Having followed through the best information hitherto laid before the public, on the heat required to produce steam, our next object must be to convert it into a form more directly useful for our purpose. For the quantity of heat which converts a liquid into vapour, requires the additional facts of the volume of the vapour, and its elastic force to render it valuable.

Of the Elastic Force of Steam.

83. To obtain a rule for determining the force of steam at any temperature, or the temperature corresponding to any given force, we must have recourse to a rule found by trial from the best experiments; it is not a satisfactory method, but we have no other means of arriving at a rule in a case where the real causes of variation are not understood. We still however may gain some assistance, from previous reasoning, in forming our conclusions. In the first place, the index of the power representing the law of variation, must be of such a simple kind as to render it probable that it is the true one. Hence the index 5.13 employed by Mr. Southern,* is not likely to represent the law of nature; Mr. Creighton's index 6;† or Dr. Young's which is 7,‡ are, either of them, more likely to be accurate. The true equation may be very complex, but this is not probable, and while we are ignorant of its nature, and can represent the results sufficiently near for practical use, by one index, it is best to adopt the simplest form, and particularly when it is equally as likely to be the true one as one of a more complex kind. In any attempt to find the index by the usual method of differences, the errors of experiment will have too great an influence.

84. Secondly. It appears probable, that there is a degree of cold at which steam

* Robison's Mechanical Phil. Vol. II. p. 172.

Natural Phil. Vol. II. p. 400.

+ Phil. Mag. Vol. LIII. p. 266.

cannot exist,* and this must be the case when it is condensed by cold, till the cohesive attraction of the particles exceeds the repellant force of the caloric interposed between them, and the change from an elastic fluid to a solid may then take place without the intermediate stage of liquidity. This physical circumstance enables us to fix another element in the calculation; for there must be a temperature when the force is nothing.

85.-Thirdly. The greatest possible force of steam must next be considered, for we are certain that onr formula must be in error if it exceeds that limit. Suppose a given quantity of water, a cubic inch for example, to be confined in a close vessel which it exactly fills; and that in this state it is exposed to a high temperature. Then, as the bulk when expanded is to the quantity the bulk is increased by expansion, without change of state, so is the modulus of elasticity of water of that temperature to the force of steam of the same density as water. If our rule therefore gives steam a greater force than this at the same density and temperature, it must be erroneous. With these limitations we must in a considerable degree be guarded against error, and the method followed is next to be explained.

86.-Let f be the elastic force of steam, in inches of mercury, and t the corresponding temperature, and let a be the temperature below which the elastic force is 0. Consider f the abscissa, and t + a the ordinate of a curve, of which the equation is Aƒ = (t + a)", whence the coefficient

A = ( t + a ) n
f

Let the abscissa increase to f ́, and the ordinate to t+a; then

Now if these points be near one extremity of the range of experiment, and two other points be taken near the other extremity, then

From four results of Mr. Southern's experiments, on steam from water, we find that a =

* An interesting paper on this subject by Mr. Faraday renders it equally so that the limit is different for different vapours; my formula had led me to the same conclusion, hence, it has another property justified by experience. See Phil. Mag. Vol. LXVIII .p. 344.

100 very nearly satisfies the conditions; and this value of a being inserted, we find n = 6 and A = 177, or its logarithm = 2.247968.

87.-If the expansion of confined water, when its temperature is raised to 1150 degrees of heat, be 0.9693 of its bulk, the force necessary to confine it to its bulk at 60°, when exposed to a heat of 1150°, the modulus of water being 22,100 atmospheres at 60°, would be about 6925 atmospheres.* Our rule gives for the force of steam at that temperature and density

The expanding power of heat, and the decrease of the modulus of elasticity, must be in the same ratio; and most probably both vary as the square of the central distances of the atoms, and consequently as the power of the volume. Hence, if e be the expansion, the original bulk being unity, and m the modulus, it must be

the force of compression capable of retaining the fluid in its original state of density.

The expansion varies as the expanding power of heat, and as the temperature, hence, it will be as the power of the temperature; and it must be 0 at 40°; consequently, A ( 1 — 40 )₺ =e, and as from 40° to 212o, it is found to The agreement of this formula with experiment is shewn in

be 04333, we have log. (-40)-5·089091 = e. the following table.

4137 atmospheres; and in the uncertainty both as to what the actual expansion of water would be in such high temperatures, and the decrease of its modulus, it is more prudent to be within that beyond the limit. But at, or near the temperature 1150° the rule will cease to be of any use because then it is simply the expansive power of compressed and it varies as the quantity water expands by a given change of tempera

water; ture.

Having thus far explained the methods by which the rules have been obtained, it only remains to give them the most simple form for use, with illustrative examples.

SS.-RULE 1. To find the force of steam from water in inches of mercury the temperature being given.

Add 100 to the temperature, and divide the sum by 177; the sixth power of the quotient is the force in inches required.

Example. To find the force of steam for the temperature 312.°

Raise this to the sixth power and it gives 159 inches for the force of the steam in inches of mercury.

Or by logarithms. Add 100 to the temperature, and from the logarithm of this sum, subtract 2.247968; and six times the difference is the logarithm of the force in inches of mercury.

Example. To find the force of steam for the temperature 250°.

89.-RULE II. The force of the steam of water being given to determine its tem

perature.

Multiply the sixth root of the force in inches by 177, and subtract 100 from the product, which gives the temperature required.

Example. Let the force of steam be eight atmospheres, equal 240 inches of mercury, to find its temperature.

The sixth root of 240 may be easily found by a table of squares and cubes, by first finding its square root, and then the cube root of the square root. Thus the square root of 240 is 15.492, and the cube root of 15.492 is 2.493; hence, (2·493 × 177) - 100 = 341-20. Mr. Southern's experiment gives 343-6.

Or by logarithms. Add one sixth of the logarithm of the force in inches to 2 247968, the sum is the logarithm of 100 added to the temperature.

Example. Let the force of steam be equal to sixty inches of mercury, which is nearly fifteen pounds on the square inch above the pressure of the atmosphere, to find its temperature.

from which subtract 100, and it gives 250-2° for the temperature. Mr. Southern's experiment gives 250-3°.

90.-When sea water is employed, as it boils at a different temperature, the force of the steam is different. The correction in the rules is easily made by finding the constant number which corresponds to a force of thirty inches of mercury, at the boiling point, with different degrees of saturation with salt. Many of the people employed about boat engines, are not yet aware that there is a difference between the temperature of steam from common water, and that from salt water, when the force is the same. I will shew in another place (Sect. IV.) the effect this has on the power of the steam engine, but at present our object is to determine the force of the steam. Mr. James Watt was the only person who had made experiments on the steam of salt water; they were made in 1774.* He does not give them as being very accurate ones, but they are sufficient to establish the fact that there is a difference; and Mr. Faraday has lately had occasion to satisfy himself, on the same point, by various experiments.†

91.-The following table gives the boiling points of solutions of different salts in

water.

* Robison's Mechanical Phil. Vol. II. p. 34.

+ Quarterly Journal of Science, Vol. XIV. p. 440.