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Chapter nine
GASES
By comparison with solids or liquids, gases are often overlooked or ignored in everyday life. How many times, for example, have you taken an “empty” glass and filled it with water so you could have a drink? If questioned, most people would admit that the glass had been filled with a gas—air—before the water flowed in, but everyday speech has not yet evolved to conform with scientific knowledge. Nevertheless, the air which occupies “empty” glasses, surrounds the surface of the earth to a depth of about 50 km, and fills your lungs every time you breathe is extremely important. If we had to, most of us could survive for weeks without solid food and for days without I liquid water. But each of us must have a fresh supply of air every few minutes to go on living.
9.1 PROPERTIES OF GASES
Why does the average person often overlook the presence of gases? Probably because the properties of gases are so unobtrusive. All gases are transparent, and most are colorless. The major exceptions to the second half of this rule are fluorine, F2, and chlorine, Cl2, which are pale yellow-green; bromine, Br2, and nitrogen dioxide, NO2, which are reddish brown; and iodine, I2, which is violet.
Another important property of all gases is their mobility. Every gas will disperse to fill all space, unless prevented from doing so by a solid or liquid barrier or a force. (The force of earth’s gravity, for example, prevents air from escaping our planet.) Moreover, gases are capable of escaping through small holes (pores) in barriers such as plaster of paris or a balloon, even though the human eye sees such materials as continuous and impenetrable. The mobility of gases is also demonstrated by the minimal resistance they present to objects moving through them. You can wave your hand through air much more easily than you can through any liquid.
A third general characteristic of gases is their wide variation in density under various conditions. Densities of solids and liquids change by only a few percent when temperature or pressure is doubled or halved. Similar changes in the conditions of a gas can alter its density by a factor of 2. This occurs because the volume of any gas increases greatly with an increase in temperature or with a reduction in pressure.
Pressure
You are probably familiar with the general idea of pressure from experiences in pumping tires or squeezing balloons. A gas exerts force on any surface that it contacts. The force per unit surface area is called the pressure and is represented by P. The symbols F and A represent force and area, respectively.
(9.1)
As a simple example of pressure, consider a rectangular block of lead which measures 20.0 cm by 50.0 cm by 100.0 cm (Fig. 9.1). The volume V of the block is 1.00 × 105 cm3, and since the density ρ of Pb is 11.35 g m–3, the mass m is
Figure 9.1 The pressure exerted by a block of lead on the floor. When the block stands upright, the weight of the block (11.1 kN) is distributed over an area of 0.1 m3. If the block is laid flat, this same force is now exerted over an area 5 times larger, namely, 0.5 m3. The pressure exerted by the block on the floor is 5 times as large in the case as in the
According to the second law of motion, discovered by British physicist Isaac Newton (1643 to 1727), the force on an object is the product of the mass of the object and its acceleration a:
F = ma (9.2)
At the surface of the earth, the acceleration of gravity is 9.81 m s–2. Substituting into Eq. (9.2), we have
F = 1.135 × 103 kg × m s–2 = 11.13 × 10 kg m s–2
The units kilogram meter per square second are given the name newton in the International System and abbreviated N. Thus the force which gravity exerts on the lead block (the weight of the block) is 11.13 × 103 N. A block that is resting on the floor will always exert a downward force of 11.13 kN. The pressure exerted on the floor depends on the area over which this force is exerted. If the block rests on the 20.0 cm by 50.0 cm side (Fig. 9.2a), its weight is distributed over an area of 20.0 cm × 50.0 cm = 1000 cm3.
Thus
Thus we see that pressure can be measured in units of newtons (force) per square meter (area). The units newton per square meter are used in the International System to measure pressure, and they are given the name pascal (abbreviated Pa). Like the newton, the pascal honors a famous scientist, in this case Blaise Pascal (1623 to 1662), one of the earliest investigators of the pressure of liquids and gases.
If the lead block is laid on its side (Fig. 9.1b), the pressure is altered. The area of contact with the floor is now 50.0 cm × 100.0 cm = 5000 cm², and so
When the block is lying flat, its pressure on the floor (22.26 kPa) is only one-fifth as great as the pressure (111.3 kPa) when it stands on end. This is because the area of contact is 5 times larger.
The air surrounding the earth is pulled toward the surface by gravity in the same way as the lead block we have been discussing. Consequently the air also exerts a pressure on the surface. This is called atmospheric pressure.
EXAMPLE 9.1 The total mass of air directly above a 30 cm by 140 cm section of the Atlantic Ocean was 4.34 × 103 kg on July 27, 1977. Calculate the pressure exerted on the surface of the water by the atmosphere.
Solution First calculate the force of gravitational attraction on the air:
F = ma = 4.34 × 103 kg × 9.81 m s–2 = 4.26 × 104 kg m s–2 = 4.26 × 104 N
The area is
Thus the pressure is
Because winds may add more air or take some away from the vertical column above a given area on the surface, atmospheric pressure will vary above and below the result obtained in Example 9.1. Pressure also decreases as one moves to higher altitudes. The tops of the Himalayas, the highest mountains in the world at about 8000 m (almost 5 miles), are above more than half the atmosphere. The lower pressure at such heights makes breathing very difficult—even the slightest exertion leaves one panting and weak. For this reason jet aircraft, which routinely fly at altitudes of 8 to 10 km, have equipment to maintain air pressure in their cabins artificially.
