Thermodynamics and Heat Engines

Carnot Cycle of Finite Range.—If we substitute J/T for F't in Carnot's equation (24) in accordance with the definition of ab solute temperature T, and also write W =JQ as required by the first law, the equation reduces to the form dQ/dT=Q/T, which tacitly involves the law of conservation of energy in addition to the adoption of Carnot's principle for the definition of the scale of temperature. The relation in this form implies that, if the range of the cycle is extended to any lower temperature, the ratio Q/T will remain constant, and the heat converted into work will increase in direct proportion to the range of temperature, since the constant ratio Q/T is equal to dQ/dT representing the heat converted into work per degree fall. Thus the formula for a Carnot cycle of finite range, in which a quantity of heat Q' is sup plied to the working substance at 7', and a quantity Q" is rejected at T", with the conversion of heat Q'–Q" into work W, reduces to the simplest possible form, namely, in which the reduction factor .1 may be omitted if Q is expressed in work units.

Entropy.—The quantity Q/T which remains constant in a Carnot cycle of any range bounded by two adiabatics, is called the entropy of the heat Q supplied at a temperature T. According to (27) the entropy of the heat supplied at T', is equal to the entropy of the heat rejected at T". The entropy difference between any two adiabatics is constant, or the adiabatics are lines of con stant entropy, and are often called isentropics. The thermal equivalent of the work done in any Carrot cycle is the product of the entropy supplied, namely Q'/T', by the range of tempera ture T'–T". This relation is frequently the most convenient for calculating the work obtainable, not only in a Carrot cycle, but also in a cycle of any form.

Clapeyron's Relations.

Clapeyron (1834) observed that Carnot's principle, as represented by equation (24), implied the existence of certain general relations between the quantity of heat absorbed in isothermal expansion and the pressure and expansion coefficients of the working substance. He employed the indicator diagram, as illustrated in fig. 3, for the deduction of these rela tions by a method equivalent to that employed by Maxwell, Theory of Heat (187o). The coordinates of the point B are the pressure p', and the volume v' of unit mass of the substance at a temperature t'. Let the substance expand from B to C along the isothermal at t' absorbing a small quantity of heat dQ, while its volume increases from v' to v", and its pressure diminishes from p' to p". Draw the line of constant volume v' through B inter secting the line of constant pressure p" through C in the point E. Through E draw the isothermal EAD at the temperature t". Com plete the Carnot cycle by drawing the adiabatics BA and CD through B and C. In the limit, when the difference of temperature t'-t" is small, it may be represented by dt as in Carnot's equa tion (24), and the form of the cycle ABCD will become a paral lelogram as indicated in the figure. The area ABCD repre sents the work dW done in the elementary cycle, and is equal to that of the rectangle BEXEC, or to (p'–p") (v"–v') . Making these substitutions in Carnot's equation (24), we obtain the gen eral relation, In order to interpret this relation in terms of the pressure coeffi cient at constant volume, or the expansion coefficient at constant pressure, as may be desired, we may in the first place transfer v"–v' to the right hand side of the equation. We then observe that p'-p" is the increase of pressure BE at constant volume corres ponding to the increase of temperature t'–t", so that the ratio represents the familiar pressure-coefficient at constant volume, de noted by (dp/dt), in the usual notation of partial differential co efficients. Similarly dQ/(v"–v'), being the ratio of the heat ab

sorbed at constant temperature to the corresponding increase of volume, may be denoted by (dQ/dv)t in the same notation. If we also substitute J/T for F't in accordance with the definition of ab solute temperature, we obtain, which is often called the first thermodynamical relation, and was in fact employed by Kelvin as a general expression for Carnot's principle in his first exposition of the equations of thermodynamics (1851), except that he retained the symbol ,u for F't, following Clapeyron, in place of substituting J/T, as in (29), according to modern practice.

Pressure and Expansion Coefficients.—The utility of a relation of this kind lies in the fact that small quantities of heat absorbed in expansion are very difficult to measure, but can usually be calcu lated with the aid of relations like (29) in terms of other coeffi cients, such as the pressure coefficient. Many examples of this will be given later. If the relation (29) itself is not directly applicable, it can usually be transformed by purely mathematical relations be tween the various coefficients into a different shape which is more convenient for the purpose required. For instance the pressure coefficient (dp/dt), is seldom directly measurable in the case of a solid or a liquid, owing to the difficulty of keeping the volume constant while the temperature is being raised. The difficulty of measuring the pressure-coefficient may be avoided by using the familiar relation, giving the required value in terms of the isothermal elasticity and the coefficient of expansion at constant pressure, both of which can be measured. Relation (3o) is one of the commonest examples in practice of the general relation between the partial differential co efficients of any three quantities, such as x, y, z, connected by a single equation, such that any one of the three may be regarded as a function of the other two. It is most easily remembered in the cyclical form, from which any required relation of this type such as (3o) may be written down by replacing p, v, t, by x, y, z, or vice versa. But as relations of this type are usually required in the form (3o), it may be as well to explain how they are deduced from first prin ciples. The volume v of unit mass of any substance depends on the temperature t and the pressure p either of which may vary independently of the other. Any expansion due to change of temperature alone, is found by multiplying the rise of tempera ture dt by the expansion per degree at constant pressure, repre sented by the coefficient (dv/dt)p. Similarly any change of volume due to increase of pressure alone, is given by the product of the increase of pressure dp by the isothermal compressibility (dv/dp) t. In the general case, when both p and t change, the whole expansion dv is the sum of the two independent effects, dv = (dv/dt)pdt-1-(dv/dp)tdp. (32) Similar formulae apply to all other properties of the substance de pending on p and t and are frequently required in thermodynamics. In the case of the volume v, it is immediately obvious from (32) that, if the compression due to increase of pressure dp is equal to the expansion due to increase of temperature dt, the volume will remain constant, or dv = o. The ratio of dp to dt at constant vol ume, or the required coefficient is obtained in terms of the other two by putting dv = o in (32), dividing by dt and trans posing, which gives the required relation in the form (3o) if we observe that the isothermal elasticity (dp/dv)t is the reciprocal of the isothermal compressibility (dv/dP)t.