Thermodynamics and Heat Engines

Equation (53), with cf." constant and equal to (1:.' may be re garded as the simplest and most useful form of the adiabatic giv ing the final value H" of the total heat for a wet vapour, when tables of 4 and H are available. It may be expressed if desired in terms of H and T only, but cannot be put in the usual form (9) or ( 1o) (as is often attempted), because the value of the index 7 varies so much with temperature and wetness as to make the equa tion difficult to use and less accurate in practice than measure ments on a diagram.

Types of Characteristic Equation.

The general relation be tween P, V, and T, which exists for any substance in various states is commonly called the characteristic equation, or equation of state, of the substance. It has often been thought possible to in clude both liquid and vapour states in a single equation, such as that of van der Waals, since the properties of the two states ap proximate to each other in the critical region, and show a con tinuous transition above the critical pressure. This will be further discussed in the article VAPORIZATION dealing with the relations between the two states. For the present purpose we may confine our attention to a few simple types of equation for the gas or vapour state, which is of primary importance in practice in rela tion to heat engines. It appears that all simple substances with stable molecules tend to approximate in the vapour state at low pressures to the ideal gas equation PV = RT, in which the value of the constant R varies inversely as the molecular weight (if V is the volume of unit mass) and is equal to the difference of the specific heats in work units. An equation of this type receives a satisfac tory physical explanation on the kinetic theory of gases, but it is the province of thermodynamics to indicate how the simple gas equation must be modified to take account of deviations from the ideal state, and to interpret the results of various experimental methods as applied to the problem.

One of the commonest methods of measuring these deviations is to observe the variation of the volume with pressure at constant temperature. The values of the product PV should then be con stant if Boyle's law is obeyed, and should give a horizontal line on the PV —P diagram. As a rule the isothermal lines thus plotted from observations at various constant temperatures, are nearly straight for a moderate range of pressure at each temperature, but have a downward slope, represented by d(PV)/dP= —c, where c diminishes with rise of temperature as the vapour approximates more closely to the ideal state. Observations of this kind could be represented by an equation of the type (42) by assuming PV to be proportional to T at low pressures, and choosing c, or c—b, to be a suitable function of the temperature. But even if all the iso

thermals were found to be horizontal, this method by itself would not prove that PV was proportional to the absolute temperature, as in PV=RT, since Boyle's law would be perfectly satisfied by an equation of the type, PV =F(T), with F(T) any arbitrary function of the temperature.

Fortunately the Joule-Thomson method, as described on page 98, affords an independent means of verifying the form of the characteristic equation. It has the additional advantages of being easy to apply and of measuring the small deviation itself, without requiring any absolute measurements of volume, which are essen tial to the Boyle's law method, and very exacting. As shown by the thermodynamic expression (4o) for the cooling effect, any substance for which C= o must have a characteristic equation of the general type V /T=F(P), in which F(P) represents any ar bitrary function of the pressure. The ideal gas, PV =RT is a special case for which F(P)=R/P. The condition C=o by itself leaves the form of F(P) indeterminate. But when the same gas also satisfies Boyle's law, which requires a characteristic equa tion of the form PV =F(T), the two conditions can be simul taneously satisfied only by the ideal gas equation PV =RT. Joule and Thomson were therefore justified in their choice of the con stant of integration R/P in equation (41), since the gases they employed also satisfied Boyle's law at low pressures.

Type of Equation Required by Condition (6).—Another case of practical interest is to find the general form of characteristic equa tion compatible with condition (6), and with the simple form of adiabatic equation (9), which follows by the first law of thermo dynamics from the assumption that the change of intrinsic energy is proportional to that of aPV as expressed in (6). Equations (6) and (9) make no mention of temperature, and it is obvious that the deduction of the relation between PV and T must essentially involve an appeal to the second law with its implicit definition of T. The most direct way of doing this is to find the two specific heats, Sp and Sy, from H and E as given by (6), and to equate the difference, Sp—Sy, thus found to the expression (35) for the dif ference of the specific heats as given by the second law. Thus by differentiating (6) at constant volume we obtain for the specific heat Sy, Similarly by adding aPV to (6) to give H—B instead of E—B, and differentiating H at constant pressure, we obtain for the specific heat Sp, Substituting the difference of these two expressions for Sp—Sy in (35), and dividing by a(dP / dT)y(dV / dT) p, we obtain the re quired expression for T in terms of P and V in the form of a dif ferential equation, namely, • • • ..