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Ice-skating and the lce-Water Equlllbrlum To the Editor:
As a chemist and a moderately accomplished figure skater 1 was fascinated hv [.eon F. Loucks article "Subtleties of Phenomena ~nvolvfngIce-Water Equilibria" [1986,63,115]. I t seems to me that the surface film of water on the ice in the case of figure skating may depend upon a combination of frictional forces and wressure effects. A figure skate is not only hollow ground h k also the bottom surface is an arc, so that onlv a few centimeters of the blade actually touch the ice a t any time. Figure skaters skate primarily on the knife edge of the figure skate blade, which typically has a much more pronounced hollow than hockey skate blades. When doing school figures the idea is in fact to stay on the knife edge. Therefore, the pressure of an 80-kg skater is applied to perhaps an area 10 cm in length and 0.001 cm in width, that is. an area of 0.01 cm2. The wressure is about 8000 kg/cm2, easily enough to melt the iceat 6 to 8 degrees below zero. Although no ice arena is in operation in central New York a t this time of the year, I did do a few simple experiments with an ice skate and water frozen in alarge baking pan. The length of the figure skate edge that touches the ice surface when the hlade is held parallel to the ice is 7 f 1cm. The temperature of the ice and the skate was -9 "C. When about 50 kilograms of pressure was placed on the skate hlade the hlade sank into the ice 1-2 mm. I examined the aroove with a 30X microscope and found little evidence for fracturing. The straieht edee of the eroove had a few small ice chips around it, wkch could have come from moving theskate from side to side, since it was difficult to hold the skate ahsolutely straight and steady. The groove made when the hlade is pushed straight down seems difficult to explain on the basis of frictional forces. It may he that plastic deformation of the ice is occurring rather than melting or fracturing and that Young's modulus calculations do not accurately represent the situation. A drawing of the observed groove is shown below.
Shape of the gmove left by a figure skate blade edge.
In figure skating an accomplished skater can tell the difference between ice a t different temperatures by how it feels to skate on it. Colder ice is referred to as hard and skates seem tosink into it less. This suggests that the small temperature variations have a noticeable effect on skating, an observation difficult to explain in terms of frictional effects. From personal experience I can state that i t is possible to tell the difference between sharp and dull skates by gliding, and it is possible to have skates that appear to be too sharp. The hollow is so deep that the edge angle is very acute. Skates of 186
Journal of Chemical Education
this sort are difficult to skate on, because they feel as if they are sticking into the ice. These observations would seem to suggest that the phenomenon of ice melting and/or deformation under pressure may he more important for some types of skating than physical chemists suppose. The last point to he made is that, a t least in figure skating, most of the skating takes place indoors so that the ice is usuallv within 6-10 degrees of freezing. In summary, both possihle mechanisms may well he operating, and, a t least in the case of indoor figure skating, ice melting and/or deforming under pressure may he an important contributor to the ease of motion. I t seems to me that the explanations associated with the physical chemistry of ice skating are still not clear. As soon as a local ice arena opens I plan to he skating and observing, I hope Loucks will join in the fun and do the same. Robed Sllberman S.U.N.Y. at Canland Cortland. NY 13045
A Comment on Slgnlflcant Figures and Propagation of
Uncedalnty To the Editor:
A recent article in this Journal [1985,62, 6931 dwelled a t length on hreakdowns in rules for significant figures, and on more complete procedures for calculating uncertainties and appropriate significant figures counts in cases where rules either do not exist or wroduce amhieuous or contradictorv results. This effort seems to have been initiated hy an incident in which an alert student noted an awvarent contradic.. tion arising from application of significant figure rules. The author's immediate reply n,a9 to the effect that significant figure rules are only approximate and sometimed lead to significant figure counts that are slightly off. I believe that this was the correct response and that the matter could well have rested there. Significant figure rules are intended only to provide a "quick and dirty"estimate of uncertainty propagation. They are aimed at finding the order of magnitude of the uncertaintv. More than this must not be exwected., and.. in light of this, I do not see the point of a' exhaustive. investigation of conditions under which these rules "fail". In fact, I'm not convinced that they should he said to fail, given their nature and intended use. The concept of significant figures and the rules governing their use are, of course, not separate from the more fundamental concepts and rules dealing directly with propagation of error (which may also he called propagation of uncertainty). Significant figures is a derived construct, which we use precisely because it is much easier and usually provides accewtahle results. Additionallv. this is crucial) the - . (and . concept and rules of significant figures will, if properly presented, convey the essential point: any datum obtained, directly or indirectly, from measurement(s) carries an uncertainty, and any result calculated from that datum will carry a corresponding uncertainty. I believe that in beginningchemistw courses. we should be satisfied with sipifrcant figure rule; as they 'are currently commonly used. We must stress-continually-to our students that the point of these rules is to take account of
uncertaintv in measurement and propagation of that uncertainty thriughcalculation. I belie"e that the greatest barrier to understanding and appreciating significant figures is that students lose sieht of this essential connection. We should also inform students that significant figure rules are approximate and will theiefore produce ambiguous or contradictory results on occasion. That there are superior methods, which nroduce unamhimrous values, might be mentioned, along with the point thai'these methohs usually require what would seem to be an inordiuate effort of ralculation for the worth of the result. If we want to do a better ioh of determining uncertainty in a result, we should turn to the more fundamekal methods of propagation of error.' These may be applied to situations in which no explicit error estimates are given, such as the examples in the cited article; significant figures convey an implied approximate uncertainty that may be handled in this way as well as an explicit value. Such an approach requires an effort comparable to that of the "General Propagation Procedure" in the cited article, and it has the advantage of being mathematicallv concise and verv straightforward. I t also dves explicitly the contribution 0%the uncertainty in each datum to the total. For a result that depends on several variables (data), R(xl,x2,. . .x,), one finds the expression for the total differential dR = (aRldx,)dx, + (aRlaz,)dx, + .. .(JRlax")dx.
