Chemistry: Chapter Five
Charles's Law: Volume and Temperature
Suppose we keep the pressure of a gas sample constant and measure its volume at a number of different temperatures. The results of several such measurements are shown in Figure 5.9▶. From the plot we can see the relationship between volume and temperature: The volume of a gas increases with increasing temperature. Looking at the plot more closely, however, reveals more—volume and temperature are linearly related. If two variables are linearly related, then plotting one against the other produces a straight line. Another interesting feature emerges if we extend or extrapolate the line in the plot backward from the lowest measured temperature. The extrapolated line shows that the gas should have a zero volume at -273.15 °C. Recall from Chapter 1 that -273.15 °C corresponds to 0 K (zero on the Kelvin scale), the coldest possible temperature. The extrapolated line indicates that below -273.15 °C, the gas would have a negative volume, which is physically impossible. For this reason, we refer to 0 K as absolute zero—colder temperatures do not exist. The first person to carefully quantify the relationship between the volume of a gas and its temperature was J. A. C. Charles (1746-1823), a French mathematician and physicist. Charles was interested in gases and was among the first people to ascend in a hydrogen-filled balloon. The direct proportionality between volume and temperature is named Charles's law after him. When the temperature of a gas sample is increased, the gas particles move faster; collisions with the walls are more frequent, and the force exerted with each collision is greater. The only way for the pressure (the force per unit area) to remain constant is for the gas to occupy a larger volume so that collisions become less frequent and occur over a larger area (Figure 5.10▲). Charles's law explains why the second floor of a house is usually a bit warmer than the ground floor. According to Charles's law, when air is heated, its volume increases, resulting in a lower density. The warm, less dense air tends to rise in a room filled with colder, denser air. Similarly, Charles's law explains why a hot-air balloon can take flight. The gas that fills a hot-air balloon is warmed with a burner, increasing its volume and lowering its density, and causing it to float in the colder, denser surrounding air. You can experience Charles's law directly by holding a partially inflated balloon over a warm toaster. As the air in the balloon warms, you can feel the balloon expanding. Alternatively, you can put an inflated balloon into liquid nitrogen and watch it become smaller as it cools (Figure 5.11◀). We can use Charles's law to calculate the volume of a gas following a temperature change or the temperature of a gas following a volume change as long as the pressure and the amount of gas are constant. For these types of calculations, we rearrange Charles's law as follows: V₁/T₁ =V₂/T₂ Where V1 and T1 are the initial volume and temperature of the gas and V2 and T2 are the final volume and temperature. The temperatures must always be expressed in kelvins (K), because, as we can see from Figure 5.9, the volume of a gas is directly proportional to its absolute temperature, not its temperature in °C. For example, doubling the temperature of a gas sample from 1 °C to 2 °C does not double its volume, but doubling the temperature from 200 K to 400 K does.
