CHAPTER VI GAS CALCULATIONS Boyle's Law. The temperature remaining constant, the volume of a gas varies inversely as the pressure to which it is subjected. Let V be the volume of gas under a pressure P, and let V' be some other volume of the same quantity of the gas and P' its corresponding pressure. The analytical expression of this law is or Charles' Law. The pressure remaining constant, the volume of a gas varies directly as the absolute temperature to which it is subjected. Let V be the volume of a gas at a temperature T, and let V' be some other volume of the same quantity of the gas and T' its corresponding temperature. Then the analytical expression of this law is Since o° C. corresponds to 273° A., the law of Charles may be stated: The pressure remaining constant, an ideal gas expands or contracts 273 of its volume at o° C., for each degree Centigrade rise or fall in temperature.3 Furthermore, the volume remaining constant, the pressure on a gas varies directly as the absolute temperature. Let P be the pressure of a gas at temperature T and let P' be some other pressure on the same quantity of the gas and T' its corresponding temperature. Then the analytical expression of this fact is = 1 P'V' k, a constant, therefore on plotting the changes of a given volume of a gas under varying pressure or temperature an hyperbola results. 2 Note that T and T' are in the absolute temperature scale. 3 2 * may be expressed as a decimal, when it becomes 0.003663. Every gas has its own coefficient of expansion, one under constant pressure, which is the coefficient usually required for gas calculations, the other under constant volume, a coefficient of a different numerical value. For air the coefficient has been found to be 0.0036706 (under constant pressure) and the decimal value 0.00367 is usually_employed for every gas where extreme accuracy is not required. See Chem. Ann., Table 52. The gas thermometer is based upon this law. The standard degree of temperature is a temperature interval such as will cause the pressure on a confined gas to change of that change in pressure which is shown by a true gas between the temperature of melting ice and the temperature of water boiling under standard pressure. Thus, the pressure exerted by a gas is used as a means of measuring temperature and is employed in the hydrogen thermometer, in which the volume is kept constant and differences of pressure caused by different temperatures are measured. This unit has been chosen for the reason that the expansion coefficient of hydrogen is very uniform over wide ranges of temperature, a property of all gases in a condition far removed from their liquefaction point. Mercury, being a liquid, does not expand with this regularity with increase of temperature, although at ordinary temperatures the difference between a temperature reading with a hydrogen thermometer and with a mercury thermometer is slight.1 Laws of Boyle and Charles Combined. The laws of Boyle and Charles are readily combined in one expression. Considering a given weight of gas, Boyle's law is in which K is a constant. The law of Charles is Ρα Τ hence PK'T, in which K' is a constant. Combining these expressions: 2 in which is a constant for the same quantity of gas. From this it follows that the same mass of gas under the conditions P', V' and T' gives 1 Gases only conform to these generalizations approximately. In general, the further a gas is removed from its liquefaction temperature, the more closely does it obey the gas laws. An "ideal" gas is a hypothetical gas which is supposed to exactly obey the gas laws. A distinction between gases and vapors is sometimes made, assigning to the former a condition of temperature above its critical temperature, and to the latter a condition of temperature below its critical temperature. 2 If the volume varies inversely as the pressure and directly as the absolute temperature, then the product of the volume into the pressure is equal to the absolute temperature into a constant. When a gram molecular volume (G.M.V.) is under consideration, this is usually expressed, PV = RT. Knowing five of these quantities, the sixth may be obtained by solving the equation. It is more to be recommended, however, that in solving gas equations the logic of the case should serve as a guide than that a formula should be used unthinkingly. For example, a volume of a gas equal to V at a temperature T is to be changed to temperature T', pressure remaining constant. Knowing that a gas expands with a rise and contracts with a fall of temperature, a ratio may be made employing T and T' which is a proper or an improper fraction according as the gas contracts or expands. This ratio is multiplied by V to obtain V'. The same may be said for changes of volume of gases with change of pressure, temperature remaining constant. Let V be the volume of a gas at pressure P, which is to be changed to pressure P'. The new volume V' will be found by using P and P' as terms of a ratio, recollecting that if the new condition of the gas is to be a smaller volume than V, the ratio must be in the form of a proper fraction; if larger, an improper fraction. This fraction is multiplied by V to obtain V'. Standard Conditions of Pressure and Temperature. As changes of temperature and pressure exert so considerable an influence upon the volume of a gas and consequently upon the weight of a unit volume, standard conditions have been adopted, which are: o° C. as standard temperature, and a pressure equal to that exerted by a column of mercury 76 cm. high when at a temperature of o° C.1 The measurement of temperature has already been discussed; it remains to take up the measurement of pressure. Pressure is usually expressed as so many units of weight per square unit of area. Atmospheric pressure is measured