EQUIVALENCE OF HEAT AND WORK. 23 the equivalence of Vis Viva and Work takes the following form:The decrease in the Vis Viva is equal to the increase in the Work done,or The sum of the Vis Viva'and Work done is constant.This latter form will be found very convenient in what follows. In the case of such forces as have an Ergal,the meaning of that quantity was defined(in s 6 of the Introduction)in such a manner that we must say,The Work done is equal to the decrease in the Ergal.'If we use the method of denot- ing work just described,we must say on the contrary,'Tne work done is equal to the increase in the Ergal;'and if the constant occurring as one term of the Ergal be determined in a particular way,we may then regard the Ergal as simply an expression for the work done. 3.Expression for the first Fundamental Principle. Having fixed as above what is to be the positive sign for work done,we may now state as follows the first main Principle of the Mechanical Theory of Heat. In all cases where work is produced by heat,a quantity of heat is consumed proportional to the work done;and inversely, by the expenditure of the same amount of work the same quantity of heat may be produced. This follows,on the mechanical conception of heat,from the equivalence of Vis Viva and Work,and is named The Principle of the Equivalence of Heat and Work. If heat is consumed,and work thereby produced,we may say that heat has transformed itself into work;and con- versely,if work is expended and heat thereby produced,we may say that work has transformed itself into heat.Using this mode of expression,the foregoing principle takes the following form:Work may transform itself into heat,and heat conversely into work,the quantity of the one bearing always a fired proportion to that of the other. This principle is established by means of many pheno- mena which have been long recognized,and of late years has been confirmed by so many experiments of different kinds,that we may accept it,apart from the circumstance of its forming a special case of the general mechanical principle of the Conservation of Energy,as being a principle directly derived from experience and observation. Google
EQUIVAL'E!IlCE OF HEAT A!IlD WORK. 23' the equivalence of Vis Viva and Work takes the following form: The decrease t'n the Vis Viva is equal to the increase in the Work done, or The sum of the Vis Viva'arrd Work done is constant. This latter form will be found very convenient in what follows. In the case of such forces as have an Ergal, the meaning' of that quantity was defined (in § 6 of the Introduction) ill such a manner that we must say, 'The Work done is equal to the decrease in the Ergal.' If we use the method of denoting work just described, we must say on the contrary, 'The work done is equal to the increase in the Ergal;' and if the constant occurring as one term of the Ergal be determined in a particular way, we may then regard the Ergal as simply an expression for the work done. - § 3. Expression for the first Fundamental Principle. Having fixed as above what is to be the positive sign for work done, we may now I!tate as follows the first main Principle of the Mechanical Theory of Heat. In all cases where work is produced by heat, a quantity of heat is consumed proportional to the work done; and inversel:/, by the e.xpenditure of the same amOltnt of work the sume quantity of heat may be produced. . This follows, on the mechanical conception of heat, from the equivalence of Vis Viva and Work, and is named The Principle of the Equivalence of Heat and Work. . If heat is consumed, aud work thereby produced, we may say that heat has transformed itself into work; and conversely, if work is expended and heat thereby produced, we may say that work has transformed itself into heat. Using this mode of expression, the foregoing principle takes the following form : Work may transform itself into heat, and heat conversely into work, the quantity of the one bearing alway.s a fixed proportion to that of the other. . This principle is established by means of many pheno-' \ ;mena which have been long recognized, and of late years f, ' has been confirmed by so many experiments of different kinds, that we may accept it, apart from the circumstance of its forming a special case of the general mechanical principle of the Conservation of Energy, as being a principle directly derived from experience and observation. Digitized by Coogle
24 ON THE MECHANICAL THEORY OF HEAT. S4.Numerical Relation between Heat and Work. While the mechanical principle asserts that the changes in the Vis Viva and in the corresponding Work done are actually egual to each other,the principle which expresses the relation between Heat and Work is one of Proportion only.The reason is that heat and work are not measured on the same scale.Work is measured by the mechanical unit of the kilogrammetre,whilst the unit of heat,chosen for convenience of measurement,is That amount of heat which is required to raise one kilogram of water from 0to 1(Centigrade).Hence the relation existing between heat and work can be one of proportion only,and the numerical value must be specially determined. If this numerical value is so chosen as to give the work corresponding to an unit of