The Effect of the Finite Volume of Gas Particles
The finite volume of gas particles becomes important at high pressure, when the particles themselves occupy a significant portion of the total gas volume (Figure 5.22▶). We can see the effect of particle volume by comparing the molar volume of argon to the molar volume of an ideal gas as a function of pressure at 500 K as shown in Figure 5.23▶. At low pressures, the molar volume of argon is nearly identical to that of an ideal gas. But as the pressure increases, the molar volume of argon becomes greater than that of an ideal gas. At the higher pressures, the argon atoms themselves occupy a significant portion of the gas volume, making the actual volume greater than the volume predicted by the ideal gas law. In 1873, Johannes van der Waals (1837-1923) modified the ideal gas equation to fit the behavior of real gases. From the graph for argon, we can see that the ideal gas law predicts a volume that is too small. Van der Waals suggested a small correction factor that accounts for the volume of the gas particles themselves: Ideal Behavior: V = nRT/P Corrected for Volume of gas particles: V = nRT/P + nb The correction adds the quantity nb to the volume, where n is the number of moles and b is a constant that depends on the gas (Table 5.4). We can rearrange the corrected equation as follows: (V-nb) = nRT/P
Mean Free Path, Diffusion, and Effusion of Gases
We have just seen that the root mean square velocity of gas molecules at room temperature is in the range of hundreds of meters per second. But suppose that your sister just put on too much perfume in the bathroom only 6 ft (1.8 m) away. Why does it take a minute or two before you smell the fragrance? Although most molecules in a perfume bottle have higher molar masses than nitrogen, their velocities are still hundreds of meters per second. Why the delay? The answer is that even though gaseous particles travel at tremendous speeds, they also travel in very haphazard paths (Figure 5.19▶). To a perfume molecule, the path from the perfume bottle in the bathroom to your nose 6 ft away is much like the path a shopper would take through a busy shopping mall during a clearance sale. The molecule travels only a short distance before it collides with another molecule and changes direction, only to collide again with another molecule, and so on. In fact, at room temperature and atmospheric pressure, a gas molecule in the air experiences several billion collisions per second. The average distance that a molecule travels between collisions is its mean free path. At room temperature and atmospheric pressure, the mean free path of a nitrogen molecule with a molecular diameter of 300 pm (four times the covalent radius) is 93 nm, or about 310 molecular diameters. If the nitrogen molecule were the size of a golf ball, it would travel about 40 ft between collisions. Mean free path increases with decreasing pressure. Under conditions of ultra-high vacuum (10-10 torr), the mean free path of a nitrogen molecule is hundreds of kilometers. The process by which gas molecules spread out in response to a concentration gradient is diffusion, and even though the particles undergo many collisions, the root mean square velocity still influences the rate of diffusion. Heavier molecules diffuse more slowly than lighter ones, so the first molecules you smell from a perfume mixture (in a room with no air currents) are the lighter ones. A process related to diffusion is effusion, the process by which a gas escapes from a container into a vacuum through a small hole (Figure 5.20◀). The rate of effusion is also related to root mean square velocity—heavier molecules effuse more slowly than lighter ones. The rate of effusion—the amount of gas that effuses in a given time—is inversely proportional to the square root of the molar mass of the gas as follows: Graham's law of effusion, named after Thomas Graham (1805-1869), addresses the ratio of effusion rates of two different gases:
The Simple Gas Laws: Boyle's Law, Charles's Law, and Avogadro's Law
A sample of gas has four basic physical properties: pressure (P), volume (V), temperature (T), and amount in moles (n). These properties are interrelated—when one changes, it affects one or more of the others. The simple gas laws describe the relationships between pairs of these properties. For example, one simple gas law describes how volume varies with pressure at constant temperature and amount of gas; another law describes how volume varies with temperature at constant pressure and amount of gas. These laws were deduced from observations in which two of the four basic properties were held constant in order to elucidate the relationship between the other two. Boyle's Law: Volume and Pressure In the early 1660s, the pioneering English scientist Robert Boyle (1627-1691) and his assistant Robert Hooke (1635-1703) used a J-tube (Figure 5.5▲) to measure the volume of a sample of gas at different pressures. After trapping a sample of air in the J-tube, they added mercury to increase the pressure on the gas. They found an inverse relationship between volume and pressure—an increase in one causes a decrease in the other—as shown in Figure 5.6▶. This relationship is now known as Boyle's law. Boyle's law: V ∝ 1 P (constant T and n) Boyle's law follows from the idea that pressure results from the collisions of the gas particles with the walls of their container. When the volume of a gas sample is decreased, the same number of gas particles is crowded into a smaller volume, resulting in more collisions with the walls and therefore an increase in pressure. Boyle's Law: V = 1/P P₁V₁ =P₂V₂ Where P1 and V1 are the initial pressure and volume of the gas and P2 and V2 are the final pressure and volume. Scuba divers learn about Boyle's law during certification because it explains why a diver should not ascend toward the surface without continuous breathing. For every 10 m of depth that a diver descends in water, she experiences an additional 1 atm of pressure due to the weight of the water above her (Figure 5.8▼). The pressure regulator used in scuba diving delivers air at a pressure that matches the external pressure; otherwise the diver cannot inhale the air because the muscles that surround the chest cavity are not strong enough to expand the volume against the greatly increased external pressure. When a diver is at a depth of 20 m below the surface, the regulator delivers air at a pressure of 3 atm to match the 3 atm of pressure around the diver (1 atm due to normal atmospheric pressure and 2 additional atmospheres due to the weight of the water at 20 m). Suppose that a diver inhaled a lungful of air at a pressure of 3 atm and swam quickly to the surface (where the pressure drops to 1 atm) while holding her breath. What would happen to the volume of air in her lungs? Since the pressure decreases by a factor of 3, the volume of the air in her lungs would increase by a factor of 3, severely damaging her lungs and possibly killing her. We can use Boyle's law to calculate the volume of a gas following a pressure change or the pressure of a gas following a volume change as long as the temperature and the amount of gas remain constant. For these types of calculations, we write Boyle's law in a slightly different way.
