Part 11

a′ − a = b − b′.

a′ is the quantity of heat required to cause the gas to rise from 1° to 100° when it occupies the space abef.

a is the quantity of heat required to cause the gas to rise from 1° to 100° when it occupies the space abcd.

The density of the air is less in the first than in the second case, and according to the experiments of MM. Delaroche and Bérard, already cited on page 87, its capacity for heat should be a little greater.

The quantity a′ being found to be greater than the quantity a, b should be greater than b′. Consequently, generalizing the proposition, we should say:

The quantity of heat due to the change of volume of a gas is greater as the temperature is higher.

Thus, for example, more caloric is necessary to maintain at 100° the temperature of a certain quantity of air the volume of which is doubled, than to maintain at 1° the temperature of this same air during a dilatation exactly equal.

These unequal quantities of heat would produce, however, as we have seen, equal quantities of motive power for equal fall of caloric taken at different heights on the thermometric scale; whence we draw the following conclusion:

The fall of caloric produces more motive power at inferior than at superior temperatures.

Thus a given quantity of heat will develop more motive power in passing from a body kept at 1 degree to another maintained at zero, than if these two bodies were at the temperature of 101° and 100°.

The difference, however, should be very slight. It would be nothing if the capacity of the air for heat remained constant, in spite of changes of density. According to the experiments of MM. Delaroche and Bérard, this capacity varies little—so little even, that the differences noticed might strictly have been attributed to errors of observation or to some circumstances of which we have failed to take account.

We are not prepared to determine precisely, with no more experimental data than we now possess, the law according to which the motive power of heat varies at different points on the thermometric scale. This law is intimately connected with that of the variations of the specific heat of gases at different temperatures—a law which experiment has not yet made known to us with sufficient exactness.

We will endeavor now to estimate exactly the motive power of heat, and in order to verify our fundamental proposition, in order to determine whether the agent used to realize the motive power is really unimportant relatively to the quantity of this power, we will select several of them successively: atmospheric air, vapor of water, vapor of alcohol.

Let us suppose that we take first atmospheric air. The operation will proceed according to the method indicated on page 70. We will make the following hypotheses: The air is taken under atmospheric pressure. The temperature of the body A is ¹⁄₁₀₀₀ of a degree above zero, that of the body B is zero. The difference is, as we see, very slight—a necessary condition here.

The increase of volume given to the air in our operation will be ¹⁄₁₁₆ + ¹⁄₂₆₇ of the primitive volume; this is a very slight increase, absolutely speaking, but great relatively to the difference of temperature between the bodies A and B.

The motive power developed by the whole of the two operations described (page 70) will be very nearly proportional to the increase of volume and to the difference between the two pressures exercised by the air, when it is found at the temperatures 0°.001 and zero.

This difference is, according to the law of M. Gay-Lussac, ¹⁄₂₆₇₀₀₀ of the elastic force of the gas, or very nearly ¹⁄₂₆₇₀₀₀ of the atmospheric pressure.

The atmospheric pressure balances at 10.40 metres head of water; ¹⁄₂₆₇₀₀₀ of this pressure equals ¹⁄₂₆₇₀₀₀ × 10^m.40 of head of water.

As to the increase of volume, it is, by supposition, ¹⁄₁₁₆ + ¹⁄₂₆₇ of the original volume, that is, of the volume occupied by one kilogram of air at zero, a volume equal to 0^{mc}.77, allowing for the specific weight of the air. So then the product,

(¹⁄₁₁₆ + ¹⁄₂₆₇) × 0.77 × ¹⁄₂₆₇₀₀₀ × 10.40,

will express the motive power developed. This power is estimated here in cubic metres of water raised one metre.

If we carry out the indicated multiplications, we find the value of the product to be 0.000000372.

Let us endeavor now to estimate the quantity of heat employed to give this result; that is, the quantity of heat passed from the body A to the body B.

The body A furnishes:

(1) The heat required to carry the temperature of one kilogram of air from zero to 0°.001;

(2) The quantity necessary to maintain at this temperature the temperature of the air when it experiences a dilatation of

¹⁄₁₁₆ + ¹⁄₂₆₇.

The first of these quantities of heat being very small in comparison with the second, we may disregard it. The second is, according to the reasoning on page 74, equal to that which would be necessary to increase one degree the temperature of one kilogram of air subjected to atmospheric pressure.