Page:EB1911 - Volume 13.djvu/156

CARNOT’S FUNCTION] reversible, that there is not, for instance, any communication between reservoirs of gas or vapour at sensibly different pressures, and that there is no waste of power in friction. If there is equilibrium both mechanical and thermal at every stage of the cycle, the ideal engine will be perfectly reversible. That is to say, all its operations will be exactly reversed as regards transfer of heat and work, when the operations are performed in the reverse order and direction. On this understanding Carnot’s principle may be put in a different way, which is often adopted, but is really only the same thing put in different words: The efficiency of a perfectly reversible engine is the maximum possible, and is a function solely of the limits of temperature between which it works. This result depends essentially on the existence of a state of thermal equilibrium defined by equality of temperature, and independent, in the majority of cases, of the state of a body in other respects. In order to apply the principle to the calculation and prediction of results, it is sufficient to determine the manner in which the efficiency depends on the temperature for one particular case, since the efficiency must be the same for all reversible engines.

16. Experimental Verification of Carnot’s Principle.—Carnot endeavoured to test his result by the following simple calculations. Suppose that we have a cylinder fitted with a frictionless piston, containing 1 gram of water at 100° C., and that the pressure of the steam, namely 760 mm., is in equilibrium with the external pressure on the piston at this temperature. Place the cylinder in connexion with a boiler or hot body at 101° C. The water will then acquire the temperature of 101° C., and will absorb 1 gram-calorie of heat. Some waste of motive power occurs here because heat is allowed to pass from one body to another at a different temperature, but the waste in this case is so small as to be immaterial. Keep the cylinder in contact with the hot body at 101° C. and allow the piston to rise. It may be made to perform useful work as the pressure is now 27.7 mm. (or 37.7 grams per sq. cm.) in excess of the external pressure. Continue the process till all the water is converted into steam. The heat absorbed from the hot body will be nearly 540 gram-calories, the latent heat of steam at this temperature. The increase of volume will be approximately 1620 c.c., the volume of 1 gram of steam at this pressure and temperature. The work done by the excess pressure will be 37.7 × 1620 = 61,000 gram-centimetres or 0.61 of a kilogrammetre. Remove the hot body, and allow the steam to expand further till its pressure is 760 mm. and its temperature has fallen to 100° C. The work which might be done in this expansion is less than th part of a kilogrammetre, and may be neglected for the present purpose. Place the cylinder in contact with the cold body at 100° C., and allow the steam to condense at this temperature. No work is done on the piston, because there is equilibrium of pressure, but a quantity of heat equal to the latent heat of steam at 100° C. is given to the cold body. The water is now in its initial condition, and the result of the process has been to gain 0.61 of a kilogrammetre of work by allowing 540 gram-calories of heat to pass from a body at 101° C. to a body at 100° C. by means of an ideally simple steam-engine. The work obtainable in this way from 1000 gram-calories of heat, or 1 kilo-calorie, would evidently be 1.13 kilogrammetre (= 0.61 × ). Taking the same range of temperature, namely 101° to 100° C., we may perform a similar series of operations with air in the cylinder, instead of water and steam. Suppose the cylinder to contain 1 gramme of air at 100° C. and 760 mm. pressure instead of water. Compress it without loss of heat (adiabatically), so as to raise its temperature to 101° C. Place it in contact with the hot body at 101° C., and allow it to expand at this temperature, absorbing heat from the hot body, until its volume is increased by th part (the expansion per degree at constant pressure). The quantity of heat absorbed in this expansion, as explained in § 14, will be the difference of the specific heats or the latent heat of expansion R′ = .069 calorie. Remove the hot body, and allow the gas to expand further without gain of heat till its temperature falls to 100° C. Compress it at 100° C. to its original volume, abstracting the heat of compression by contact with the cold body at 100° C. The air is now in its original state, and the process has been carried out in strict accordance with Carnot’s rule. The quantity of external work done in the cycle is easily obtained by the aid of the indicator diagram ABCD (fig. 5), which is approximately a parallelogram in this instance. The area of the diagram is equal to that of the rectangle BEHG, being the product of the vertical height BE, namely, the increase of pressure per 1° at constant volume, by the increase of volume BG, which is rd of the volume at 0° C. and 760 mm., or 2.83 c.c. The increase of pressure BE is, or 2.03 mm., which is equivalent to 2.76 gm. per sq. cm. The work done in the cycle is 2.76 × 2.83 = 7.82 gm. cm., or .0782 gram-metre. The heat absorbed at 101° C. was .069 gram-calorie, so that the work obtained is .0782/.069 or 1.13 gram-metre per gram-calorie, or 1.13 kilogrammetre per kilogram-calorie. This result is precisely the same as that obtained by using steam with the same range of temperature, but a very different kind of cycle. Carnot in making the same calculation did not obtain quite so good an agreement, because the experimental data at that time available were not so accurate. He used the value for the coefficient of expansion, and .267 for the specific heat of air. Moreover, he did not feel justified in assuming, as above, that the difference of the specific heats was the same at 100° C. as at the ordinary temperature of 15° to 20° C., at which it had been experimentally determined. He made similar calculations for the vapour of alcohol, which differed slightly from the vapour of water. But the agreement he found was close enough to satisfy him that his theoretical deductions were correct, and that the resulting ratio of work to heat should be the same for all substances at the same temperature.

