CHAPTER XX.

FORMULE RELATING TO EXPANSION.

194. Measure of Expansion, Factor of Expansion.-When a substance expands, so that its volume changes from V to V'=V+v, the ratio is the numerical measure of the expansion per unit volume, and is usually called simply the expansion of the substance.

Another ratio which it is frequently necessary to consider, is

v

that is to say, the ratio of the final to the initial volume. This may conveniently be called the factor of expansion, or the expansion factor. If m denote the expansion, 1+m will be the factor of expansion, and we have

195. Coefficient of Expansion.-It often happens that when the temperature of a body is increased by successive equal amounts, the successive increments of volume are also equal. In this case the body is said to expand uniformly between the extreme temperatures, and if, V denote the volume of the body at 0° Cent., and V' its volume at to, then we have

where K is a constant number, called the coefficient of expansion per degree Centigrade.

The coefficient of expansion per degree Fahrenheit will be of this value of K, and the coefficient of expansion for the interval from 0° to 100° C. (if the expansion be uniform through this range) will be 100 K.

196. It is usual to employ the name coefficient of expansion to denote the coefficient of expansion per degree, the thermometric scale referred to being stated in the context.

CUBIC, LINEAR, AND SUPERFICIAL EXPANSION.

?

265

Example. The volume of a glass vessel is 450 cubic inches at 0° C.; find its volume at 80° C., the coefficient of expansion for glass being '00002. By (1) we have

V'=450 (1+80 × ⋅00002) = 450·72 cubic inches.

197. Cubic, Linear, and Superficial Expansion. Thus far we have considered only expansion of volume. When a solid body expands, we may consider separately the increase of one of the linear dimensions of the body; this is called the linear expansion. Or we may consider the increase in area of any portion of its surface, which is called the superficial expansion; or finally, the increase of volume, which is called expansion of volume, or cubical expansion. By substituting the words "length" and "surface" for "volume" in § 194 we obtain definitions of linear and superficial expansion as numerically expressed, and we can demonstrate the two following propositions:(1.) The cubical expansion is three times the linear expansion. (2.) The superficial expansion is twice the

linear expansion.

Suppose Fig. 202 to represent a cube formed of any substance, and let the length of each edge of the cube be unity at zero; the volume is consequently equal to 1, and the area of any one of the faces is also represented by 1. If the body be heated to any temperature t, each of the edges will increase by a certain quantity l, and the area of each face will become (1+7)2=1+27+72, while the volume of the cube will become

(1 + 7)3 = 1 + 37 +312+13.

Fig. 202.

But as the quantity l, which represents the linear expansion, is always very small, we may, without sensible error, neglect its second and third powers in comparison with its first power. We thus see that the increase in area of one of the faces of the cube is sensibly equal to 21, and that the increase in volume is sensibly equal to 31, which proves the propositions. These propositions evidently hold for the coefficients of expansion, so that we may say that the coefficient of linear expansion is equal to one-third of the coefficient of cubical expansion, and to one-half of the coefficient of superficial expansion.

The above demonstration supposes the body to remain similar to itself during expansion, which is not the case with bodies of a fibrous

or laminated structure, nor with crystals, except those belonging to the cubic system.

198. Various Formulæ. From equations (1) and (2) we may find the value of V in terms of V', thus,

that is to say, given the volume of a body at a certain temperature, the volume at zero is found by dividing the given volume by the factor of expansion.

Formulæ (1), (2), (3), and (4) are particular cases of a more general formula. Let V and V' be the volumes of the same body at temperatures t and ť respectively, U the volume at zero, K the coefficient of expansion, and m and m' the respective expansions between 0 and t, and 0 and t'. We then have, by formulæ (1) and (2),

whence by division

V=U(1+m)=U(1+Kt)
V=U(1+m)=U(1+Kt);

1+Kt

V 1+m
V=1+m2-1+Kť'

(5)

or the volumes of the same body at different temperatures are proportional to the factors of expansion.

The above formulæ are evidently applicable also to linear and superficial expansions.

199. Influence of Temperature upon Density.-As the density of a substance is inversely as the volume occupied by unit mass, it follows from last section that the densities of the same substance at different temperatures are inversely as the factors of expansion, so that if D, D', D, denote the densities at the temperatures t, t', 0, then

200. Correction of Specific Gravity for Temperature.—Let d be the density of a substance at temperature t° Cent., and do its density at 0° C. Also, let D be the density of water at t° C., and D1 its density at 4° C. (the temperature of maximum density). Then if the specific

4

d

gravity of the substance be computed by comparing its density with that of water at the same temperature, as in the ordinary methods of experimental determination, the specific gravity thus obtained is and has different values according to the temperature at which the determination is made.

D'

d

The specific gravity as commonly given in tables is the value of do D; that is to say, is the ratio of the density of the substance at 0° C. D, 4 to that of water at the temperature of maximum density. The tabular specific gravity is easily derivable from the observed specific gravity D' if the law of expansion of the substance be known; for if 1+m be the factor of expansion of the substance from 0° C. to t° C., and 1+e the factor of expansion of water from 4° C. to t° C., then

In the case of solid bodies this correction is generally of little importance; but in determining the specific gravities of liquids, especially those which are very expansible by heat, it cannot be neglected.

201. Formulæ for the Expansion of Gases.-The volume of a gas depends both on the temperature and on the pressure to which it is subjected; hence the formulæ of expansion for this class of bodies are somewhat more complicated. Suppose we wish to find the relation between the volumes V and V' of the same mass of gas at temperatures t and t', and under pressures P and P' respectively. Let U be the volume of the given mass of gas at pressure P and temperature t', and let a be the coefficient of expansion of the gas.

The two volumes V and U, being under the same pressure, are proportional to the expansion factors corresponding to their temperatures (§ 198), which gives

The volumes U and V', at the same temperature t', are, by Boyle's law, inversely proportional to the pressures; whence we have

U P'
V P

From these two equations we obtain by multiplication

which means that the volumes assumed by the same mass of gas are inversely proportional to the pressures, and directly proportional to the expansion factors corresponding to the temperatures.

From equation (9) we may easily deduce another equation, by remarking that the densities of the same quantity of gas must be inversely as the volumes occupied. Thus we have

which signifies that the density of a gas varies directly as the pressure, and inversely as the expansion factor corresponding to the temperature.