by the height of a column of mercury which will balance this downward pressure of the atmosphere. Then if the height of the mercury column is taken to be standard at 76 cm. at a temperature of 0° C. and the specific gravity of mercury is 13.596 at this temperature, the pressure per square centimeter is 13.596 × 76 1033.3 g. As atmospheric pressure is always measured by a mercury barometer or a barometer standardized against a mercury barometer, it is only necessary to indicate the height of the column of mercury. = Correction of Barometer for Temperature. — Mercury expands and contracts with rise and fall of temperature; consequently its specific gravity increases and decreases. So, merely measuring the height of the column of mercury is not an exact measurement of the pressure. As o°C. is the standard of temperature of a gas, so also is this same temperature taken at which the height of the mercury column is standard. For example, at o° C. the specific gravity of mercury is 1 See Chem. Ann., Table 4. = 13.596, then the pressure of 76 cm. of mercury at this temperature is 76 X 13.596 1033.296 g. per square centimeter. At 15° C. the specific gravity of mercury is 13.560, then the pressure of 76 cm. of mercury at this temperature is 76 X 13.560 = 1030.56 g. per square centimeter. The height of the column of mercury is usually measured by graduations on the glass, or the glass tube is mounted in a brass jacket which carries the graduations. If the expansion of the mercury and of the substance carrying the graduations were the same, no correction for temperature in reading the barometer would be necessary; because, as the mercury expanded, though its specific gravity would be lowered, the material carrying the graduations would expand by an equal amount and these expansions would neutralize each other, as the graduations would register a greater length than the true length.1 As an expansion of the mercury is accompanied by an expansion of the material carrying the graduations, it is only necessary to determine the difference between the coefficient of expansion of the mercury and its containing tube. This is the apparent expansion of the mercury, and for measuring this height on a glass tube 0.00017 of the column at o° C. for each degree departure from this temperature is the correction. For a brass scale it is 0.00016. Hence 1±at is the length of the apparent column as compared with the column at o° C. In this expression a is the apparent coefficient of expansion and t is the number of degrees from o° C. Then if P is the observed height of the baP 2 rometer, the corrected reading is I at Moist Gases. Volumes of gases are often measured over liquids which may or may not exert an appreciable vapor pressure. The vapor pressure of a saturated vapor depends upon the temperature only, and is independent of the pressure or the presence or absence of an inert gas. If a sufficient amount of a volatile liquid is introduced into a Torricellian vacuum above the mercury in a barometer or into a barometer tube containing a gas, the height of the mercury column will be depressed by an amount which is independent of all conditions except the temperature. If, then, the volume of a confined gas is measured over a volatile liquid such as water, the volume will appear greater than the volume of the same amount of dry gas by an amount corresponding to the vapor pressure of the water (if this be the liquid 1 For the very accurate reading of some forms of the mercury barometer, capillarity must also be taken into account. This correction will not be considered here as it varies with the form of the instrument and the diameter of the tube. This correction is usually supplied by the manufacturer with each instrument. 2 For a table of Barometric corrections, see Chem. Ann., Table 51. employed)1 at that temperature. If this vapor pressure were a constant quantity or increased regularly with rise in temperature, it would be a simple matter to correct for it; but such not being the case, the vapor pressures corresponding to various temperatures are determined experimentally and tabulated.2 Let Then P+Pw is the pressure of the dry gas, the volume of which is When measuring a gas over mercury, whether moist or not, a common procedure is to bring the mercury to the same level inside and outside the tube, the pressure of the gas being that indicated by the barometer. It may not be convenient to do this; then, to measure the pressure of a gas confined in this way, the height of the mercury in the tube must be subtracted from the barometric reading.3 Let Рь = the pressure indicated by the barometer, Ph the height of mercury in the tube. = Then the pressure of the moist gas is P Ph and the pressure of the dry gas is (P- Ph) - Pw, or Pb (Ph + Pu), and the volume of the dry gas is Pb − (Ph + Pw) Vo+w." 1 Every liquid has a vapor pressure which is peculiar to that liquid. To determine the variation of vapor pressure of a liquid with change of temperature is a matter of experiment. See Chem. Ann., Tables 174-180. 2 For the values for water, see Chem. Ann., Tables 174-180. 3 If the gas is measured over some liquid other than mercury and the level inside and outside the tube is not the same, the height of the liquid must be reduced to the equivalent height of a column of mercury. This necessitates a knowledge of the specific gravities of the mercury and the liquid. 4 Tables are constructed for reducing the volume of a gas, moist or dry, to the volume which the gas would occupy dry at standard conditions (0° C. and 760 mm.). For example, 300 c.c. of a gas is measured moist at 18° C. and 765 mm. What is the volume, dry, at o° C. and 670 mm.? Using the decimal coefficient, the expression to solve is V, = I I + (18 x 0.00367) 760 (1+0.00367 t) X 765 - Pw X 300. The logarithm |