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but it is most frequently in this cafe) very small, and even eyanescent in comparison of the resistance arising from the mutual cohesion of the molecula. The contrary effect has place in the course of the paffage from the liquid to the gaseous or aeriform state;- the cohesion of the fluid moleculæ being extremely small, the elasticity of the caloric has scarcely any thing to furmount to produce volatilisation besides the pressure of the atmosphere, or gas which actually compreffes it.
55. Hence it results that the same liquid under different preffures ought to volatilise at different temperatures. M. Lavoisier proved the truth of this result, by placing ether under the receiver of an air-pump and producing volatilisation solely by taking off a part of the pressure of the atmosphere. See Chymie, tome !. pa. 9. And we know by many experiments of M. Deluc and others, that water boils the more speedily as it is less preffed by the weight of the atmosphere.
Lavoisier notices a curious consequence of what has been here faid ; which is, that if our planet revolved upon its axis with such a velocity as to leffen the pressure of the atmofphere, or if the temperature of the air were raised, then several fluids which we now fee under a liquid state would only exist in the aeriform state ; for example, if under the temperature of summer the pressure of the atmofphere were only equivalent to 20 or 24
inches of the barometrical tube, that pressure would not retain ether in the fluid ftate, it would be changed into gas; and the like would happen, if while the preffure of the air was equivalent to 28 or 30 inches of the mercury the habitual temperature were 105 or 110 degrees on Fahrenheit's scale.
56. The principles which have been here exhibited are sufficient for the understanding of all which relates to the action of water or other fluids reduced to vapour. Now, it has appeared from frequent experiments that water heated in common air volatilises ať 80° of Reaumur's thermometer, or 212° of Fahrenheit's, the height of the barometer being 28 French, or 29.0 English inches : and spirits of wine under a like pressure volatilises at between 639 and 64 of Reaumur, or nearly 175° of Fahrenheit. The expansive force of the vapour must, therefore, in both these cases, according to the principles juft ex plained, be measured by a column of mercury of 28 French, or 29'9 English inches, in like manner as such a column measures the preffure of the atmosphere, or the elasticity of common air. And at any more eleyated temperatures the elastic force of the vapour will surpass the preffure of the atmosphere by a quantity which has a certain relation with the excess of the temperaturę above those just stated.
57. Till lately there was wanting on this important subject a series of exact and direct experiments by means of which, having given the temperature of the heated fuid, the expansive force of the steam rising from it might be known, and vice versa. There was likewise wanting an analytical theorem exprefsing the relation between the temperature of the heated fluid and the pressure with which the force of the steam was in equilibrio. These desiderata have, however, been lately supplied by M. Bettancourt, an ingenious Spanish philosopher, after a method which we shall now concisely explain.
58. M. Bettancourt's apparatus consists of a copper vefsel or boiler, with its cover firmly soldered on: this cover has three orifices which close up with screws: at the first the water or other fluid is put in and out; through the second passes the ftem of a thermometer which has the whole of its scale or graduations above the vessel, and its ball within, where it is immersed either in the fluid or in the steam according to the different circumstances; through the third hole pafles a tube, making a communication between the cavity of the boiler and one branch of an inverted fyphon, which contains mercury, and acts as a barometer for measuring the pressure of the elastic vapour within the boiler. In the side of the vessel there is a fourth hole into which is inserted a tube with a turncock, making a communication with the receiver of an air-pump, in order to extract the air from the boiler and to prevent its return.
The apparatus being prepared in good order, and distilled water introduced into the boiler at the first hole, and then stopped, as well as the end of the inverted fyphon or barometer, M. Bettancourt surrounded the boiler with ice, to lower the temperature of the water to the freezing point, and then, having extracted all the air from the boiler by means of the airpump, the difference between the columns of mercury in the two branches of the barometer shewed the measure of the elastic force of the vapour arising from the water in that temperature. Then lighting the fire below the boiler, he gradually raised the temperature of the water from o to 110° of Reaumur's thermometer, that is, from 32° to 27910 of Fahrenheit's thermometer; and for each degree of elevation in the temperature he observed the height of the mercurial column which measured the elasticity or pressure of the vapour.
These experiments were repeated various times and with different quantities of water in the vessel; their results were arranged in different columns for the sake of comparison, and it appeared that the pressures for different temperatures agreed yery nearly, however much the quantity of fluid in the vefel was varied. It was also seen that the increase in the expansive force of the vapour is at first very low; but increases gradually
tinto the higher temperatures, where the increase becomes very rapid, as will be obvious from an examination of the tables in fome of the following pages.