17. Carnot’s Function. Variation of Efficiency with Temperature.—By means of calculations, similar to those given above, Carnot endeavoured to find the amount of motive power obtainable from one unit of heat per degree fall at various temperatures with various substances. The value found above, namely 1.13 kilogrammetre per kilo-calorie per 1° fall, is the value of the efficiency per 1° fall at 100° C. He was able to show that the efficiency per degree fall probably diminished with rise of temperature, but the experimental data at that time were too inconsistent to suggest the true relation. He took as the analytical expression of his principle that the efficiency W/H of a perfect engine taking in heat H at a temperature t° C., and rejecting heat at the temperature 0° C., must be some function Ft of the temperature t, which would be the same for all substances. The efficiency per degree fall at a temperature t he represented by F′t, the derived function of Ft. The function F′t would be the same for all substances at the same temperature, but would have different values at different temperatures. In terms of this function, which is generally known as Carnot’s function, the results obtained in the previous section might be expressed as follows:—

“The increase of volume of a mixture of liquid and vapour per unit-mass vaporized at any temperature, multiplied by the increase of vapour-pressure per degree, is equal to the product of the function F′t by the latent heat of vaporization.

“The difference of the specific heats, or the latent heat of expansion for any substance multiplied by the function F′t, is equal to the product of the expansion per degree at constant pressure by the increase of pressure per degree at constant volume.”

Since the last two coefficients are the same for all gases if equal volumes are taken, Carnot concluded that: “The difference of the specific heats at constant pressure and volume is the same for equal volumes of all gases at the same temperature and pressure.”

Taking the expression W = RT log er for the whole work done by a gas obeying the gaseous laws pv = RT in expanding at a temperature T from a volume 1 (unity) to a volume r, or for a ratio of expansion r, and putting W′ = R log er for the work done in a cycle of range 1°, Carnot obtained the expression for the heat absorbed by a gas in isothermal expansion

He gives several important deductions which follow from this formula, which is the analytical expression of the experimental result already quoted as having been discovered subsequently by Dulong. Employing the above expression for the latent heat of expansion, Carnot deduced a general expression for the specific heat of a gas at constant volume on the basis of the caloric theory. He showed that if the specific heat was independent of the temperature (the hypothesis already adopted by Laplace and Poisson) the function F′t must be of the form

where C and t0 are unknown constants. A similar result follows from his expression for the difference of the specific heats. If this is assumed to be constant and equal to C, the expression for F′t becomes R/CT, which is the same as the above if t0 = 273. Assuming the specific heat to be also independent of the volume, he shows that the function F′t should be constant. But this assumption is inconsistent with the caloric theory of latent heat of expansion, which requires the specific heat to be a function of the volume. It appears in fact impossible to reconcile Carnot’s principle with the caloric theory on any simple assumptions. As Carnot remarks: “The main principles on which the theory of heat rests require most careful examination. Many experimental facts appear almost inexplicable in the present state of this theory.”

Carnot’s work was subsequently put in a more complete analytical form by B. P. E. Clapeyron (Journ. de l’Éc. polytechn.,