Considering this thermometer as a vessel consisting of a gla.s.s bulb and a long gla.s.s stem of fine and uniform bore, hermetically sealed and containing only mercury and mercury vapour, he explained the numerical relation between the temperature as shown by the instrument and the volumes of the mercury and vessel. The scale is really defined by the method of graduation adopted. Two points of reference are marked on the stem at which the top of the mercury stands when the vessel is immersed (1) in melting ice, (2) in saturated steam under standard atmospheric pressure. The stem is divided into parts of equal volume of bore between these two points and beyond each of them. For a centigrade thermometer the bore-s.p.a.ce between the two points is divided into 100 equal parts, and the lower point of reference is marked 0 and the upper 100, and the other dividing marks are numbered in accordance with this along the stem. Each of these parts of the bore may be called a degree-s.p.a.ce.

Now let the instrument contain in its bulb and stem, up to the mark 0, N degree-s.p.a.ces, and let v be the volume of a degree-s.p.a.ce at that temperature. The volume up to the mark 0 will be Nv, at that temperature; and if the substance of the vessel be quite uniform in quality and free from stress, N will be the same for all temperatures.

If v0 be the volume of a degree-s.p.a.ce at the temperature of melting ice the volume of the mercury at that temperature will be Nv0. If G be the expansion of the gla.s.s when the volume of a degree-s.p.a.ce is increased from v0 to v by the rise of temperature, then v = v0(1 + G). The volume of the mercury has been increased therefore to (N + n)v0(1 + G) by the same rise of temperature, if the top of the column is thereby made to rise from the mark 0 so as to occupy n degree-s.p.a.ces more than before. But if E be the expansion of the mercury between the temperature of melting ice and that which has now been attained, the volume of the mercury is also Nv0(1 + E). Hence N(1 + E) = (N + n)(1 + G). This gives n = N(E - G)?(1 + G).

If we take, as is usual, n as measuring the temperature, and subst.i.tute for it the symbol t, we have, since N = 100(1 + G100)?(E100 - G100),

t = 100 {(1 + G100)?(1 + G)} {(E - G)?(E100 - G100)} ... (D)

In this reckoning the definition of any temperature, let us say 37 C., is the temperature of the vessel and its contents when the top of the mercury column stands at the mark 37 above 0, on the scale defined by the graduation of the instrument; but the numerical signification with relation to the volumes is given by equation (D). This shows that the numerical measure of any temperature involves both the expansion of the vessel and that of the gla.s.s vessel between the temperature of melting ice and the temperature in question. This result may be contrasted with the erroneous statement frequently made that equal increments of temperature correspond to equal increments of the volume of the thermometric substance. It also shows that different mercury-in-gla.s.s thermometers, however accurately made and graduated, need not agree when placed in a bath at any other temperature than 0 C. or 100 C. This fact, and the results of the comparison of thermometers made with different kinds of gla.s.s with the normal air thermometer, which was carried out by Regnault, were always insisted on by Thomson in his teaching when he dealt with the subject of heat. The scale of a mercury-in-gla.s.s thermometer is too often in text-books, and even in Acts of Parliament regarded as a perfectly definite thing, and the expansion of a gas is not infrequently defined by this indefinite scale, instead of being used as it ought to be, as the basis of definition of the scale of the gas thermometer. The whole treatment of the so-called gaseous laws is too often, from a logical point of view, a ma.s.s of confusion.

In his article on Heat Thomson gave two definitions of the scale of absolute temperature. One is that stated on p. 126 above, namely, that the temperature of the source and refrigerator are in the ratio of the heat taken in from the source to the heat given to the refrigerator, when the engine describes a Carnot cycle consisting of two isothermal and two adiabatic changes.

The other definition is better adapted for general use, as it applies to any cycle whatever which is reversible. Let the working substance expand under constant pressure by an amount dv (AB' in Fig. 12), and let heat H be given to the substance at the same time. The external work done is pdv. Thomson called pdv?H the work ratio. Now let the temperature be raised by dT without giving heat to the substance or taking heat from it, and let the corresponding pressure rise be dp; and call dp?p the pressure ratio. The temperature ratio dT?T is equal to the product of the work ratio and the pressure ratio, that is,

dT?T = dvdp?H

This is clearly true; for dvdp is the area of a cycle like AB'C'D, represented in Fig. 12, for which an amount of heat H is taken in, though not in this case strictly at one temperature. And clearly, since in Fig. 12 the change from B' to B is adiabatic, H is the heat which would have to be taken in for the isothermal change AB in the Carnot cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of the cycle is dvdp?H, and this by the former definition is dT?T.

