful experiments with equal-length pendulums with different bobs, that the factor g in eqns. (14)-(16) is evldently a universal constant independent of an object's size, shape, and chemical composition. In summary, for free and constrained descents there are changes in both the velocity and the altitude of a single object. For nearly balanced seesaws (and other machines), there are changes only in altitude, of two (or more) ohjects. Finally to be examined are instances in which there are changes only in velocity. Collisions
The table summarizes (after Mach) the implications of Galileo's second assumption. Galileo supposed (correctly) that, with the assumption u = gt, and starting from rest, the distance traversed is the elapsed time t times an average velocity u,, (u,,,,,,, ur,,,,)/2 = (0 + gt)/2 = (1k)gt. Thus s = (1h)gtz. With a water-clock, Galileo verified experimentally that, for retarded motion, starting from rest, s is proportional to t2. Six algebraically equivalent ways of expressing the motion of an object falling freely near the earth's surface, starting from rest from height ho, are
-
gh
+
+ (1/2)u2 = constant, ghn
324 /Journal of Chemical Education
(16)
As most men and many women know, when two billiard balls collide head on with equal speeds, they reverse their velocities, receding from each other a t the same relative speed with which, initially, they approached each other, Figure 4a. Similarly, when a travelling hillard ball strikes a stationary hall head on, the colliding partners exchange velocities and, once more, recede from each other with the same relative speed with which they approached each other, Figure 4b. And, as nearly everyone knows, when an elastic hall bounces off an elastic portion of the earth, there is again a reversal of relative velocity. Experimentally, the reversal of relative velocity holds for all elastic collisions (ones in which there is no net change in the temperature or shapes of the colliding ohjects). If V and u are the initial, V' and u' the respective final velocities of two objects, then, regardless of their respective masses, M and m, in an elastic collision, head on
Taken with the conservation of linear momentum, eqn. (17) yields, with algebraic manipulation, the result that in elastic collisions uis uiua is conserved M V 2 + m d Z =
+ mu2. In elastic collisions, as Huygens stressed and evidently assumed)
dV
MV2
+ mv2 = constant
(W)
The eqns. (la), (16), and (9) share several features: v2 appears in two of them, h in two. Beyond the constants g and I,$,, there appear only the factors Wand M (or m). Weight "Weight" appeared (without comment) in the expression of Archimede's Law of Levers, ( I ) . That Law's physical content would appear to hinge on prior knowledge of procedures concerning the assignment of numerical values to weights and lengths. Let us suppose that, following accepted conventions, we know how to assign numerical values to lengths (and volumes). To assign a W-value to an object, we place it on a delicate, carefully-made, "equal-arm" seesaw and "balance it" with a set of weights. To produce a "set of weights" we need two conventions: a Reference Convention giving the W-value assigned to a standard object; and an Additivity Convention stating that if n objects separately, two-by-two, balance each other and if, jointly, they balance an object whose weightvalue is W, their individual W-values are W/n. W-values so defined are found to satisfy two laws: Archimede's Lever Law and the familiar law that for homogeneous bodies a t stated temperature and pressure, Wvalues are proportional to volume-values.
Alternatively, we could define W-values with the aid of the Reference Convention and expression (19), as did Newton in his much-maligned Definition I: "The quantity of matter is the measure of the same, arising from its density and bulk conjointly" (3). W-values so defined would be found, experimentally, to satisfy two "laws": the Additivitv Statement (formerlv a convention) and Archimede's lever Law. Or, we could define W-values via the Reference Convention and Archimede's expression (1) rearranged to W = Wo(Lo/L), by determining the ratio of two lengths (LoIL), a common practice in laboratories equipped with "singleheam" balances. W-values so defined would be found to satisfy, again, two laws: the Additivity Law and the Density'Law, eqn. (17). Anv one of the three statements. the Lever-. Additivitv-. or Density-statement, may be designated a Convention. U ~ o nex~erimentation.the other two statements would he denominated Laws of ~ a t u r e . Like the ~ostulatesand theorems of a formal axiomatic system, Conventions and Laws of Nature are, to an extent, interchangeable. Were we to designate as Conventions two of our three statements (e.g.: the Lever- and Additivity-Statements; or the Lever- and Density-Statements). . . some numerical ~rocedureformerlv viewed as determined by Convention (e.g., length-ratios; or volumemeasure from linear-measure) would be denominated a Law of Nature. 0
on prior knowledge concerning the assignment of numerical values to masses and velocities. Let us suppose that, following accepted conventions, we know how to assign numerical values to velocities (and accelerations). How, then, are numerical values assigned to M and m? How is "mass" defined? Physical quantities are defined to make the expression of physical laws simple. T(the index of thermal behavior of an ideal gas) was defined to make expression of Charles' Law simple. T(in "K) was defind to make expression of Carnot's theorem simple. P(an index of volume behavior) was defined to make a summary of Boyle's experiments on the "spring of air" simple. "Acceleration" and "velocity" (instantaneous) were defined (by Galileo) to make a summary of experience with descending objects simple. Mindful, therefore, that identical objects colliding impart to each other identical changes in lvelocityl (Fig. 4), and that relatively ponderous objects tend to "barge on" with relatively small changes in velocity (the earth, e.g., colliding with a hall, which "bounces"), Mach introduced this celebrated definition of mass: If we take A as our unit, we assign to that body the mass m which imparts to A m times the acceleration that A in the reaction imparts to it. To put it another way, "the ratio of the masses is the negative inverse ratio of the counter-accelerations" (2).