Temperature and Molecular Velocities
According to kinetic molecular theory, particles of different masses have the same average kinetic energy at a given temperature. The kinetic energy of a particle depends on its mass and velocity according to the equation: KE = 1/2mv^2 The only way for particles of different masses to have the same kinetic energy is for them to travel at different velocities, as we saw in Conceptual Connection 5.6. In a gas mixture at a given temperature, lighter particles travel faster (on average) than heavier ones. In kinetic molecular theory, we define the root mean square velocity (urms) of a particle as follows: The root mean square velocity, as we have seen, is a kind of average velocity. Some particles are moving faster and some are moving slower than this average. The velocities of all the particles in a gas sample form a distribution like the ones shown in Figure 5.17▲. We can see from these distributions that while some particles are indeed traveling at the root mean square velocity, many particles are traveling faster and many slower than the root mean square velocity. For lighter molecules, the velocity distribution is shifted toward higher velocities and the curve becomes broader, indicating a wider range of velocities. The velocity distribution for nitrogen at different temperatures is shown in Figure 5.18◀. As the temperature increases, the root mean square velocity increases and the distribution becomes broader.
Pressure: The result of Molecular Collisions
Air can hold up a jumbo jet or knock down a building. How? Air contains gas molecules in constant motion that collide with each other and with the surfaces around them. Each collision exerts only a small force, but when these forces are summed over the many molecules in air, they add up to a substantial force. As we have just seen, the result of the constant collisions between the atoms or molecules in a gas and the surfaces around them is pressure. Because of pressure, we can drink from straws, inflate basketballs, and breathe. Variation in pressure in Earth's atmosphere creates wind, and changes in pressure help us to predict weather. Pressure is all around us and even inside us. The pressure that a gas sample exerts is the force that results from the collisions of gas particles divided by the area of the surface with which they collide: Pressure = force/area The pressure exerted by a gas sample, therefore, depends on the number of gas particles in a given volume—the fewer the gas particles, the lower the force per unit area and the lower the pressure (Figure 5.2◀). Since the number of gas particles in a given volume generally decreases with increasing altitude, pressure decreases with increasing altitude. Above 30,000 ft, for example, where most commercial airplanes fly, the pressure is so low that you could pass out for lack of oxygen. For this reason, most airplane cabins are artificially pressurized. You may sometimes feel the effect of a drop in pressure as a brief pain in your ears (Figure 5.3▲). When you ascend a mountain, the external pressure (the pressure that surrounds you) drops, while the pressure within your ear cavities (the internal pressure) remains the same. This creates an imbalance—the greater internal pressure forces your eardrum to deform, causing pain. With time, and with the help of a yawn or two, the excess air within your ear's cavities escapes, equalizing the internal and external pressure and relieving the pain.