heat,it is called the Mechanical Equivalent of Heat;if on the contrary it gives the heat corresponding to an unit of work,it is called the Thermal Equivalent of Work.We shall denote the former by E,and 1 the latter by The determination of this numerical value is effected in different ways.It has sometimes been deduced from already existing data,as was first done on correct principles by Mayer (whose method will be further explained hereafter), although,from the imperfection of the then existing data,his result must be admitted not to have been very exact.At other times it has been sought to determine the number by experiments specially made with that view.To the dis- tinguished English physicist Joule must be assigned the credit of having established this value with the greatest cir- cumspection and care.Some of his experiments,as well as determinations carried out at a later date by others,will more properly find their place after the development of the theory;and we will here confine ourselves to stating those of Joule's experiments which are the most readily understood, and at the same time the most certain as to their results. Joule measured,under various circumstances,the heat generated by friction,and compared it with the work con- sumed in producing the friction,for which purpose he employed descending weights.As accounts of these experi- Google
24 ON THE HECHANICAL THEORY OF HEAT. § 4. Numerical ReZatioo between Heat and Work. While the mechanical principle asserts that the changes in the Vis Viva and in the corresponding Work done are actually equal to each other, the principle which expresses the relation between Heat and Work is one of Proportioo only. The reason is that heat and work are not measured on the same scale. Work is measured by the mechanical unit of the kilogrammetre, whilst the unit of heat, chosen for convenience of measurement, is That amount of heat which is required to raise ooe kilogram of water from 00 to 10 ( Centigrade). Hence the relation existing between heat and work can be one of proportion only, and the numerical· value must be specially determined. If this numerical value is so chosen as to give the work corresponding to an unit of heat, it is called the Mechanical Equivalent of Heat; if on the contrary it gives the heat corresponding to an unit of work, it is called the Thermal Equivalent of Work. We shall denote the former by E, and 1 the latter by E' The determination of this numerical value is effected in different ways. It has sometimes been deduced from already existing data, as was first done on correct principles by Mayer (whose method will be further explained hereafter), although, from the imperfection of the then existing data, his result must be admitted not to have been very exact. At other times it has been sought to determine the number by experiments specially made with that view. To the distinguished English physicist Joule must be assigned the credit of having established this value with the greatest circumspection and care. Some of his experiments, as well as determinations carried out at a later date by others, will more properly find their place after the development of the theory; and we will here confine ourselves to stating those of Joule's experiments which are the most readily understood. and at the same time the most certain as to their results. Joule measured, under various circumstances, the heat generated by friction, and compared it with the work consumed in producing the friction, for which purpose he employed descending weights. As accounts of these experiDigitized by Coogle
EQUIVALENCE OF HEAT AND WORK. 25 ments are given in many books,they need not here be described;and it will suffice to state the results as given in his paper,published in the Phil.Trans.for 1850. In the first series of experiments,a very extensive one, water was agitated in a vessel by means of a revolving paddle wheel,which was so arranged that the whole quantity of water could not be brought into an equal state of rotation throughout,but the water,after being set in motion,was continually checked by striking against fixed blades,which' occasioned numerous eddies,and so produced a large amount of friction.The result,expressed in English measures,is that in order to produce an amount of heat which will raise 1 pound of water through 1 degree Fahrenheit,an amount of work equal to 772695 foot-pounds must be consumed. In two other series of experiments quicksilver was agitated in the same way,and gave a result of 774083 foot-pounds. Lastly,in two series of experiments pieces of cast iron were rubbed against each other under quicksilver,by which the heat given out was absorbed.The result was 774987 foot- pounds. Of all his results Joule considered those given by water as the most accurate;and as he thought that even this figure should be slightly reduced,to allow for the sound pro- duced by the motion,he finally gave 772 foot-pounds as the most probable value for the number sought. Transforming this to French measures we obtain the result that,To produce the quantity of heat required to raise 1 kilogramme of water through I degree Centigrade,work must be consumed to the amount of 42355 kilogrammetres.This appears to be the most trustworthy value among those hitherto determined,and accordingly we shall henceforward use it as the mechanical equivalent of heat,and write E=42355..(1) In most of our calculations it will be sufficiently accurate to use the even number 424. 5.The Mechanical Unit of Heat. Having established the principle of the equivalence of Heat and Work,in consequence of which these two may be Google
