THERMODYNAMICS 287 But the last term on the right is, by definition, cdt ; so that dt f# + |^0. dt dv with the condition Thus K-C= -i- dv dt I dv which is a perfectly general expression. As the most important case, let /represent the pressure, then we see, by 10, that d<f> dp dv^ dt and the formula becomes 13. Properties of an Ideal Substance which follows the Laws of Boyle and Charles. Closely approximate ideas of the thermal behaviour of a gas such as air, at ordinary temperatures and pres sures, may be obtained by assuming the relation which expresses the laws of Boyle and Charles. formula of last section, we have at once Thus, by the a relation given originally by Carnot. Hence, in such a substance, dt , T d(f> = c + (K- t dv In terms of volume and pressure, this is tf> - = c log^/R + k log v , the equation of the adiabatics on Watt s diagram. This is (for $ constant) the relation between p and v in the pro pagation of sound. It follows from the theory of wave-motion (HYDROMECHANICS) that the speed of sound is &~T7 where t is the temperature of the undisturbed air. This expres sion gives, by comparison with the observed speed of sound, a very accurate determination of the ratio kjc in terms of R. The value of R is easily obtained by experiment, and we have just seen that it is equal to k - c ; so that k and c can be found for air with great accuracy by this process, a most remarkable instance of the indirect measurement of a quantity (c) whose direct determination presents very formidable difficulties. 14. Effect of Pressure on the Melting or Boiling Point of a Sub stance. By the second of the thermodynamic relations in 10, above, we have so that dt dv But, if the fraction e of the working substance be in one molecular state (say liquid) in which V is the volume of unit mass, while the remainder 1 - e is in a state (solid) where Yj is the volume of unit mass, we have obviously p-eVo+a-dVj. Let L be the latent heat of the liquid, then td<j> L /<ty V dv ) ~ t Also, as in a mixture of the same substance in two different states, the pressure remains the same while the volume changes at con stant temperature, we have dp/dv=0, so that finally which shows how the temperature is altered by a small change of pressure. In the case of ice and water, V x is greater than V , so the temperature of the freezing-point is lowered by increase of pressure. When the proper numerical values of V , V a , and L are introduced, it is found that the freezing point is lowered by about 0074 C. for each additional atmosphere. When water and steam are in equilibrium, we have V much greater than V 1} so that the boiling-point (as is well known) is raised by pressure. The same happens, and for the same reason, with the melting point, in the case of bodies which expand in the act of melting, such as beeswax, paraffin, cast-iron, and lava. Such bodies may therefore be kept solid by sufficient pressure, even at temperatures far above their ordinary melting points. This is, in a slightly altered form, the reasoning of James Thomson, alluded to above as one of the first striking applications of Carnot s methods made after his work was recalled to notice. 15. Effect of Pressure on Maximum Density Point of Water. One of the most singular properties of water at atmospheric pres sure is that it has its maximum density at 4 C. Another, first pointed out by Canton in 1764, is that its compressibility (per atmosphere) is greater at low than at ordinary temperatures, being, according to his measurements, 000,049 at 34 F., and only 000, 044 at 64 F. It is easy to see (though it appears to have been first pointed out by Puschl in 1875) that the second jof these properties involves the lowering of the maximum density point by increase of pressure. To calculate the numerical amount of this effect, note that the expansibility, like all other thermal properties, may be expressed as a function of any two of the quantities p, v, t, < ; say in the present case p and t. Then we have for the expan sibility dv Also the compressibility may be expressed as 1 (dv f d . t= ( -=- = - (-,- }logv . vdpj dpj The relation between small simultaneous increments of pressure and temperature, which are such as to leave the expansibility unchanged, is thus Now the expansibility is zero at the maximum density point, for which therefore this equation holds. But the equations above so that The volume of water at low temperatures under atmospheric pres sure varies approximately as 144,000 ~rff) = 79 nno near ly> and from Canton s experi mental result above stated we gather that (roughly at least) Thus we have ( 1 = _ 0-000,005%^= -0-000,000,3 ; dt J oO from which the formula gives - 02 C. nearly for the change of the maximum density point due to one additional atmosphere. Recent investigations, carried out by direct as well as by indirect methods, seem to agree in showing that the true value is somewhat less than this, viz., about -0 018 C. ; so that water has its maximum density at C. when subjected to about 223 atmo spheres. Thus, taking account of the result of 14 above, we find that the maximum density point coincides with the freezing point at - 2" 8 C. under an additional pressure of about 377 atmospheres, or (say) 2 5 tons weight per square inch. 16. Motivity and Entropy, Dissipation of Energy. The motivity of the quantity H of heat, in a body at temperature t, is where t is the lowest available temperature. The entropy is expressed simply as TT // Xi/w being independent of any limit of temperature. If the heat pass, by conduction, to a body of temperature tf (less than t, but greater than t ), the change of motivity (i.e., the dis sipation of energy) is which is, of course loss; while the corresponding change of entropy is the gain Hf^-i The numerical values of these quantities differ by the factor t , so that, if we could have a condenser at absolute zero, there could be no dissipation of energy. But we see that Clausius s statement that the entropy of the universe tends to a maximum is practically merely another way of expressing Thomson s earlier theory of the dissipation of energy.
When heat is exchanged among a number of bodies, part of it
