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In a second experiment the external work done amounted in an hour to 20750 metre-kilograms, while 112.2 grams of oxygen were consumed, and 255.6 units of heat produced. These 112.2 grams of oxygen would, in a state of rest, have afforded 585.7 units of heat; thus 585.7- 255.6=330.1, and, consequently, 330.1 units of heat more than the experiment shows. These 330.1 units of heat have been hence expended in mechanical work; but the quantities of heat consumed in both experiments for mechanical work, 436.68 and 330.1 stand, in fact, in relation to the external work done; for, from the proportion
330.1 : 436.68=20750 : x results x=27450, which coincides quite nearly with the observed number 27448.
A strong horse, when he remains standing at rest in a stable, is amply nourished with 7.5 kilograms of hay, and 2.5 kilograms of oats, which, together, contain 4 kilograms of carbon. But as soon as the horse is put to work this amount of nourishment does not suffice; it is necessary, if he is to be kept in good condition, to add 5.5 kilograms of oats, which contain 2.2 kilograms of carbon. On a day of work, therefore, 6.2 kilograms of carbon are supplied to this animal's body.
As the power of one horse executes, in a second, a work of 75 metre-kilograms, the work done in 8 hours equals 75 · 60 · 60 · 8, or 2,160,000 metre-kilograms, According to the experiments of Favre and Silbermann, by the oxidation of one kilogram of carbon 8080 units of heat are developed, which corresponds to a mechanical work of 8080 · 424, or 3,425,920 metre-kilograms. Hence the day's work performed by a horse, 2,160,000 metro-kilograms, corresponds to a consumption of
=0.63 kilogram of carbon.
3425920 Of the nutriment, therefore, supplied to the horse's body for a day's work, namely, 6.2 kilograms of carbon, only 0.63, being about it, is expended for the performance of mechanical labor, the rest being partly used for the sustentation of animal life, for the production of heat, and in part passing unoxidized through the body.
According to Boussingault only 65 per cent. of the carbon introduced into the body is oxidized, while 35 per cent is given off unconsumed. Of the four pounds of carbon, therefore, which the horse at rest takes for his daily nourishment, only 2.6 pounds, and of the additional 2.2 kilograms of carbon allotted for days of labor only 1.4 kilogram arrive at oxidation in the body of the animal. Hence in a day of rest there are produced in the horse's body 8080 • 2.6=21008 units of heat. Of the 1.4 kilogram of carbon, further oxidized on days of labor, 0.63 are consumed in mechanical work, while the remaining 0.77 kilogram (1.4-0.63) go to supply the increased heat production of 8080.0.77=6221 units. Thus on a day of work the heat developed in the horse's body ascends to 27229 units, while only 8080 · 0.63=5090 units of heat are converted into work.
VI.-ELEMENTS OF THE MECHANICAL THEORY OF HEAT. If we conceive heat to be a molecular movement, the temperature of a body is to be taken as proportional to the vis-viva inherent in the material atoms which move in some way, perhaps vibrate around their position of equilibrium. An increase of temperature consists ,therefore, in an augmentation of this vis-viva, and hence in an enhanced velocity of the molecular movement.
Not all the heat, however, added to a body contributes to the raising of its temperature; and hence not all the heat added to it is employed in the augmentation of the active vis-viva of its molecular vibrations, for a part of the heat may, under conditions, be consumed in order to overcome the molecular forces which exert an action between the several atoms of the body and present an impediment to their free movement. This last heat, which Clausius denotes as that consumed in internal work, is usually called latent heat.
If we designate the free heat of a body by T, the latent heat, which it may contain, by L, the whole quantity of heat in the body will be U=T+L.
W. Thomson has proposed for this quantity, first introduced into the doctrine of heat by Clausius, the name of energy of the body; but Clausius has recently designated the two components of U as store of heat, T, (Wärmeinhalt,) and store of work, L, (Werkinhalt.)
If now so much heat be added to a body of the temperature t and the volume v, in which the collective quantity of heat, U, (energy,) is contained, that its temperature be increased by t'°, its volume by v cubic metres, and the quantity of heat contained within it by U', it will not answer to ascribe to it this quantity of heat U', because through the simultaneous expansion of the body by v an external work has been done which consumes a corresponding quantity of heat.
If p be the pressure under which the body stands, then the external work which corresponds to the enlargement of volume u will be p v', supposing the pressure p to remain unaltered during the whole expansion ; but the quantity of heat corresponding to that work is w=A pv. Thus the quantity of heat which must be supplied to the body in question in order to increase its temperature from t to t+t', its volume from v to v+v', and the heat contained in it from U to U+U' is, q=U'+A pui
I an equation which corresponds to the differential equation
dQ=dU+A p du in which d Q designates the very small quantity of heat which must be supplied to the body, in order for the interior heat of the body to undergo the small augmentation d U and its volume to be increased by the small magnitnde d v.