That (V' - V) and (u' - u) have always opposite signs, "that there are therefore, by our definition, only positive masses," adds Mach, "is a point that experience teaches, and experience alone can teach" (2). Laws and Definitions On rearrangement, eqn. (21) yields identically eqn. (20). Mach's definition of mass does to the Law of Momentum Conservation what the single-beam balance definition of weight did to the Lever Law. It appears "to substitute a truism for a great dynamical fact" (8). "It seems remarkable," writes Feynman, "that we can transform physical laws into mere definitions" (9). Definitions into which physical laws have been transformed are not, however, merely empty, arbitrary defini-
After
Before
,
Mass "Mass" was introduced implicitly in the sentence immediately above eqn. (18). If, indeed, linear momentum is conserved in all worldly events, then, in our previous notation, we have that, for a two-body, head-on collision, elastic or inelastic (b) Knock-On
The physical content of eqn. (20) would appear to hinge
Figure 4. Elastic collisions. Two demonstrations with a popular toy that in elastic collisions velocities of approach and recession are equal in magnitude. apposite in sign.
Volume 50, Number 5,May 1973 / 325
tions. They are existence theorems. That there really are relations amongst bodies corresponding to our definitions is not, remarks Rankine, a bare truism, but a physical fact (10). Additionally, in the present instance, as Mach (2) and Feynman (9) note, that two bodies (A,B) in mutual action with a third body (C) act as equal masses (MA= MG M, = Mc) act similarly toward each other (MA = M,) is not a logical necessity. I t is a physical fact. "It appears to me," writes Rankine, "that the making of a physical law wear the appearance of a truism, so far from being a ground of objection to the definition of a physical term is rather a proof that such definition has been framed in strict accordance with reality" (10). Weight and Mass
Mach-defined mass-values satisfy a familiar law. For a homogeneous material at stated temperature and pressure, M-values are proportional to the volume.
Taken with eqn. (19), eqn. (22) yields Mm,
a W
~
~
h
~
~
~
(23) d ~
This useful relation was checked personally hy the Master of Mechanics himself. "In experiments made with the greatest accuracy [on pendulums of different materials] I have always found," wrote Newton, "the quantity of matter in bodies to be proportional to their weight" (3).
ing. Aristotle, not Galileo and Newton, seems best to have summarized the natural state of objects: not perpetual, rectilinear motion; rather, rest. M.E. seems more often to be lost than got. It is seldom constant. Can M.E. be annihilated without a trace? T o shorten a long story of controversy and doubt, Joule discovered through careful measurement a useful conjunction. With a lose in M.E. is always, somewhere, an increase in temperature. Rub something and it gets hot. T o express Joule's discovery compactly, a definition (as usual) is convenient. Define 8 1Thermal Surroundings
(30)
Then we may say, with Joule: As M.E. decreases To increases proportionately. In the A-notation AT,
-A(M.E.)
(31)
Or,
The numerical value of the constant of proportionality in eqn. (32) is a function of the physical state and chemical composition .of the thermal surroundings and those ~ conventions by which numerical values are assigned to A(M.E.) and ATB.For a "universe" (an isolated system) comprised solely of a mechanical system and its thermal surroundings, we have that, by definition
Constants of Frictionless Motion
In summary, for machine-moderated changes in altitude [eqns. (1-71, with (23)l. Zrnih; = constant For collision-produced changes in velocity [eqn. (la)],
Cg is a measure of the capacity of 0 to keep constant its temperature as M.E. is annihilated. Traditionally, it is called 8's "heat capacity." Typically, its units are calories per degree. It is a rate: the rate of change of -M.E. with respect to Tg. Equation (32) may be written in the algebraically equivalent form
Zrn,",? = constant For earth-induced changes in altitude and velocity [eqn. (Wl
A glance a t eqns. (24)-(29) suggests this definition
ghi + (1/2)u,2 = constant The above expressions can be combined (with some loss of information) into one compact expression. Multiply the first one by g, the second by Ij$, the third by mi summing over all i. And define P.E.
