ELEMENTARY THERMODYNAMICS.

CHAPTER I.

THE CONSERVATION OF ENERGY.

1. Units of Measurement.-As it is now becoming universal to express all dynamical quantities in terms of a very convenient set of units based on the Metric, or French, system of weights and measures, we shall give an account of the method, and also make a comparison with the less simple, but more familiar, English system.

In the metric system, which was established in France by law in 1795, and is now widely adopted for commercial purposes, the standards of length and mass are the Metre and the Kilogramme, respectively. The Metre is the distance between the ends of a rod of platinum made by Borda, when the temperature is that of melting ice: the Kilogramme is the mass of a piece of platinum, also made by Borda.

The subsidiary measures of the metric system are formed as follows:

The Litre (used for liquids) is the same as the cubic decimetre.

The standards of the metric system were originally chosen so that the metre should be the ten-millionth part of the distance from the pole to the equator, and the kilogramme the mass of a litre of distilled water, at 4o C., the temperature of maximum density, as nearly as could be then determined.

In the centimetre-gramme-second, or C. G. S., system of absolute units, now so generally used in dynamics, the fundamental units of length, mass, and time, are chosen to be the centimetre, gramme, and mean solar second, respectively. The C.G.S. absolute unit of force, called a Dyne, is then defined to be that force which, acting for one second on the mass of a gramme, will generate a velocity of one centimetre per second. The C.G.S. absolute unit of pressure is a pressure of one dyne per square centimetre.

In conjunction with these absolute units, we frequently employ arbitrary units, as the gramme-weight for force, and the millimetre of mercury and the atmo for pressures.

As the weight of a given mass is not quite the same in all parts of the world, whenever we speak of a grammeweight as a measure of force, we shall mean a force equal to the weight of a gramme at Paris. The acceleration of any body falling freely at Paris under its own weight being 980-868 centimetres per second, we see that the weight of a gramme at Paris is 980-868 dynes. To determine the weight of a gramme at any other place, we may use Clairaut's formula

g = G (1 – '0025659 cos 2x) (1 – 1·323),

where X is the latitude of the place, z its height above the mean level of the sea, r the radius of the earth, g the number of dynes in a gramme-weight at the place, and G the value of g at the mean level of the sea in latitude 45o.

The Atmo is defined to be the pressure produced (at Paris) by a column of mercury 760 millimetres high, when the temperature is 0o C., and the top of the mercury subjected to no force but the pressure of its own vapour. It is found that an atmo is equal to the weight (at Paris) of 1033-279 grammes per square centimetre, or to a pressure of 1013510 dynes per square centimetre.

The Atmo, the name of which is due to Prof. J. Thomson, is about equal to the average pressure of the atmosphere in ordinary places and is chiefly used for measuring high pressures: the gramme-weight is very convenient in estimating small forces.

In the English, or practical, system of dynamical units, the fundamental units of length, mass, and time, are the foot, the pound avoirdupois, and the mean solar second, respectively. The absolute unit of force, which Prof. J. Thomson calls a Poundal, is that force which, if it acted for a second on the mass of a pound, would generate in it a velocity of one foot per second. The accelerating effect of gravity being 32-1889 feet per second at London, it follows that the weight of a pound at London is 32.1889 poundals. Hence, for rough purposes, we may consider a poundal equal to the weight of half an ounce in any part of the world.

We shall always work in the C.G.S. system of units, but it will be easy to express our results in the English practical method by means of the following data:

1 kilogramme=2.2046 lbs. avoir.=15432 grains.

Hence we have, roughly:

Weight of 1 lb. avoir.=444900 dynes.

1 lb. per sq. inch=70·3 grammes per sq. centimetre=69000 dynes per sq. centimetre.

1 gramme per sq. centimetre 0142 lb. per sq. inch.

1 atmo=14·7 lbs. per sq. inch.

2. The Conservation of Energy.-We are now prepared to explain the principle of the Conservation of Energy. This is a deduction, by means of experiment and experience, from a more obvious result known as the principle of Work, which will now be obtained as a direct theoretical consequence of Newton's three Laws of Motion.

In order to apply these laws correctly, it is necessary to conceive the bodies which we are considering to be made up of a large number of very small pieces, or particles, each of which is so small in all its dimensions that for all dynamical purposes it may be treated as a mathematical point; the relative motions of its parts with respect to one another being negligible in comparison with its motion of translation as a whole.

Before we are able to consider real bodies of finite size, we must take the case of a single particle.

3. When a particle, which is moving about in any manner under the action of a force of p dynes, receives a

small displacement of ds centimetres, in a direction which is not at right angles to the force, the force is said to do Work on the particle, or the particle is said to do work against the force, and the amount of work done on the particle is defined to be the product of the displacement into the resolved part of the force along the displacement. If e be the angle between the positive directions of p and ds, the work done by the force will therefore be p cos e. ds, and may obviously be either positive or negative. It is also clear that the work done by the force is equal to the product of the force and the projection of the displacement along the force.

The absolute unit of work in the c.G.S. system of units is called an Erg, and is the work done by a force of one dyne when it moves its point of application one centimetre in its positive direction. The absolute unit of work in the English system is the Foot-poundal, which is the work done by a force of one poundal when its point of application moves one foot in the positive direction of the force.

Work is also reckoned in arbitrary units, as the gramme-centimetre and the foot-pound. These different methods may easily be compared by means of the following relations:

1 gramme-centimetre (at Paris)=980·868 ergs.

1 foot-pound=1356 x 104 ergs, roughly.

If several forces act simultaneously on the particle, the work which they do is defined to be the work done by their resultant. And since the resolved part of the resultant along the displacement is equal to the sum of the resolved parts of the component forces in the same direction, it is evident the work done by the resultant is