Second Relation.—From the first thermodynamical relation, as given in equation (29), Clapeyron deduced his second relation, for the heat evolved in isothermal compression, by substituting for the expression given in (3o), which leads immediately to the corresponding expression for (dQ/dP)t, T(dv/d0p= (dQ/dP) t (33) in terms of the coefficient of expansion at constant pressure. The negative sign in this relation indicates that heat is evolved by an increase of pressure, whereas heat is absorbed when the volume increases. There are several other thermodynamical relations, which are in effect equivalent to Carrot's equation (24), but the two originally given by Clapeyron are the most useful, and suffice for the majority of practical requirements. It will be observed that they are not independent relations, since one can be deduced from the other by a purely mathematical transformation, but either may be employed as required with the certainty of obtaining re sults consistent with the laws of thermodynamics.
Specific Heats.—The specific heats are among the most familiar and useful coefficients in practice since they are commonly required for calculating quantities of heat. They are usually measured by observing the quantity of heat given up by a body in cooling through a large range of temperature, and are often treated as constants. But in addition to the variation with temper ature, as illustrated in the articles HEAT and CALORIMETRY, the results found may depend to a great extent on other conditions, such as pressure, in a manner subject to the laws of thermo dynamics and amenable to calculation. In dealing with variable specific heats, the general expression for the specific heat of any substance at a temperature T is the ratio dQ/dT of the small quantity of heat dQ supplied per unit mass to the corresponding rise of temperature dT produced; but it is also necessary to specify the conditions under which the measurement is made, as these may affect the result. The two simplest cases are those in which the specific heat is measured, (a) at constant pressure, (b) at constant volume, though the latter condition can seldom be realised satisfactorily in the case of a solid or liquid. For this reason it is useful to have a relation giving the specific heat at constant volume in terms of that at constant pressure. The re quired relation may be obtained most directly by the aid of Clapeyron's relation (33) in the following manner, which also gives incidentally the expression for any other variety of specific heat.
Difference of Specific Heats.—To find a general expression for the specific heat dQ/dT under any condition, write down the gen eral expression for dQ in terms of dt and dp, exactly as for v in (32), by simply replacing the letter v in (32) by the letter Q, and divide each term in the expression by dt, thus we obtain, dQ/dt=(dQ/dt)P+(dQ/dP)tdp/dt. (34)
The first term on the right, namely (dQ/dt)p, representing the heat absorbed per unit mass per degree rise of temperature at constant pressure, is obviously the specific heat as ordinarily meas ured at constant pressure, which may be denoted by Sp. The next coefficient, (dQ/dp)t, represents the heat evolved per unit mass per unit increase of p in isothermal compression. The value of this is given by Clapeyron's second relation (33) in terms of the coefficient (dv/dt)p. Since the specific heat is usually required in thermal units, it is best to retain the reduction factor 1/J, ex plicitly in equations expressed in thermal units, in which it is usually represented by the single letter a for convenience in writ ing or printing. The last factor dp/dt must be taken to correspond with the condition under which the specific heat dQ/dt is desired. Thus if we require the specific heat at constant volume (dQ/c/t)v, or we must also append the suffix v to dp/dt implying that it must be taken at constant volume. Making these substitutions in (34) we obtain the required expression for Sv, = (dQ/d0v= Sp— aT(dv/d0p(dP/dt)v (35) Saturation Specific Heat.—As a further illustration of the same method, if we require the specific heat of a wet vapour main tained in the state of saturation, implied by the suffix s, we have merely to replace the suffix v in (35) by the suffix s, thus, 8 (36) in which (dp/dt) 8 represents the rate of increase of saturation pressure p with temperature, which is usually much larger than (dp/dt), for the dry vapour. For this reason is often negative, whereas Sv though smaller than Sp, is always positive. Thus, in the case of wet steam at ioo° C, —1.04.
By means of these and similar relations the variation of any specific heat can be calculated if one specific heat, such as Sp or is known by experiment at the required temperature. In some cases these variations may be considerable, especially near the critical point of any substance. In general, if the path on the indicator diagram along which the specific heat is to be measured approaches the isothermal, the specific heat will become very large. If the path coincides with the isothermal, the specific heat becomes infinite, because there is finite absorption of heat while dt=o. When the path on the diagram is a little steeper than the iso thermal the specific heat becomes negative, changing from posi tive to negative infinity in crossing the line. For any path between the isothermal and the adiabatic the specific heat is negative, fall ing to zero when the path coincides with the adiabatic (dQ=o), and changing to positive again on the other side. Such changes would be very troublesome to deal with in experimental work, if the thermodynamic relations did not afford a complete method of taking them into account.