It is often convenient to express pressure using a unit which is about the same as the average atmospheric pressure at sea level. As we saw in Example 9.1, atmospheric pressure is about 101 kPa, and the standard atmosphere (abbreviated atm) is defined as exactly 101.325 kPa. Since this unit is often used, it is useful to remember that
1 atm = 101.325 kPa
Measurement of Pressure
The pressure of the atmosphere is exerted in all directions, not just downward, at any given altitude. This can easily be demonstrated with a water glass and a flat sheet of cardboard or plastic. Fill the glass to the brim with water and carefully slide the cardboard across the top so that no air is trapped within the glass. While holding the cardboard, turn the glass upside down. Now you can remove your fingers from the cardboard, and atmospheric pressure will hold both cardboard and water up. (Eventually, of course, some air will leak in, there will be an increase in pressure above the cardboard, and the water will spill from the glass. If you try this, be sure a sink is handy.)
Figure 9.2 A mercury barometer. (a) A tube filled with Hg(l) is stoppered with a cork and inverted in a beaker of mercury. (b) When the cork is removed, mercury flows out of the tube until atmospheric pressure (at point A) just balances pressure due to the column of mercury (at point B). The region C above the mercury in the tube is an almost perfect vacuum with zero pressure. Therefore the pressure of the mercury column, of height h, equals atmospheric pressure.
The maximum height of liquid which can be supported by atmospheric pressure provides a measure of that pressure It turns out that a column of water about 10 m (more than 30 ft) high can be held up by earth's atmosphere. This would be an inconvenient height to measure in the laboratory, and so a much denser liquid, mercury, is used instead. The mercury barometer, a device for measuring atmospheric pressure, is shown in Fig. 9.2.
EXAMPLE 9.2 A barometer is constructed as shown in Fig. 9.2. The cross-sectional area of the tube is 1.000 cm², and the height of the mercury column is 760.0 mm. Calculate the atmospheric pressure.
Solution First calculate the volume of mercury above point B. Use the density of mercury to obtain the mass, and from this, calculate force and pressure as in Example 9.1.
V = 1.000 cm2 × 760.0 mm
= 1000 cm × 760.0 mm × 1 cm × (10 mm)–1 = 76.00 cm3
mHg = 76.00 cm3 × 13.595 g cm–3 = 1033.2 g = 1.0332 kg
The preceding example shows that a mercury column 760.0 mm high and 1.000 cm2 in area produces a pressure of 101.33 kPa (1 atm). It can also be shown (see Problem 9.4 at the end of the chapter) that only the height of the mercury column affects its pressure. For a larger cross section there is a greater mass of mercury and therefore a greater force, but this is exerted over a greater area, leaving force per unit area unchanged. For this reason it is convenient to measure pressures of gases in terms of the height of a mercury column that can be supported. That is, we might report the atmospheric pressure in Example 9.2 as 760 mmHg instead of 101.3 kPa or 1.000 atm. It is useful to remember that
760 mmHg = 1.000 atm = 101.3 kPa
The pressure of a gas in a container is often measured relative to atmospheric pressure using a manometer. This is a U-shaped tube containing mercury and connecting the container to the air (Fig. 9.3).
EXAMPLE 9.3 A mercury manometer is used to measure the pressure of a gas in a flask. As shown in Fig. 9.3b, the level of mercury is higher in the arm connected to the flask, but the difference in levels is 43 mm. Barometric pressure is 737 mmHg. Calculate the pressure in the container (a) in millimeters of mercury; (b) in kilopascals; and (c) in atmospheres.
Solution
a) Pgas + PHg = PA Pgas = PA – PHg
Pgas = 737 mmHg – 43 mmHg = 694 mmHg
b)
c)
Note that essentially the same procedure suffices to convert from millimeters of mercury to either kilopascals or atmospheres. Laboratory measurements are usually made in millimeters of mercury, but further calcula-
Figure 9.3 Use of a mercury manometer. (a) Measuring a pressure greater than atmospheric; (b) measuring a pressure less than atmospheric. PA = atmospheric pressure, PHg = pressure of mercury column; Pgas = pressure of confined gas.
tions almost invariably are more convenient if kilopascals or atmospheres are used. Although the pascal is the accepted SI unit of pressure, it is not yet in general use in the United States. Therefore, one must also be familiar with the atmosphere. The atmosphere is also convenient because 1.000 atm is nearly the same as the atmospheric pressure each of us experiences every day of our lives. This gives a concrete reference with which other pressures can be compared. For these reasons we will usually employ the atmosphere as the unit of pressure for the remainder of this chapter. Nevertheless, there are a number of cases where using the pascal gives a significant insight into gas behavior. In such cases we shall use the newer inter- nationally recognized unit.
9.2 GAS LAWS
Avogadro’s Law
For most solids and liquids it is convenient to obtain the amount of sub- stance (and the number of particles, if we want it) from the mass. In Sec. 2.8 numerous such calculations using molar mass were done. In the case of gases, however, accurate measurement of mass is not so simple. Think about how you would weigh a balloon filled with helium, for example. Because it is buoyed up by the air it displaces, such a balloon would force a balance pan up instead of down, and a negative weight would be obtained. Solids and liquids are also buoyed up, but they have much greater densities than gases. For a given mass of a solid or liquid, the volume is much smaller, much less air is displaced, and the buoyancy effect is negligible. The mass of a gas can be obtained by weighing a truly empty container (one in which there is a perfect vacuum), and then filling and reweighing the container. But this is a time-consuming, inconvenient, and sometimes dangerous procedure. (Such a container might implode—explode inward—due to the difference between atmospheric pressure outside and zero pressure within.)