Method Of Calculating Heights From Observations With The Mountain Thermometer

Having considered how to make observations with the proper care and accuracy, it becomes necessary to know how to deduce the height by calculation. That a constant intimate relation exists between the boiling temperature of water and the pressure of the air, we have already learned. This knowledge is the result of elaborate experiments made by several scientific experimentalists, who have likewise constructed formulæ and tables for the conversion of the boiling temperatures into the co responding pressures of vapour, or, which is equivalent, of the atmosphere, when the operation is performed in the open air. As might be expected, there is not a perfect accord in the results arrived at by different persons. Regnault is the most recent, and his experiments are considered the most reliable.

From Regnault’s table of vapour tension, we can obtain the pressure in inches of mercury at 32°, which corresponds to the observed boiling-point; or vice versa, if required. From the pressure, the height may be deduced by the method for finding heights by means of the barometer.

The following table expresses very nearly the elevation in feet corresponding to a fall of 1° in the temperature of boiling water:—

Boiling Temperatures between.	Â	Elevation in Feet for each Degree.
214° and 210—	Â	520
210   and 200—	Â	530
200   and 190	Â	550
190   and 180	Â	570

These numbers agree very well with the results of theory and actual observation. The assumption is that the boiling-point will be diminished 1° for each 520 feet of ascent until the temperature becomes 210°, then 530 feet of elevation will lower it one degree until the water boils at 200°, and so on; the air being at 32°.

Let H represent the vertical height in feet between two stations; B and b, the boiling-points of water at the lower and upper stations respectively; f, the factor found in the above table. Then

H = f (B - b)

Further, let m be the mean temperature of the stratum of air between the stations. Now, if the mean temperature is less than 32°, the column of air will be shorter; and if greater, longer than at 32°. According to Regnault, air expands 1⁄491·13 or ·002036 of its volume at 32°, for each degree increase of heat. Calling the correction due to the mean temperature of air C, its value will be found from the equation,

C = H (m - 32) ·002036

Calling the corrected height H′, it will be found from the formula,

H′ = H + H (m - 32) ·002036

that is,

H′ = H {1 + (m - 32) ·002036}

and substituting the value of H,

H′ = f (B - b) {1 + (m - 32) ·002036}

Strictly, according to theoretical considerations, there is a correction due to latitude, as in the determination of heights by the barometer; but its value is so small that it is practically of no importance.

If a barometer be observed at one of the stations, the table of vapour tensions (p. 62) will be useful in converting the pressure into the corresponding boiling-point, or vice versa; so that the difference of height may be found either by the methods employed for the boiling-point thermometer or the barometer.

In conclusion, it may be remarked that observers who have good instruments at considerable elevations, as sites on mountains or plateaus, would confer a benefit to science, by registering for a length of time the barometer along with the boiling temperature of water, as accurately as possible. Such observations would serve to verify the accuracy of theoretical deductions, and fix with certainty the theoretical scale with the barometer indications.

Example, in calculating Heights from the Observations of the Boiling-point of Water.—1. At Geneva the observed boiling-point of water was 209°·335; on the Great St. Bernard it was 197°·64; the mean temperature of the intermediate air was 63°·5; required the height of the Great St. Bernard above Geneva.

Method by formula:—

H′ = f (B - b) {1 + (m - 32°) ·002036}

In this case f is between 530 and 550, or 540.

B =	209·335	Â	Â	m =	63·5
b =	197·64	Â			32
Â	11·695	Â			31·5
f =	540	Â			·002036
Â	6315·3	Â			0·0641340
Â	1·064	Â			1
H′ =	6719·5	feet.	Â		1·064

Method by Tables supplied with boiling-point apparatus made by Messrs. Negretti and Zambra:—

209·335	gives	1464	in Table I.
197·64	"	7736	"
Â		6272
63·5	"	1·07	in Table II.
Height	Â	6711