Avogadro's Law
Avogadro's law states that, at constant temperature and pressure, the volume of a gas is proportional to the number of particles. According to kinetic molecular theory, when we increase the number of particles in a gas sample, the number of collisions with the sur- rounding surfaces increases. Since the greater number of collisions would result in a greater overall force on surrounding surfaces, the only way for the pressure to remain constant is for the volume to increase so that the number of particles per unit volume (and thus the number of collisions) remains constant.
Density of a Gas
Because one mole of an ideal gas occupies 22.4 L under standard temperature and pressure, we can use this information to calculate the density of an ideal gas under standard temperature and pressure. Since density is mass/volume, and since the mass of one mole of a gas is its molar mass, the density of a gas under standard temperature and pressure is given by the relationship: Density = molar mass/molar volume Notice that the density of a gas is directly proportional to its molar mass. The greater the molar mass of a gas, the more dense the gas is. For this reason, a gas with a molar mass lower than that of air tends to rise in air. For example, both helium and hydrogen gases (molar masses of 4.00 and 2.02 g>mol, respectively) have molar masses that are lower than the average molar mass of air (approximately 28.8 g>mol). Therefore a balloon filled with either helium or hydrogen gas floats in air. We can calculate the density of a gas more generally (under any conditions) by using the ideal gas law. For example, we can arrange the ideal gas law as follows: PV = nRT or n/V = P/RT Since the left-hand side of this equation has units of moles/liter, it represents the molar density. We can obtain the density in grams/liter from molar density by multiplying by the molar mass (M): Molar density (moles/liter) ---> molar mass (grams/mole) ---> density in grams/liter (grams/liter) d = PM/RT
Boyle's Law
Boyle's law states that, for a constant number of particles at constant temperature, the volume of a gas is inversely proportional to its pressure. If we decrease the volume of a gas, we force the gas particles to occupy a smaller space. It follows from kinetic molecular theory that, as long as the temperature remains the same, the result is a greater number of collisions with the surrounding surfaces and therefore a greater pressure.
Charles's Law
Charles's law states that, for a constant number of particles at constant pressure, the vol- ume of a gas is proportional to its temperature. According to kinetic molecular theory, when we increase the temperature of a gas, the average speed, and thus the average kinetic energy, of the particles increase. Since this greater kinetic energy results in more frequent collisions and more force per collision, the pressure of the gas would increase if its volume were held constant (Gay-Lussac's law). The only way for the pressure to remain constant is for the volume to increase. The greater volume spreads the collisions out over a greater area, so that the pressure (defined as force per unit area) is unchanged.
Dalton's Law
Dalton's law states that the total pressure of a gas mixture is the sum of the partial pressures of its components. In other words, according to Dalton's law, the components in a gas mixture act identically to, and independently of, one another. According to kinetic molecular theory, the particles have negligible size and they do not interact. Consequently, the only property that distinguishes one type of particle from another is its mass. However, even particles of different masses have the same average kinetic energy at a given temperature, so they exert the same force upon collision with a surface. Consequently, adding components to a gas mixture—even different kinds of gases—has the same effect as simply adding more particles. The partial pressures of all the components sum to the overall pressure.
Breathing: Putting Pressure to Work
Every day, without even thinking about it, you move approximately 8500 L of air into and out of your lungs. The total mass of this air is about 11 kg (or 24 lb). How do you do it? The simple answer is pressure. You rely on your body's ability to create pressure differences to move air into and out of your lungs. Pressure is the force exerted per unit area by gas particles (molecules or atoms) as they strike the surfaces around them (Figure 5.1▶ on the next page). Just as a ball exerts a force when it bounces against a wall, so a gaseous molecule exerts a force when it collides with a surface. The result of all these collisions is pressure—a constant force on surfaces exposed to any gas. The total pressure exerted by a gas depends on several factors, including the concentration of gas particles in the sample; the higher the concentration, the greater the pressure. When you inhale, the muscles that surround your chest cavity expand the volume of your lungs. The expanded volume results in a lower concentration of gas molecules (the number of molecules does not change, but since the volume increases, the concentration goes down). This in turn results in fewer molecular collisions, which results in lower pressure. The external pressure (the pressure outside of your lungs) remains relatively constant and once you inhale, it is now higher than the pressure within your lungs. As a result, gaseous molecules flow into your lungs (moving from higher pressure to lower pressure). When you exhale, the process is reversed. The chest cavity muscles relax, which decreases your lung volume, increasing the pressure within the lungs and forcing air back out. In this way, over your lifetime, you will take about half a billion breaths and move about 250 million L of air through your lungs. With each breath you create pressure differences that allow you to obtain the oxygen that you need to live.