Zm,gh,
K.E. 1 Z ( ~ / Z ) ~ . U , ~
(24)
(n)
One may then write that, for frictionless motion, i.e., in a (hy definition) purely mechanical system
With eqn. (351, eqn. (34) takes this form A(M.E.)
+ AE, = 0
(36)
Equation (36) indicates merely that, for a universe comprised of a mechanical system and its thermal surroundings (B), we agree to set AE, = -A(M.E.)
(37)
(29)
This definition, eqn. (37), like all definitions, is neither true nor false. It makes no assertions. I t merely says, "let such a phrase [AERI be used in such a sense [= a(M.E.)]" (10). That, in Rankine's words, it is a useful rather than a fantastic definition, that it corresponds in some way to observed phenomena, is assured, in this instance, by Joule's experiments. Finally, for uniformity in the notation, replace "M.E." by "Em,,, " For a universe = mech. sys. + 8, the expression of &onservation, eqn. (36), is then easy to set down. Merely place AE before each part
Friction-free systems are, like ideal gases, physical-if useful-fictions. Even carefully oiled machines, remote satellites, and super bouncing balls eventually stop mov-
Equation (38) expresses compactly-and (with recall of a host of conventions) perhaps usefully-numerous observations on machines, trajectories, collisions, and friction.
P.E.
+ K.E. = constant
(26)
T o obtain a still more compact expression, define M.E. I P.E. + K.E. Then, for purely mechanical changes, one may write M.E. = constant Or, A(M.E.1 = 0
(27)
(28)
..
Joule's Discovery
326 /Journal of Chemical Education
Lavoisier's Discovery
Even in the absence of Joulean friction, TOcan change, as every youngster who's played with matches (or icicles) knows. Lavoisier-among others-discovered this law of Nature: As a chemical reaction (or phase change) advances, Te changesproportionntely. T o express compactly Lavoisier's discovery, these definitions are useful chemical system. A place where chemical changes occur (7). C-Degree of advancement of a chemical reaction (or phase change) in o. Far example, the number of moles of a product produced ( 1 1 ) . With the above definitions, we may write, in the Anotation, that, according to Lavoisier AT,
Or AT,
Or
-
=
At
(39)
hAt
(40)
GAT" - k'A[
= 0.
AER- k'AZ
=0
(h'
= Cnh)
(41)
Or (eqn. 32) Define
sines, cosines, and similar triangles. It's the problem of density and velocity; atomic weights and concentrations; slopes and differential quotients. It's the problem of units. The problem of scales and measurement. It divides society, intellectually, into two cultures: mathematicians, scientists, and engineers whose work per permeates and those for whom per in any form is perplexing. A Declaration of Dependence
The laws of Archimedes, Galileo, Huygens, Joule, and Lavoisier are, in essence, humble recognition that we cannot do only one thing. We cannot change only the altitude, or only the velocity, or only the temperature of one object. Many things we cannot do. "The blowing wind I let it blow," writes the American poet Robert Francis in "Old Man's Confession of Faith." "I let it come, I let i t go./ Always it has my full permission. Such is my doctrinal position./ . . . Blow east, blow west-I let it blow. I never tell it No" 1121. No matter what happens in a system and its surroundings (thermal and mechanical), by the confessions of experience summarized in eqns. (38) and (44)
(42)
AE* = +'At (43) Then, for a universe 9 + a (e.g., a bomb calorimeter) we may write AE, + AE. = 0 (44) A Weed in the Field of Thermodynamics
The numerical value of the proportionality constant h' in eqn. (41) is a function of the nature of the chemical reaction (or phase change) advancing in a, a's temperature, pressure, and composition, and those conventions by which numerical values are assigned to AE, and A t . For universes 9 + a, we have that, by definition
h' is a measure of the capacity of a change in a to produce a change in TO.Traditionally, it is called "the heat of reaction." Typically, it is assigned the units "calories per mole" (of something produced in a). Usually, it is signified by the symbol " A E (of a or 9; sign-conventions regarding the "heat of reaction" vary from book to book; in some books, from page to page). Unfortunately, contrary to a neophyte's legitimate expectations, based on eqn. (2)