The True Meanlng of isothermal
~~~~
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For small uncertainties, 6x;, we may approximate 6%; dx;. Thus, 6R
--
k~~lax,)l~x,
(2)
To the Editor.
In a recent paper [1985,62,847] M. F. Granville discusses the conditions needed to make the inequality AG < 0 applicable for a spontaneous process. The author states that "this is true for isothermal, constant pressure changes". In my opinion there exists a quite widespread misconception among chemists about the validity conditions of this inequality. The abovementioned paper does not bring about enough clarification on this particular point and might be misleading. Contrary to what the word "isothermal" seems to imply, the system temperature in these "isothernial, constant pressure changes" is not necessarilv a constant. The system n e e d s o n l G hein rontact with aheat reservoir at aconstant temnerature. At the beginning and at the end of the transformation the system t&nper&ure is that of the heat reservoir. During the transformation the system temperature may vary and may even be inhomogeneous. This can he derived from the second law, which states that the entropy of the system plus its surroundings increases when a spontaneous process occurs:
Consider asystem a t a constant pressure P i n contact with a heat source at temperature T. Suppose this system undergoes an irreversible transformation during which its temperature is temporarily different from T. The work exchanged by the system is
,=1
(The absolute values assure that we are finding the maximum range of uncertainty-that terms of opposite sign do not canrel fortuitousl\..~"One has onlv to evaluate the several partial derivatives "sing the givenbata, multiply each by the appropriate (explicit or implied) uncertainty, and take the sum. This produces the same result as the "General Propagation Procedure," except possibly in nonlinear cases with large relative uncertainties. (In the example of a pH calculation used, one finds (pH) = 0.0063, exactly as determined bv the author's nrocedure.) This uncertaintv determines thk appropriate &nificant figure count in the result; correct . orooaeation of significant figures is subsunied bv .this procedure. Given the calculus content of the propagation-of-errors 8 approach, it may be used in physical chemistry and suhsequent courses, if desired. Certainly, chemistry graduates should be familiar with it. At lower levels, significant figure rules should suffice. If for our own gratification or amusement, we want a "better" result, we may use the calculusbased approach. I see no need for novel approaches to deal with correct propagation of significant figures.
The heat exchanged by the system is
- Wws = AH,,
Q,,=
The heat exchanged by the heat source is
The entropy change of the latter is
Using eq 1we obtain AH,,
- T ASimt < 0
I t should be emphasized that T represents here the temperature of the heat source only. This is also the initial and final system temperature and since we may finally write
'
See, for example, Shoemaker, David P.; Garland, Carl W.: Steinfeld. Jeffrey I.; Nibler. Joseph W. Experiments in Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1981; pp 46-50. To take account of the factthat random errors in several data will naturally tend to offseteach other, one may do, in effect,a root-meansquare calculation, squaring each term on the right of eq 2, summing. and taking the square root. This gives what might be termed a statistically "most probable" uncertainty range (the (pH) value turns out to be 0.0047 in example used) rather than the maximum uncertainty range.
Thus this inequality holds even if the system temperature varies in the course of the transformation. Similar remarks can be made about "constant pressure". Manv exothermic chemical reactions are rapid, and rlearly the system cannot be a t each instant in equilibrium with its surroundings. It would be incorrect to consider that such transformations do not obey that AGSmt < 0 inequality. 0. Fain
Boyd L. Earl
lnstitut Superieur des Sciences et Techniques de I'Univemite de Picardie Saint-OuentinCedex, France
University of Nevada LBS Vegas, NV 89154
Volume 65
Number 2
February 1988
187