Kinetic Molecular Theory: A model for gases
In Chapter 1, we discussed how the scientific approach to knowledge proceeds from obser- vations to laws and eventually to theories. Remember that laws summarize behavior—for example, Charles's law summarizes how the volume of a gas depends on temperature— while theories give the underlying reasons for the behavior. A theory of gas behavior explains, for example, why the volume of a gas increases with increasing temperature. The simplest model for the behavior of gases is the kinetic molecular theory. In this theory, a gas is modeled as a collection of particles (either molecules or atoms, depending on the gas) in constant motion (Figure 5.15▶). A single particle moves in a straight line until it collides with another particle (or with the wall of the container). Kinetic molecu- lar theory has three basic postulates (or assumptions): 1. The size of a particle is negligibly small. Kinetic molecular theory assumes that the particles themselves occupy no volume, even though they have mass. This postulate is justified because, under normal pressures, the space between atoms or molecules in a gas is very large compared to the size of the atoms or molecules themselves. For example, in a sample of argon gas under STP conditions, only about 0.01% of the volume is occupied by atoms and the average distance from one argon atom to another is 3.3 nm. In comparison, the atomic radius of argon is 97 pm. If an argon atom were the size of a golf ball, its nearest neighbor would be, on average, just over 4 ft (1.2 m) away at STP. 2. The average kinetic energy of a particle is proportional to the temperature in kelvins. The motion of atoms or molecules in a gas is due to thermal energy, which distributes itself among the particles in the gas. At any given moment, some particles are moving faster than others—there is a distribution of velocities—but the higher the temperature, the faster the overall motion, and the greater the average kinetic energy. Notice that kinetic energy (½ mv^2)—not velocity—is proportional to temperature. The atoms in a sample of helium and a sample of argon at the same temperature have the same average kinetic energy but not the same average velocity. Since the helium atoms are lighter, they must move faster to have the same kinetic energy as argon atoms. 3. The collision of one particle with another (or with the walls) is completely elastic. This means that when two particles collide, they may exchange energy, but there is no overall loss of energy. Any kinetic energy lost by one particle is completely gained by the other. This is the case because the particles have no "stickiness," and they are not deformed by the collision. In other words, an encounter between two particles in kinetic molecular theory is more like the collision between two billiard balls than between two lumps of clay (Figure 5.16▼). Between collisions, the particles do not exert any forces on one another. If we start with the postulates of kinetic molecular theory, we can mathematically derive the ideal gas law. In other words, the ideal gas law follows directly from kinetic molecular theory, which gives us confidence that the assumptions of the theory are valid, at least under conditions where the ideal gas law works. Let's see how the concept of pressure and each of the gas laws that we have examined in this chapter follow conceptually from kinetic molecular theory.