"AE," regretably, is not a difference. It is a rate, an instantaneous rate (eqn. (45)) analogous to "beat capacity" (eqn. (33)). It is the rate a t which E (subscript a or 9) changes as 5 changes. To determine "AE," it is not necessary-nor, indeed, usually desirable-to allow an entire mole of material to react, anymore than it is necessary to travel an hour to decide. one's speed is sixty miles per hour. To paraphrase Mach (2), to determine accurately density and velocity; atomic weights and concentrations; of advancement vanishingly small and such only; otherwise the system may be carried over into a state for which h' is entirely different. A study of the thermodynamics of themochemistry, phase equilibria, chemical equilihria, and electrochemistry is like a conversation with Humpty Dumpty. Nowhere else does "A" stand for "per." Most of the A's should be uprooted and replaced by the operator (cf. eqn. (45)) a / a t . A Thorn in the Side of Nonscientists
The problem of per transcends thermodynamics. It's what makes the hard sciences hard. It's the problem of Division, of fractions, ratios, proportions, and percent; of
Energy is conserved. We can't create it. We can't destroy it. We onlv transform it. That is our doctrinal Dosition. ~nergywise,life is a zero-sum game. Limits of Science
Creation of an energy-function is a branch of phenomenology, a treatment of the universal qualities of phenomena. Its purpose, writes the philosopher Charles Pierce, is "to unravel the tangled skein of appearances and to wind it into distinct forms," not an easy task (13). For the phenomenologist "must see what stares him in the face, which is the faculty of the artist" (13). An artist [and phenomenologist].' writes Hans Hoffman, "must learn to see (It is amazing what people do not see), and he must learn to express his experience Plastically [i.e. Mathematically]" (14). "Sensory raw material [experimental data] blended to a spiritual unity [a selfconsistent theory] through legitimate use of the medium [e.g., mathematics] is Art [Science]" (14). A vase gives form to a void (15), music to silence, d he no me no low ".to haooenines. .. " Phenomenology simply scrutinizes direct appearances (13). I t endeavors to combine minute accuracv with bmad generalizations (13). I t doesn't make moral jidgments. It doesn't draw distinctions between good and had (13). Phenomenology doesn't say "should or "shouldn't." It says "will" or "won't." I t doesn't sav a released a . n.d e should fall. I t savs i t will fall. Phenomenology does not say whether we should, or will, release the apple-or strip mine a Kentucky hillside. That's a more complicated matter, Can we, should we, will we ever reduce every "should" to will? Such would be the scientist's ideal: perfect law and order in the Universe. Literature Cited (1) Professor Milton Bunon, in a Symporium on Teaching Thermodynamics a ~ ~ ~ ~ o r e d
by the Division of Chemical Education al the 141rt Meelinp of tho ACS, Washinnon. D. C.. April. 1962. (2) Mach, E., "The Science of Mechanics: A critics1 and Historical Account of Its
Deveiopmonf." (Tranxlaior: Mccormack, T. J I The o,,en Court Puhlirhine CO..
I31 Neutun. I.. "Plincipia." IT~ansloror:Cajori. F.) University of Cslifornia Press. Berkeley, UnIlf.. IPbo. I41 Whowell. W., " H i = h y of the Inductive Scienc=,"'Frank C ~ J& J Co.. London. England. 1961, Part 11.
'Brackets enclose editorial comments. Volume 50, Number 5, May 7973 / 327
151 Glles. R.. "Mathematical Foundation. of Thermodynamics," Pexgamon Press. The MacmillanCo..NewYnk. N.Y.. 1964. (61 Csllen. H. B. "Thermdynamin."John Wilsy and Sons. Inc., New York,1960. 171 Bent, H. A , 'Tho Second Law." Oxford University Press, N e r York. N. Y.. 1965: alsoJ. CHEM. EDUC.. 47.337 119701. lJohn Herschel. "Familiar Leefunson Scientific Subjectr." Quoted in ref. 110). 18) S 191 Feynman, R. P., Loighton. R. B., and Sands. M.. "The Feynman lecture^ on Physics," Addison-Wesiey Publishing Co.. Reading, Mass. 1963. Val. 1, p. 10-
.L4..
110) Rsnkine. "Miscellanems Scientific Papen." Chsriea Grifin and Co.. London. Eng-
328 /Journal of Chemical Education
land. 1881,Ch. 13.
rill himsine, I. and Defay, R.. "Chemical Thermodynamics," Vmnslotor: Everett. H.1 Longmans Green andCa.. London, 1954.
eb. R.. "Came Out Into the Sun." University of Masaaehusefrr Press, A ~ nt. Mars.. 1968. eman, J. K.. to the Philodoghy of Charles S. Pierce." . "An . . . Intmduction .. M.1.1. mesa. ~ a m m d g e , ~ a r sIYW. ., 1141 Hoffman. H.,"SosrchfortheResl."M.I.T. PI-. Cambridge,Masa., 1987. 115) Brsque. G.. "Illustrated Nofebmks 1917-1955." ITran~lolor: Appelbaum, S.) Dover Publication.Inc.. NewYork. 1971.
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