The equation I or rather a differential equation corresponding to it, is the mathematical expression of the first law of the mechanical theory of heat. By the help of this equation we can calculate the quantity of heat which disappears through a given change of volume of a body submitted to a given pressure, provided we know the whole work done thereby; which is, however, only the case when we have to do exclusively with external work without the accession of any internal work proceeding from molecular forces, and which evades a direct measurement. The first law, therefore, of the mechanical theory of heat suffices only for the solution of correspondent problems when in the magnitude U' of the equation I or in d U of the equation I a an internal work is not comprehended, a condition which is only satisfied by bodies of a completely gaseous form.
The following examples will illustrate the application of the first law of the mechanical theory of heat to gaseous bodies. In a hollow cylinder, (Fig. 9,) havFig. 9.
ing a transverse section of one square metre, is situated at the distance of one metre from the bottom an easily movable, but air-tight, piston K; the space of one cubic metre, shut in by this piston, is filled with atmospheric air at 0°, while the pressure of the atmosphere weighs on the piston; and hence the total pressure which tends to sink the piston is 10333 kilograms.
If this mass of air, the pressure unchanged, be heated to 273° it will be expanded to double its original volume; the piston, during this expansion, will be thrust one metre higher, and thus be brought into the position K'. The work so done is 10333 metre-kilograms, and the quantity of heat, A pv', consumed in doing this work is, in that case,
=24.37 units of heat.
424 The total heat, q; which must be supplied to a cubic metre of air at 0° and under atmospheric pressure
(1.293 kilograms of air) in order to raise it to a temperature of 273°, while, with pressure unchanged, it is expanded to double its volume, is
273 · 1.293 · 0.2377=83 units of heat, since the specific heat of the air under constant pressure is equal to 0.2377; hence we have 83=U'+24.37, or U'=83–24.37=58.63.
Thus, of the 83 units of heat which we supply to the 1.293 kilogram of air, in order to raise its temperature from 0° to 273°, while with unaltered pressure it undergoes expansion to twice its original volume, 58.63 units of heat have gone over into this air, while the remaining 24.37 units of heat were expended in doing the work involved in its expansion. In order, then, to raise the temperature of 1.293 kilograms of air from 0° to 273°, while the volume of the air remains unaltered, and so no external work is done, only 58.63 units of heat are necessary. The specific heat of the air under constant pressure stands, theres fore, to the specific heat of the air under constant volume as 83 : 58.63, or as 1.415 : 1; while this ratio has been found, in another manner, to be as 1.421 to 1.
We have here supposed the mechanical equivalent of heat to be known, and from this derived the ratio of the specific heat of the air under constant pressure and constant volume, while in § 4 the inverse process was followed, inasmuch as we assumed this last ratio to be known, and from thence derived the mechanical equivalent of heat.
The quantity of heat q which must be supplied to a body in order to raise its temperature from t to t+t', to increase the heat contained in it from U to U+U', and to enlarge its volume from v to v+v', is by no means the same under als circumstances; for, with a like condition at the beginning and the ending, the work done during the transition from the first to the last may be very different. The equation I is properly constructed only for a special case; for the case, namely, in which the pressure p remains unaltered while the volume of the body enlarges from v to vtu. When the pressure p is variable the equation I can only so long be recognized as valid, as the augmentation v of volume is small enough to be regarded as the differential of space; as is the case with the differential equation I a corresponding to the equation I.
But when, with a variable value of p, the enlargernent of volume v' is somewhat considerable, the work done during the expansion from v to v+v can no longer be expressed simply by the product pu'. Here the case presents itself when a higher method of calculation must indispensably be put in practice, if the object be an exact expression for the work done.
With elementary expedients we can, in such cases, only attain, by special calculations, to approximative values.
Let us proceed, in order to make this more intelligible, to the consideration of a special case. We have above calculated the quantity of heat q which is requisite to raise a cubic metre of air of 0° and sustaining atmospheric pressure, to 273°, while the air expands under an unvarying pressure to double its original volume. Here is the final condition : two cubic metres of air of 273° temperature and an elasticity of one atmosphere. The same final condition can, however, be also reached, beginning with the same incipient condition in another manner. Let the piston (Fig. 9) be again in its original position K, and, under it in the cylinder, one cubic metre of air at 0° sustaining atmospheric pressure, the burden of the piston being thus 10333 kilograms. If this weight be now slowly and regularly diminished to one-half, the air will gradually expand and push the piston upwards; thus there should, in the first place, be so much heat supplied to the air that, with an unchanged temperature of 0°, the piston is heaved upwards one metre, and the volume of air therefore doubled.