Gases in Chemical Reactions: Stoichiometry Revisited
In Chapter 4, we discussed how we use the coefficients in chemical equations as conversion factors between number of moles of reactants and number of moles of products in a chemical reaction. We can use these conversion factors to determine, for example, the mass of product obtained in a chemical reaction based on a given mass of reactant, or the mass of one reactant needed to react completely with a given mass of another reactant. The general conceptual plan for these kinds of calculations is: Mass A → amount of A (in moles) ---> Amount B ( in moles) ---> mass B where A and B are two different substances involved in the reaction and the conversion factor between amounts (in moles) of each comes from the stoichiometric coefficients in the balanced chemical equation. In reactions involving gaseous reactants or products, we often specify the quantity of a gas in terms of its volume at a given temperature and pressure. As we have seen, stoichiometric relationships are always between amounts in moles. For stoichiometric calculations involving gases, we can use the ideal gas law to find the amounts in moles from the volumes, or find the volumes from the amounts in moles. n = PV/RT V = nRT/P The general concept is: P,V,T, of gas A ---> amount A (in moles) ---> amount B (In moles) ---> P,V,T of gas B Molar Volume and Stoichiometry: In Section 5.5, we saw that, under standard temperature and pressure, 1 mol of an ideal gas occupies 22.4 L. Consequently, if a reaction is occurring at or near standard temperature and pressure, we can use 1 mol = 22.4 L as a conversion factor in stoichiometric calculations, as demonstrated in Example 5.13.
The Effect of Intermolecular Forces
Intermolecular forces, which we will cover in more detail in Chapter 11, are attractions between the atoms or molecules in a gas. These attractions are small and therefore do not matter much at low pressure, when the molecules are too far apart to "feel" the attractions. They also do not matter much at high temperatures because the molecules have a lot of kinetic energy. When two particles with high kinetic energies collide, a weak attraction between them does not affect the collision very much. At lower temperatures, however, the collisions occur with less kinetic energy, and weak attractions do affect the collisions. We can understand this difference with an analogy to billiard balls. Imagine two billiard balls that are coated with a substance that makes them slightly sticky. If they collide when moving at high velocities, the stickiness will not have much of an effect—the balls bounce off one another as if the sticky substance was not even present. However, if the two billiard balls collide when moving very slowly (say barely rolling) the sticky substance would have an effect—the billiard balls might even stick together and not bounce off one another. The effect of these weak attractions between particles is a decrease in the number of collisions with the surfaces of the container and a corresponding decrease in the pressure compared to that of an ideal gas. We see the effect of intermolecular forces when we compare the pressure of 1.0 mol of xenon gas to the pressure of 1.0 mol of an ideal gas as a function of temperature and at a fixed volume of 1.0 L, as shown in Figure 5.24▼. At high temperature, the pressure of the xenon gas is nearly identical to that of an ideal gas. But at lower temperatures, the pressure of xenon is less than that of an ideal gas. At the lower temperatures, the xenon atoms spend more time interacting with each other and less time colliding with the walls, making the actual pressure lower than the pressure predicted by the ideal gas law. Ideal Behavior: P = nRT/V Corrected: P = nRT/V -a(n/v)^2 The correction subtracts the quantity a(n>V) 2 from the pressure, where n is the number of moles, V is the volume, and a is a constant that depends on the gas (see Table 5.4). Notice that the correction factor increases as n>V (the number of moles of particles per unit volume) increases. This is because a greater concentration of particles makes it more likely that particles will interact with one another. We can rearrange the corrected equation as follows: P + a(n/v)^2 = nRT/V
Real Gases: The effects of Size and intermolecular forces
One mole of an ideal gas has a volume of 22.41 L at STP. Figure 5.21▼ shows the molar volume of several real gases at STP. As we can see, most of these gases have a volume that is very close to 22.41 L, meaning that they act very nearly as ideal gases. Gases behave ideally when both of the following are true: (a) the volume of the gas particles is negligible compared to the space between them; and (b) the forces between the gas particles are not significant. At STP, these assumptions are valid for most common gases. However, these assumptions break down at higher pressures or lower temperatures.
The Nature of Pressure
P = F/A According to kinetic molecular theory, a gas is a collection of particles in constant motion. The motion results in collisions between the particles and the surfaces around them. As each particle collides with a surface, it exerts a force upon that surface. The result of many parti- cles in a gas sample exerting forces on the surfaces around them is a constant pressure.