Or we may regard the matter thus:--The amount of heat H which corresponds to an infinitesimal expansion dv may be used in equation (A) whether the expansion is isothermal or not, if we take T as the average temperature of the expansion. Hence we have dp?dT = H?(dv.T), that is, dT?T = dpdv?H. The theorem on p. 128 is obtained by what is virtually this process.

COMPARISON OF ABSOLUTE SCALE WITH SCALE OF AIR THERMOMETER

The comparison which Joule and Thomson carried out of the absolute thermodynamic scale with the scale of the constant pressure gas thermometer has already been referred to, and it has been shown that the two scales would exactly agree, that is, absolute temperature would be simply proportional to the volume of the gas in a gas thermometer kept at the temperature to be measured, if the internal energy of the gas were not altered by an alteration of volume without alteration of temperature, that is, if the de - ?e of p. 107 above were zero. Joule tested whether this was the case by immersing two vessels, connected by a tube which could be opened or closed by a stopc.o.c.k, in the water of a calorimeter, ascertaining the temperature with a very sensitive thermometer, and then allowing air which had already been compressed into one of the vessels to flow into the other, which was initially empty. It was found that no alteration of temperature of the water of the calorimeter that could be observed was produced. But the volume of the air had been doubled by the process, and if any sensible alteration of internal energy had taken place it would have shown itself by an elevation or a lowering of the temperature of the water, according as the energy had been diminished or increased.

Thomson suggested that the gas to be examined should be forced through a pipe ending in a fine nozzle, or, preferably, through a plug of porous material placed in a pipe along which the gas was forced by a pump, and observations made of the temperature in the steady stream on both sides of the plug. The experiments were carried out with a plug of compressed cotton-wool held between two metal disks pierced with holes, in a tube of boxwood surrounded also by cotton-wool, and placed in a bath of water closely surrounding the supply pipe. This was of metal, and formed the end of a long spiral all immersed in the bath. Thus the temperature of the gas approaching the plug was kept at a uniform temperature determined by a delicate thermometer; another thermometer gave the temperature in the steady stream beyond the plug.

In the case of hydrogen the experiments showed a slight heating effect of pa.s.sage through the plug; air, oxygen, nitrogen and carbonic acid were cooled by the pa.s.sage.

The theory of the matter is set forth in the original papers, and in a very elegant manner in the article on Heat. The result of the a.n.a.lysis shows that if ?w be the positive or negative work-value of the heat which will convert one gramme of the gas after pa.s.sage to its original temperature; and T be absolute temperature, and v volume of a gramme of the gas at pressure p, and the difference of pressure on the two sides of the plug be dp, the equation which holds is

(1?T) (?T??v) = 1?{v + (?w?dp)} ... (E)[18]

It was found by Joule and Thomson that ?w was proportional to dp for values of dp up to five or six atmospheres. At different temperatures, however, in the case of hydrogen the heating effect was found to diminish with rise of temperature, being .100 of a degree centigrade at 4 or 5 centigrade, and .155 at temperatures of from 89 to 93 centigrade for a difference of pressure due to 100 inches of mercury.

If there is neither heating nor cooling ?w = 0, and we obtain by integration T = Cv, where C is a constant.

Elaborate discussions of the theory of this experiment will be found in modern treatises on thermodynamics, and in various recent memoirs, and the differential equation has been modified in various ways, and integrated on various suppositions, which it would be out of place to discuss here.

The cooling effect of pa.s.sing a gas such as air or oxygen through a narrow orifice has been used to liquefy the gas. The stream of gas is pumped along a pipe towards the opening, and that which has pa.s.sed the orifice and been slightly cooled is led on its way back to the pump along the outside of the pipe by which more gas is approaching the orifice, and so cools slightly the advancing current. The gas which emerges later is thus cooler than that which emerged before, and the process goes on until the issuing gas is liquefied and falls down into the lower part of the pipe surrounding the orifice, whence it can be drawn off into vessels constructed to receive and preserve it.

