and 
Level: introductory (ex. 1-2), introductory+ (ex. 3)
Reference: Joseph-Louis Gay-Lussac, "The Expansion of Gases by Heat," Annales de Chimie 43, 137 (1802)
Notes: Joseph-Louis Gay-Lussac (1778-1850) carried out important research on the physical and chemical properties of gases. He is probably best known for his work on the combining volumes of gases, showing that the volumes in which gases react have small integer ratios. He helped establish that the volume of a gas at constant pressure varies simply with its temperature, the result on which these exercises are based. His study of gases was not confined to the laboratory: he ascended several thousand meters in a baloon to make meteorological and magnetic measurements, and to collect samples of gas for chemical analysis. His work on the acids hydrogen fluoride, hydrogen chloride, hydrogen iodide, and hydrogen cyanide led to the realization that acids need not contain oxygen. He was also first to isolate the element boron.
The variation of gas volume with temperature is usually called Charles' law after Jacques Charles (1746-1823), a French natural philosopher and balloonist. Charles did not publish his work, which was done some 15 years before his Gay-Lussac's, but Gay-Lussac learned of them and acknowledged them. Gay-Lussac also noted that his own work was more extensive than Charles', and that they disagreed in some respects.
So much for the name of the law; now for the content. As currently taught, Charles' law expresses the direct proportionality between the volume of a gas and its absolute temperature. Students are asked in exercise 1 to use Charles' law in the form familiar to them to assess (with hindsight) Gay-Lussac's experiments. At the start of the 19th century, however, there were no absolute temperature scales. In fact, the temperature dependence of gases' volume provides a way of defining the zero of an absolute temperature scale: extrapolate the volume/temperature relationship to find the temperature at which the gas volume would vanish. (This is explored in exercise 2b.) So naturally Gay-Lussac did not write his results in the now-familiar form of Charles' law. In fact, he did not even write the law as a linear variation of volume with temperature, as students are asked to do in exercise 2a, or establish a linear relationship for that matter. What Gay-Lussac did establish was the factor by which gases expanded when heated from 0° to 100°, and he computed the coefficient of expansion that would hold if the expansion is linear (namely DV/DT. He also established that all the gases he tested have expand to the same extent.
Exercise 3 asks students to quantify the precision of Gay-Lussac's measurements and to make inferences from those measures. Formal education in statistics is not usually required of chemistry majors, although it is more frequently required of the biology majors that populate many introductory chemistry courses. At any rate, no formal statistical tests are needed to solve the problem--just the observation that there is no discernible difference between the expansion of air and oxygen. The solution goes into formal statistics only so far as to mention the significance of the standard deviation with respect to the normal distrubtion. (For the interested instructor, a formal statistical test for exercise 3b is given below in these notes.) Exercise 3 is not too difficult for an introductory course. It may be better suited to the analytical chemistry course, however, because of its emphasis on data analysis.
Formal statistical test for exercise 3b:[1] We have two sample means (Vavg1 and Vavg2) with sample sizes n1 and n2 and (sample) standard deviations s1 and s2. We are trying to judge whether the population means (that is the true means of the population of measurements that we have sampled) are in fact different. The appropriate test is the t-test for two population means. The statistic to be computed is called t, defined as
t = (Vavg1 - Vavg2)/(s12/n1 + s22/n2)1/2 .We must also calculate the "degrees of freedom" n, defined as
n = (s12/n1 + s22/n2)2 / [s14/n12(n1+1) + s24/n22(n2+1)] - 2[HTML is pretty poor for mathematical typesetting, so here are graphics files of these equations.]
Plugging in the numbers yields t = 0.5 and n = 5. We must refer to a table of critical values for the t distribution. If t is greater than the critical value, then the conclusion is that the population means are unequal. One can also assign a significance level to the conclusion. The critical value at the 0.3 significance level for five degrees of freedom (n rounded to the nearest integer) is 0.559, meaning that if t were greater than or equal to the critical value, we would conclude that the population means are different (that oxygen and atmospheric air really have different expansions), and we would expect such a conclusion to be correct 70% of the time. Since t is less than this critical value, we cannot be even that confident of the conclusion. (Typically, significance at 0.05 or less, corresponding to probabilities of 95% or greater, are needed before a difference is said to be statistically significant.) The conclusion stands that there is no basis to conclude that there is any real difference.
Historical note: The French Revolution was, of course, a watershed event in the history of Europe and beyond. A chemistry instructor interested in making connections to his students' other courses can mention some points of contact between chemical history and the world-historical French Revolution. That Antoine Lavoisier was executed in the revolutionary reign of terror is undoubtedly the best known example. A literal translation of the whole title of Gay-Lussac's paper is itself a reminder that this work was done during the first French republic: "On the expansion of gases and vapors, read at the National Institute on 11 pluviôse, year 10, by citizen Gay-Lussac" Pluviôse was one of the nature-descriptive names of months used in the republican calendar. It is the "rainy" month corresponding to late January and early February. The Revolutionary government instituted a new calendar, counting years from the establishment of the republic. And the title Citizen was used to level distinctions of rank.
Solutions: To download solutions, go to:
http://web.lemoyne.edu/giunta/classicalcs/gaygas.doc
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