EQUIVALENCE OF BEAT AND WORK. 25 ments are given in many books, they need not here he described; and it will suffice to state the results as given in his paper, published in the Phil. Trans. for 1850. In the first series of exyeriments, a very extensive one, water was agitated in a vesse by means of a revolving paddle wheel, which was so arranged that the whole quantity of water could not be brought into an equal state of rotation throughout, but the water, after being set in motion, was continuaUy checked by striking against fixed blades, which' occasioned numerous eddies, and so produced a large amount of friction. The result, expressed in English measures, is that in order to prod~ce an amount of heat which will raise 1 pound of water through 1 degree Fahrenheit, an amount of work equal to 772'695 foot-pounds must be consumed. In two other series of experiments quicksilver was agitated in the same way, and gave a result of 774'083 foot-pounds. Lastly, in two series of experiments pieces of cast iron were rubbed against each other under quicksilver, by which tbe heat given out was absorbed. The result was 774'987 footpounds. Of all his results Joule considered those given by water as the most accurate j and as he thought that even this figure should be slightly reduced, to allow for the sound produced by the motion, he finally gave 772 foot-pounds as the most probable value for the number sought. Transforming this to French measures we obtain the result that, To produce the quantity of heat required to raise 1 kilogramme of water through 1 de,qree Oentigrade, work must be consumed to the amount of 423'55 kz·logrammetres. This appears to be the most trustworthy value among those hitherto determined, and accordingly we shall henceforward use it as the mecbanical equivalent of heat, and write E = 423·55 .......................... (1). In most of our calculations it will be sufficiently accurate to use the even number 424. § 5. The Mechanical Unit of Heat. Having established the principle of the equivalence of Heat and Work, in consequence of which these two may be Digitized by Coogle
26 ON THE MECHANICAL THEORY OF HEAT. opposed to each other in the same expression,we are often in the position of having to sum up quantities,in which heat and work enter as terms to be added together.As, however,heat and work are measured in different ways,we cannot in such a case say simply that the quantity is the sum of the work and the heat,but either that it is the sum of the heat and of the heat-equivalent of the work,or the sum of the work and of the work-eguivalent of the heat.On account of this inconvenience Rankine proposed to employ a different unit for heat,viz.that amount of heat which is equivalent to an unit of work.This unit may be called simply the Mechanical Unit of Heat.There is an obstacle to its general introduction in the circumstance that the unit of heat hitherto used is a quantity which is closely connected with the ordinary calorimetric methods (which mainly depend on the heating of water),so that the reductions required are slight,and rest on measurements of the most reliable character;while the mechanical unit,besides need- ing the same reductions,also requires the mechanical equivalent of heat to be known,a requirement as yet only approximately fulfilled.At the same time,in the theoretical development of the Mechanical Theory of Heat,in which the relation between heat and work often occurs,the method of expressing heat in mechanical units effects such important simplifications,that the author has felt himself bound to drop his former objections to this method,on the occasion of the present more connected exposition of that theory.Thus in what follows,unless the contrary is expressly stated,it will be always understood that heat is expressed in mechanical units. On this system of measurement the above mentioned first main Principle of the Mechanical Theory of Heat takes a yet more precise form,since we may say that heat and its corresponding work are not merely proportional,but equal to each other. If later on it is desired to convert a quantity of heat expressed in mechanical units back again to ordinary heat units,all that will be necessary is to divide the number given in mechanical units by E,the mechanical equivalent of heat. 5 t Google
