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
~~~~
-
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
