where p is one atmosphere expressed in dynes per square centimetre. Now (§ 261) Therefore = Now is very small compared to 273, so that we shall not produce any appreciable error in omitting the term in the denominator of the right-hand member. Thus 0.0907 × 273 xp=3.352 × 109t p = 1.35 × 10st. If is one atmosphere, or 1013260 dynes per square centimetre, This number agrees with the results of experiment. 264*. Irreversible Cycles.-The cycles which we have up to now considered have all been reversible, that is, if they are worked backwards, so that all the various operations are performed in the reverse order and sense, the physical and mechanical changes are also reversed. There are, however, many cycles of operations in which, for various reasons, the operations cannot be reversed, or, if they are, the mechanical changes are not reversed. Thus if during any cycle any of the energy is employed in producing motion against friction, such a cycle cannot be reversible, for, as we have seen in § 110, although we reverse the direction of motion, the conversion of mechanical energy into heat due to friction always takes place. Thus, when working direct, the engine working in such a cycle may do a certain quantity of mechanical work owing to the expenditure of a certain quantity of heat-energy; yet if we reverse the engine and do work on it to the same extent as it did before, since some of this energy is employed in doing work against friction, we shall not completely reverse the thermal processes. Again, if during any part of a cycle there is conduction of heat from one part of the engine to any other, since heat only flows by conduction from bodies at higher temperatures to those at lower, on reversing the engine the heat that passed by conduction from a higher to a lower temperature is not made to pass in the reverse direction. It was to avoid the conduction of heat that, in describing Carnot's reversible cycle, we had to suppose that the walls of the cylinder were composed of a perfect non-conductor of heat. Also during a reversible process, when there is passage of heat from one body to another, as, for instance, in the Carnot cycle during the isothermal expansion, when the working substance is taking heat from the Y source, it is necessary to suppose that the transference takes place so slowly that the temperature of the working substance never differs by more than an infinitesimal amount from that of the source. If this were not so, when we reversed the cycle, in order to reverse the conditions exactly, we should still require to have the temperature of the working substance higher than that of the source by the same amount as before, and yet have heat flowing from the source to the working substance, i.e. from a cold to a hot body. 265. Dimensions of Thermal Quantities. We have used two distinct units of quantity of heat. One of these, the calorie, depends on the thermal capacity of water, and on the scale of temperature adopted, as well as on the unit of mass. The other, the erg, simply depends on the fundamental units of mass, length, and time, and has the dimensions [ML2T=2]. If Q represents a certain quantity of heat measured in calories, and H the same quantity measured in ergs, then by the first law of thermo-dynamics we have H=JQ, where is the mechanical equivalent. Therefore [H]=[MLT-2]=[/Q]· Now in § 251 we have seen that the value of depends on the scale of temperature adopted, since the value of Q depends on this scale. Hence the dimensions of depend on the temperature scale. We do not, however, know the dimensions of temperature, as measured on the ordinary gas-thermometer scale, in terms of the fundamental units of length, mass, and time, and so we are reduced to using a symbol [0] for the unknown dimensions of temperature. Since the thermal unit depends on the mass of water taken, as well as on the unit of temperature, we have Hence The symbol here plays the part of a fourth fundamental unit, and Professor Rücker has proposed to call it a secondary fundamental unit. There is no doubt that it is only the limit of our knowledge as to the nature of temperature which prevents our expressing [0] in terms of [Z], [M], and [7]. For instance, we have in § 257 supposed that in the case of a gas the temperature is proportional to the mean kinetic energy of translation of the molecules. Hence we might measure temperatures by the mean kinetic energy of a molecule of a gas when at that temperature, and we should on this scale have [0]=[ML2T-2]. As yet, such a method of measuring temperature is not warranted by our knowledge of the molecular conditions of gases, to say nothing of fiquids and solids. It is, therefore, better to retain, when dealing with dimensional formulæ involving temperature, the symbol [0] for the dimension of the unit of temperature. Since specific heat is the quantity of heat required to raise unit mass through a temperature of one degree, So that specific heat has no dimension, and is therefore a mere number. This is at once evident, if we remember that specific heat may also be defined as the ratio of the heat required to raise a given mass of the substance through a given range of temperature, to the heat required to raise an equal mass of water through the same range. Latent heat being the quantity of heat required to convert unit mass of the substance from one state to the other, [4]=[2]/[M]=[0]. Since the coefficients of expansion are we have Original length (or volume) Increase in Temperature' [a]=[L]/[LO]=[0-1] [y]=[Z3]/[Z30]=[0−1]. The quantity of heat, Q, which passes in a time, t, through a slab of area A, thickness d and conductivity k, when the difference of temperature between the opposite faces is 0, is given by BOOK III WAVE-MOTION AND SOUND 7 8 PART I-WAVE-MOTION CHAPTER I WAVE-MOTION AND WATER WAVES 266. Wave-Motion.—We have in Book I. chap. vii. considered the periodic motion of a single particle or rigid body; we have now to consider in some detail the resultant motion when the various particles of a medium are executing periodic motions, but the phase ($50) of the motions of the various particles is not the same for all, but are related to one another in certain definite ways. Suppose we have a number of particles arranged, when at rest, at equal distances along a line AB (Fig. 214), and that these particles all execute S.H.M.'s (§ 50) of equal amplitude and period along lines at right angles to AB, but in such a way that the phase of each successive particle, counting from A, differs from that of the preceding particle by a constant amount. Thus if the constant difference in phase is 30°, when the particle I is at its median position, the position of the others will be as shown by the dots in the figure. The displacement of particle 2 at any moment equal to the displacement of particle I at 1/12 of the periodic time (7) |