59. To express the restation between the degrees of temperature of the vapour and its elastic force, this philosopher employs a method suggested by M. Prony, which consists in imagining the heights of the columns of mercury measuring the expansive force to represent the ordinates of a curve, and the degrees of heat the corresponding absciflæ of that curve; making the ordinates equal to the sum of feveral logarithmic ones which contain two indeterminates, and ascertaining these quantities in such manner that the curve may agree with a tolerable number of observations taken throughout the whole extent of the change of temperature, from the lowest to the highest extreme of the experiments. Then a formula or equation to a curve is investigated, and when the curve corresponding to that equation is constructed, if it coincide (with the exception of a few trifling anomalies) with the curve constructed by the refults of the experiments, the formula may be looked upon as correct, and furnishing a true analytical representation of the phenomena. This was done by M. Bettancourt, and the curve constructed from his equation has a point of inflexion at about the 102° of Reaumur, as it ought to have, because the second differences of the barometrical meafures of the elastic force became negative at that temperature.
60. In a similar mannér M. Bettancourt made experiments on the strength of the vapour from alcohol or spirit of wine; constructing the curve and deducing the requisite analytical formula. This curve had likewise a point of inflexion at about 880 of Reaumur, the second differences in the table of barometrical measures becoming then negative. From a comparison of the experiments on the vapour of water with those on the vapour of alcohol, a remarkable conclusion was derived: for it appeared that, after the first 20° of Reaumur, the strength of the vapour of spirit of wine was to that of the vapour of water, nearly in the same constant ratio of 23 to 10, or 7 to 3, for any one and the fame degree of heat. Thus, at the temperature of 40° of Reaumur, the strength of the steam of water is reasured by 2-9711 Paris inches in the barometer, and that of vapour of alcohol by 6.9770, the latter being about 24 times the former.
61. The equations to the curve of temperature and preffure; denoting the relation between the abscissä and ordinates, or be
ween the temperature and the elasticity of the vapour, as given by M. Bettancourt, were of the following form.
fe tax utax ox-g o'x- & d. For water, y
te e tax etax
? oxop 2. -- alcohol, y=e te
A. Where y represents the height of the column of mercury which measures the expansive force, * the corresponding degrees of Reaumur's thermometer, and the other letters certain values which are assigned to them in the investigation.
62. But M. Prony, in the 2d volume of his Architecture Hydraulique, has thrown these equations into a rather more convenient form, though analogous to thofe of Bettancourt. His formula for the vapour of water is this,
y=k, 8.*+*+ Su+ Nov for The method which he followed consisted in satisfying the results between oo and 80°, by means of the two first terms, and to interpolate by means of the other two, the differences between the observed values, and those computed by the two first terms, from 80° up to 110°. In this manner he succeeded to exprefs so exactly the observations in their whole extent, that the curves of the calculus and the experiments were only distinguishable the one from the other by such little anomalies, as were manifeftly the effect of some trifling though inevitable errors in the observations, and in the graduations of the scales in the apparatus. He afterwards employed an equation of three terms, giving to the different coefficients the following values : 8 =1'172805
loge =0'0692259 l=1'047773
log. 8, =0'0202661 = =1'028189
log.ex=O'OT 20736 «. = -0'00000072460407
log. = 7.8601007 W;;= +0.8648188803 ·
log. u=7-9369271 f = -0.8648181057
. log. = 1.9369248 Substituting these several values in the equation
y=M, 8,* tu,,,* + 30," it satisfies not only the numbers employed in its formation, but all the intermediate observations, as may be concluded from the following table, which exhibits to every 10 degrees of Reaumur's thermometer the barometrical results both of observation. and the calculus.
The anomalies are generally much more minute than in the formulæ of four terms: we may therefore regard the equation just preceding the table, which is more fimple than that of Bettancourt, as representing the phenomena and measuring the effects of the expansive force of the steam of water with all deGrable accuracy. M. Prony remarks, that the smallness of the coefficient jy will allow the term 4,8x to be neglected in reckoning between oo and 80°; and thus from the temperature of ice up to that of boiling water, the equation of two terms alone will fuffice, that is to say ..::9=86,* +, B,*.
63. M. Prony's equation for the vapour of alcohol comprises 5 terms originally : but in most cases three of those terms will give results sufficiently accurate. The numeral values of the coefficients are as below: & - = I'11424
log. e = 0.04697771 : SHI = 1'05714
log. , = 0·02413079 So, 0'79943
log. ,,= 19027776 fe, - O'0021293
log. M = 3.3282330 ples, = + Oʻ9116186
log. , = T-9598132 = -+ 0'2097778
log. = 1-3217595 Hiv=- l'1192671
These numbers cause the experiments and calculus to coincide very nearly, when introduced into the equation
y=r, 8% M8, +p, But +Mivo The magnitude of the anomalies will be seen by inspecting the following table.