Variation of H with P.—The cooling-effect C at constant H as thus defined, is the ration of the two partial differential coefficients of H with respect to P and T, thus, C= (dT / dP) H = — (dH / (dH / dT) (38) This does not involve thermodynamics, but is merely a special case of the formal relation (3o), as is easily verified by writing H in place of V in (3o). Putting S for (dH/dT) p the specific heat at constant pressure, we have the useful relation (dH/dP), = —SC, giving the variation of H with P at constant T in terms of the specific heat S and the cooling effect C, which are the most easily measured of all coefficients. Another expression for the same coefficient may be obtained by applying the two laws of thermo dynamics. From the first law in the form dH=dQ-1--aVdP, as given in (5), dividing by dP at constant T, we find, (dH/dP) =(dQ/dP)T-1-aV = —aT (dV / dT) p (39) in which the second law is involved in the substitution for (dQ/dP) from Clapeyron's second thermodynamical relation (33)• Joule-Thomson Equation.—The utility of relation (39) lies in the fact that it supplies the necessary and sufficient condition which must be satisfied by any expressions selected to represent H and V (the two most important thermodynamical properties of any substance) in order to render such expressions consistent with the laws of thermodynamics. Joule and Thomson made use of (39) in the form in order to deduce an expression for V consistent with their ob servations on the cooling-effect. Their experiments showed that C was approximately independent of the pressure, but varied as with temperature for air and CO2 over the range o° to joo° C. Integrating (4o) at constant P, with S constant, and they obtained the .olution: in which the constant of integration R/P was determined by the condition that the equation must approximate to the form aPV = RT when T is very large. The small term which may be written SC/3T, represents the deviations of the actual gas, air or from the ideal state, in terms of the observed values of S and C for the gas employed. Thus equation (4i) makes it possi ble to deduce the absolute temperature T from the observed tem perature by air thermometer as defined in terms of P and V, but the process of reduction (for which see the article THERMOM ETRY), is far from being as simple as it might appear to be at first sight. Moreover the original Joule-Thomson equation as given in (41), though it showed that the deviations of air from the ideal state must be very small, was somewhat unsatisfactory in other respects. It failed to explain the heating-effect observed in the case of hydrogen, and made no allowance for the known varia tion of specific heat in the case of CO2. These difficulties may be avoided in practice by reversing the procedure. Assume a con venient type of equation for V, differentiate to find the expres sion for SC as in (39), and determine the constants by comparison with experimental results for S and C.
Modified Equation.—A suitable equation of a type similar to (41), is the following, where c= The small constant b, called the'covolume, may be regarded as representing the limiting volume of the molecules at high tem peratures and pressures. The small correction term c, called the coaggregation volume, represents the defect from the ideal volume caused by coaggregation or pairing of molecules. This is assumed to vary inversely as the nth power of T, and is the value of c at any convenient temperature T such as o. C. On this assump tion c is a function of the temperature only, and dc/dt= —nc/T. Differentiating equation (42) in order to find SC as given by (39) or (4o), we obtain, SC =aT(dV /dT) p — aV = a(n+ i)c — ab (43) from which the values of the constants c, n, and b, may be de duced, when the values of S and C are known by observations taken over a sufficient range. It is easy in this way to take account of any variations of S with temperature or pressure. The slope of the isothermals on the Amagat diagram, in which PV is plotted against P, is represented by d(PV)/dP= and is positive in the case of hydrogen (for which PV increases with P) because b is so much larger than c at ordinary temperatures. For the same
reason the cooling effect C is negative, as found by Joule and Thomson, who observed a rise of temperature with drop of pres sure in throttling. This result is explained by (42), if b exceeds (n+ )c, as is actually the case with hydrogen.
in which the constant of integration B is determined by reference to any known value of H at some definite point, such as ioo° C and atmospheric pressure. In the case of steam, S. may be taken as constant and equal to S..
Variation of the Specific Heat S with Pressure.—According to (45) the specific heat S at constant pressure will be a function of both temperature and pressure. The required expression for S is easily obtained by differentiating (45) at constant pressure, thus, This shows that, with an equation of the type (41) or (42), S cannot be constant, as assumed by Joule and Thomson in the in tegration of (4o), and invalidates their method of deducing (4i), but the same objection does not apply to the reverse procedure em ployed in deducing (42). The variation of S with pressure at constant temperature for any equation of this type can also be obtained from the consideration that H is a definite function of P and T depending only on the state, so that if we differentiate S as found in (46) with regard to P at constant T, we must obtain identically the same result, namely as by differentiat ing (or —SC) with regard to T at constant P. Using the general expression for SC given in (39), we obtain, - (dSC/dT) /dr)p (47) which shows that S cannot be independent of the pressure, if SC is a function of the temperature. From another point of view (47) represents the condition that dH as given in (44) should be the exact differential of a definite function of P and T, as must be the case if H is a property of the substance depending only on the state as defined by P and T. On the other hand if we apply the same condition to the general expression for dQ, namely, dQ= SdT —aT(dV /dT)pdP we observe that (47) cannot be satisfied in the case of dQ con sistently with the second law of thermodynamics, because the heat added in any transformation is not simply a property of the substance depending on the initial and final states, but depends essentially on the process by which the transformation is effected. In other words, dQ is not the exact differential of any function of P and T, and cannot be integrated without knowing the relation between P and T defining the process. But if the path is given, Q can always be found from (48). Thus if the path on the indicator diagram is a straight line defined by dP=kdT, dQ/dT=S—akT (dV and the required value of Q can be found for any substance for which the specific heat and the coefficient of ex pansion are known. The heat required for any transformation is often required in practice, and may always be obtained in this way, if the path is given.