26 ON THE MECHANICAL THEORY OF HEAT. opposed to each other in the same expression, we are often in the position of having to flum up quantities, in which heat and work euter as terms to be added together. As, however, heat and work are measured in different ways, we cannot in such a case say simply that the quantity is the sum of the work and the heat, but either that it is the sum of the heat and of the heat-equivalent of the work, or the sum of the work and of the work-equivalent of the heat. On account of this inconvenience Rankine proposed to employ a different unit for heat, viz. that amount of heat which is equivalent to an unit of work. This unit may be called simply the Mechanical Unit of Heat. There is an obstacle to its general introduction in the circumstance that the unit of heat hitherto used is a quantity which is closely connected with the ordinary calorimetric methods (which mainly depend on the heating of water), so that the reductions l'equired are slight, and rest on measurements of the most reliable character; while the mechanical unit, besides needing the same reductions, also requires the mechanical equivalent of heat to be known, a requirement as yet only approximately fulfilled. At the same time, in the theoretical development of the Mechanical Theory of Heat, in which the relation between heat and work often occurs, the method of expressing heat in mechanical units effects such impOrtant simplifications, that the author has felt himself bound to drop his former objections to this method, on the occasion of the present more connected exposition of that theory. Thus in what follows, unless the contrary is expressly stated, it will be always understood that heat is expressed in mechanical units. On this system of measurement the above mentioned first main Principle of the Mechanical Theory of Heat takes a yet more precise form, since we may say that heat and its corresponding work are not merely pl'oportional, but equal to each other. If later on it is desired to convert a quantity of heat expressed in mechanical units back again to ordinary heat units, all that will be necessary is to divide the number given in mechanical units by E, the mechanical equivalent of heat. Digitized by Coogle
EQUIVALENCE OF HEAT AND WORK. 27 S6.Development of the first main Principle. Let any body whatever be given,and let its condition as to temperature,volume,&c.be assumed to be known.If an indefinitely small quantity of heat d is imparted to this body, the question arises what becomes of it,and what effect it produces.It may in part serve to increase the amount of heat actually existing in the body;in part also,if in conse- quence of the imparting of this heat the body changes its condition,and that change includes the overcoming of some force,it may be absorbed in the work done thereby.If we denote the total heat existing in the body,or more briefly the Quantity of Heat of the body,by H,and the indefinitely small increment of this quantity by dH,and if we put dl for the indefinitely small quantity of work done,then we can write do=dd+al....). The forces against which the work is done may be divided into two classes:(1)those which the molecules of the body exert among themselves,and which are therefore dependent on the nature of the body itself,and (2)those which arise from external influences,to which the body is subjected.According to these two classes of forces,which have to be overcome,the work done is divided into internal and external work.If we denote these two quantities by dJ and dw,we may put aL=a+aw..................(2), and then the foregoing equation becomes do=dH+a+aw...............(II. 7.Different conditions of the Quantities J,W,and H. The internal and external work obey widely different laws.As regards the internal work it is easy to see that if a body,starting from any initial condition whatever,goes through a cycle of changes,and finally returns to its original condition again,then the internal work done in the whole process must cancel itself exactly.For if any definite amount,positive or negative,of internal work remained over at the end,there must have been produced thereby either an Google
A"' ....... '''''' OF HEAT 27 § level:op11~ent of the first Let any body whatever be given, and let its condition as to temperature, volume, &c. be assumed to be known. If an indefinitely small quantity of heat dQ is imparted to this body, the question arises what becomes of it, and what effect it produces. It may in part serve to increase the amount of heat actually existing in the body; in part also, if in consequence the imparting of this heat the changes its condition, that change includes of some force, it absorbed in the work If we denote heat existing in more briefly the Quantity of the body, indefinitely small this quantity by we put dL for the indefinitely small quantity of done, then we can write: dQ= dH + dL ........................ (1). The forces against which the work is done may be divided into two classes: (1) those which the molecules of the body exert among-' themselves, and which are therefore dependent nature of the body and (2) those which external to the body is subjected. to these two forces, which have to work done into internal and external If we denote quantities by dJ and may put dL=dJ +dW ..................... (2), and then the foregoing equation becomes dQ = dH + dJ + dW ............... (II). • § 7. W, andH. The and external work different laws. internal work see that if a body, from any initial whatever, goes through changes, .and its original condition again, then the internal in the whole process must cancel itself exactly. For if any definite amount, positive or negatiye, of internal work remained over at the end, there must have been produced thereby either an