It is possible thus to liquefy hydrogen, which shows that at the low temperature at which the process is usually started (an initial cooling is applied) the pa.s.sage through the orifice has a cooling effect as in the other cases.

Another idea, that of _thermodynamic motivity_, on which Thomson suggested might be founded a fruitful presentation of the subject of thermodynamics, may be mentioned here. It was set forth in a letter written to Professor Tait in May 1879. If a system of bodies be given, all at different temperatures, it is possible to reduce them to a common temperature, and by doing so to extract a certain amount of mechanical energy from them. The temperatures must for this purpose be equalised by perfect thermodynamic engines working between the final temperature T0, say, and the temperatures of the different parts of the system.

This process is one of the levelling up and the levelling down of temperature; and the temperature T0 is such that exactly the heat given out at T0 by certain engines, receiving heat from bodies of higher temperature than T0, is supplied to the engines which work between T0 and bodies at lower temperatures. The whole useful work obtained in this way was called by Thomson the motivity of the system. Of course equalisation of temperature may be obtained by conduction, and in this case the energy which might be utilised is lost. With two equal and similar bodies at absolute temperatures T, T' the temperature to which they are reduced when their motivity is extracted is v(TT'). If the temperatures are equalised by conduction the resulting temperature is higher, being (T + T'). Thus, if only the two bodies are available for engines to work between, the motivity is the measure of the energy lost when conduction brings about equalisation of temperature.

A very suggestive paper on the subject was published by Lord Kelvin in the _Trans. R.S.E._, vol. 28, 1877-8.

DISSIPATION OF ENERGY

In connection with the theory of heat must be mentioned Thomson's great generalisation, the theory of the dissipation of energy.[19] Most people have some notion of the meaning of the physical doctrine of conservation of energy, though in popular discourses it is usually misstated. What is meant is that in a finite material system, which is isolated in the sense that it is not acted on by force from without, the total amount of energy--that is, energy of motion and energy of relative position (including energy of chemical affinity) of the parts--remains constant. The usual misstatement is that the energy of the universe is constant. This may be true if the universe is finite; if the universe is infinite in extent the statement has no meaning. In any case, we know nothing about the universe as a whole, and therefore make no statements regarding it.

But while there is thus conservation or constancy of amount of energy in an isolated and finite material system, this energy may to residents on the system become unavailable. For useful work within such a system is done by conversion of energy from one form to another and the total amount remains unchanged. But if this conversion is prevented all processes which involve such conversion must cease, and among these are vital processes.

The unavailable form which the energy of the system with which we are directly and at present concerned, whatever may become of us ultimately, is taking, according to Thomson's theory, is universally diffused heat.

How this comes about may be seen as follows. Even a perfect engine, if the refrigerator be at the lowest available temperature, rejects a quant.i.ty of heat which cannot be utilised for the performance of the work. This heat is diffused by conduction and radiation to surrounding bodies, and so to bodies more remote, and the general temperature of the system is raised. Moreover, as heat engines are imperfect there is heat rejected to the surroundings by conduction, and produced by work done against friction, so that the heat thrown on the unavailable or waste heap is still further increased.

Conduction of heat is the great agency by which energy is more and more dispersed in this unavailable form throughout the totality of material bodies. As has been seen, available motivity is continually wasted through its agency; and in the flow of heat in the earth and in the sun and other unequally heated bodies of our system the waste of energy is prodigious. Aided by convection currents in the air and in the ocean it continually equalises temperatures, but does so at an immense cost of useful energy.

Then in our insanely wasteful methods of heating our houses by open fires, of half burning the coal used in boiler furnaces, and allowing unconsumed carbon to escape into the atmosphere in enormous quant.i.ties, while a very large portion of the heat actually generated is allowed to escape up chimneys with heated gases, the store of unavailable heat is being added to at a rate which will entail great distress, if not ruin, on humanity at no indefinitely distant future. It will be the height of imprudence to trust to the prospect, not infrequently referred to at the present time, of drawing on the energy locked up in the atomic structure of matter. He would be a foolish man who would wastefully squander the wealth he possesses, in the belief that he can recoup himself from mines which all experience so far shows require an expenditure to work them far beyond any return that has as yet been obtained.