The quantity of heat necessary for this is only to be determined by liigher processes of calculation, but an approximate value may be obtained in an elementary way. Let us conceive the pressure p, which weighs upon the piston
when it occupies the position K, to be not constantly diminished, but to be with
1 drawn at suitable intervals, each time some P, the pressure bearing upon the
20 piston will then conform to the succession of values .exhibited in the first column of the following table under D:
If we denote by 1 the height of the piston above the floor, when it occupies its position at the commencement, it will ascend by the succession of diminished pressurés to the respective heights indicated in the second column.
The height, therefore, through which the piston rises at each succeeding diminution of pressure has the value given in the third column under h.
Without sensible error, we can now assume that the pressure of the enclosed mass of air acting upon the piston from beneath remains unaltered during its ascent through one of the heights indicated in the third column under h. For this pressure we may assign, as a first approximation, the value of D, standing in the first column in the same horizontal row, which we must multiply into the corresponding value of h, in order to obtain the value of the work which is done in the ascent of the piston through one of the divisions in question. The products thus obtained are grouped together in the last column under h D.
The total work which is done while the piston rises, under the indicated circumstances, from K to K', is therefore the sum of the values exhibited in the last vertical row of the above table, namely,
1 1 1 1 1 1 1 1 1 L'= + + + + trt +
14 15 16 17 18 19 20 The sum of the fractions standing between parentheses, which is most readily obtained if they be changed into decimal fractions and then added, is 0.668, and since p=10333, while l is 1 metre, there results for the total work L'=0.668. 10333=6902 metre kilograms.
This value of the total work is, howeyer, manifestly too small; for we have multiplied each of the heights consigned to the third column into the pressure which acts against the under surface of the piston when it stands at the upper end of the corresponding division. If we multiply each of the values of h into the pressure which acts against the piston when it is at the lower end of the division, the result will be 1 1 1 1 1 1
hence I'=0.718pl, or putting for p and I their numerical value, L'=0.718. 10333=7419 metre-kilograms. But this value is evidently too great ; the true value of the total work L is, at any rate, very nearly equal to the mean between L/ and I'; hence
1+1' 6902+7419 L=
or L=7160 metre-kilograms; 2
424 424 The exact value of w' is found by equation (1) in § 3, if we take y=10333, and a'=2; the result is then L=2.3026 · 10333 • log 2=7153, a value from which that obtained above in an approximative way differs but inconsiderably. When, now,
the piston has become fixed, so that no further expansion of the air is possible, 58.63 units of heat are necessary to raise the temperature of the included air from 0° to 273°, whereby its elasticity also is enhanced from onehalf to one atmosphere. Thus the final condition of the air is exactly the same as in the case above considered, in which the air expanded under a constant pressure. The quantity of heat, however, requisite for the attainment of the final condition in question is, in the last case, only 58.63+16.88=75.51, while in the first case it was equal to 83.
Thus the quantity of heat which must be supplied to a body, in order that, starting from a given condition, it shall pass over into a determinate final condition, is by no means an invariable magnitude, but is dependent on the magnitude of the mechanical work which is done during that transition.
VII.-APPLICATION OF THE MECHANICAL THEORY OF HEAT TO AQUEOUS
Suppose that at the bottom of a hollow cylinder, of which the transverse section is one square metre, there is a litre of water at 0°, and
Fig. 10. that directly upon this is placed a piston on which a pressure p is exerted, (Fig, 10.) This pressure p is that which is equal to the elasticity of the saturated vapor of t°C. The table on a following page, contains, according to Regnault's experiments, the values of p for the temperatures given in the first column, p being the pressure which the saturated vapor of the corresponding temperature exerts on one square metre.
Let the water under the piston be now heated from 0° to t°; it will thus expand to a magnitude which, for our present purpose, may remain unknown. There needs for this elevation of tenperature a quantity of heat expressed by W=t+0.000027 +0.0000003t", if we take
K into consideration the variableness of the specific heat of water; while W=t would be the expression, were the specific heat of water taken as constant and equal to one. Hence the quantity of heat to be taken as a unit is that which is required to raise the temperature of one kilogram (one litre) of water from 0° to 1° C. During this exaltation of temperature from 0° to to no steam can be formed.
But if we continue the supply of heat, the formation of steam commences, and the steam has forthwith the elasticity of p; it pushes back the piston, and the space made free is continually filled with fresh vapor, until finally all the water is converted into vapor. At this moment the end is attained ; the heat which must be supplied to the water of t° during the formation of steam with