It is not apart from our present theme to urge that it is high time the question of the national economy of fuel, and the desirability of utilising by afforestation the solar energy continually going to waste on the surface of the earth, were dealt with by statesmen. If statesmen would but make themselves acquainted with the results of physical science in this magnificent region of cosmic economics there would be some hope, but, alas! as a rule their education is one which inevitably leads to neglect, if not to disdain of physical teaching.

From the causes which have been referred to, energy is continually being dissipated, not destroyed, but locked up in greater and greater quant.i.ty in the general heat of bodies. There is always friction, always heat conduction and convection, so that as our stores of motional or positional energy, whether of chemical substances uncombined, the earth's motion, or what not, are drawn upon, the inevitable fraction, too often a large proportion, is shed off and the general temperature raised. After a large part of the whole existent energy has gone thus to raise the dead level of things, no difference of temperature adequate for heat engines to work between will be possible, and the inevitable death of all things will approach with headlong rapidity.

THERMOELASTICITY AND THERMOELECTRICITY

In the second definition of the scale of absolute temperature just discussed, stress of any type may be subst.i.tuted for pressure, and the corresponding displacement s for the change of volume. Thus for a piece of elastic material put through a cycle of changes we may subst.i.tute dS for dp and Ads for dv; where A is such a factor that AdSds is the work done in the displacement ds by the stress dS. As an example consider a wire subjected to simple longitudinal stress S. Longitudinal extension is produced, but this is not the only change; there is at the same time lateral contraction. However, s within certain limits is proportional to S.

Let heat dH in dynamical measure be given to the wire while the stress S is maintained constant, and let the extension increase from s to s + ds.

The stress S will do work ASds _on the wire_, and the work ratio will be -ASds?dH. Now let the stress be increased to S + dS while the extension is kept constant, and the absolute temperature raised from T to T + dT.

The stress ratio (as we may call it) is dS?S and the temperature ratio dT?T. Thus we obtain (p. 134 above)

-(dS?dT) = (1?TA) (dH?ds)

In his Heat article Thomson used the alteration e of strain under constant stress (that is ds?l, where l is the length of the wire) corresponding to an amount of heat sufficient to raise the temperature under constant stress by 1. Hence if K be the specific heat under constant stress, and le be put for ds in the sense just stated, we have

dT = -(TedS?K?) ... (F)

where ? is the density, since dH = K?lA.

The ratio of dH to the increase ds of the extension is positive or negative, that is, the substance absorbs or evolves heat, when strained under the condition of constant stress, according as dS?dT is negative or positive. Or we may put the same thing in another way which is frequently useful. If a wire subjected to constant stress has heat given to it, ds is negative or positive, in other words the wire shortens or lengthens, according as dS?dT is positive or negative, that is, according as the stress for a given strain is increased or diminished by increase of temperature.

It is known from experiment that a metal wire expands under constant stress when heat is given to it, and thus we learn from the equation (F) that the stress required for a given strain is diminished when the temperature of the wire is raised. Again, a strip of india-rubber stretched by a weight is shortened if its temperature is raised, consequently the stress required for a given strain is increased by rise of temperature.

These results, from a qualitative point of view, are self-evident. But from what has been set forth it will be obvious that an equation exactly similar to (F) holds whether the change ds of s is taken as before under constant stress, or at uniform temperature, or whether the change dS of S is effected adiabatically or at constant strain.

In all these cases the same equation

dT = -T (edS?K?) ... (G)

applies, with the change of meaning of dT involved.

This equation differs from that of Thomson as given in various places (_e.g._ in the _Encyclopaedia Britannica_ article on Elasticity which he also wrote) in the negative sign on the right-hand side, but the difference is only apparent. According to his specification a pressure would be a positive stress, and an expansion a positive displacement, and in applying the equation to numerical examples this must be borne in mind so that the proper signs may be given to each numerical magnitude.

As an example of adiabatic change, a sudden extension of the wire already referred to by an increase of stress dS may be considered. If there is not time for the pa.s.sage of heat from or to the surroundings of the wire, the change of temperature will be given by equation (G).