Common Units of Pressure:
Pascal (1 N/m^2), Pa. 1-1.325 (average air pressure at sea level) Pounds per square inch: psi, 14.7 psi. Torr(2mmHg) , torr, 760 torr inches of mercury, in Hg, 29.92 Atmosphere, atm, 1 atm
Avogadro's Law: Volume and Amount (in Moles)
So far, we have discussed the relationships between volume and pressure, and volume and temperature, but we have considered only a constant amount of a gas. What happens when the amount of gas changes? The volume of a gas sample (at constant temperature and pressure) as a function of the amount of gas (in moles) in the sample is shown in Figure 5.12▶. We can see that the relationship between volume and amount is linear. As we might expect, extrapolation to zero moles shows zero volume. This relationship, first stated formally by Amedeo Avogadro, is Avogadro's law: When the amount of gas in a sample is increased at constant temperature and pressure, its volume increases in direct proportion because the greater number of gas particles fill more space. You experience Avogadro's law when you inflate a balloon. With each exhaled breath, you add more gas particles to the inside of the balloon, increasing its volume. Avogadro's law can be used to calculate the volume of a gas following a change in the amount of the gas as long as the pressure and temperature of the gas are constant. For these types of calculations, we express Avogadro's law as So far, we have discussed the relationships between volume and pressure, and volume and temperature, but we have considered only a constant amount of a gas. What happens when the amount of gas changes? The volume of a gas sample (at constant temperature and pressure) as a function of the amount of gas (in moles) in the sample is shown in Figure 5.12▶. We can see that the relationship between volume and amount is linear. As we might expect, extrapolation to zero moles shows zero volume. This relationship, first stated formally by Amedeo Avogadro, is Avogadro's law: V₁/n₁ =V₂/n₂ Where V1 and n1 are the initial volume and number of moles of the gas and V2 and n2 are the final volume and number of moles. In calculations, we use Avogadro's law in a manner similar to the other gas laws, as shown in Example 5.4.
The Ideal Gas Law
The ideal gas law contains within it the simple gas laws that we have discussed as summarized in Figure 5.13◀. The ideal gas law also shows how other pairs of variables are related. For example, from Charles's law we know that V ∝ T at constant pressure and constant number of moles. But what if we heat a sample of gas at constant volume and constant number of moles? This question applies to the warning labels on aerosol cans such as hair spray or deodorants. These labels warn against excessive heating or incineration of the can, even after the contents are used up. Why? An "empty" aerosol can is not really empty but contains a fixed amount of gas trapped in a fixed volume. What would happen if we were to heat the can? We can rearrange the ideal gas law by dividing both sides by V to clearly see the relationship between pressure and temperature at constant volume and constant number of moles This relationship between pressure and temperature is also known as Gay-Lussac's law. As the temperature of a fixed amount of gas in a fixed volume increases, the pressure increases. In an aerosol can, this pressure increase can blow the can apart, which is why aerosol cans should not be heated or incinerated. They might explode. We can use the ideal gas law to determine the value of any one of the four variables (P, V, n, or T) given the other three. However, each of the quantities in the ideal gas law must be expressed in the units within R: pressure (P) in atm moles (n) in mol volume (V) in L temperature (T) in K
Molar Volume at Standard Temperature and Pressure
The volume occupied by one mole of a substance is its molar volume. For gases, we often specify the molar volume under conditions known as standard temperature (T = 0 °C or 273 K) and pressure (P = 1.00 atm), abbreviated as STP. Using the ideal gas law, we can determine that the molar volume of an ideal gas at STP is: V = nRT/P = 22.4 L The molar volume of an ideal gas at STP is useful because—as we saw in the Check steps of Examples 5.5 and 5.6—it gives us a way to approximate the volume of an ideal gas under conditions that are close to STP.
Molar Mas of a Gas
We can use the ideal gas law in combination with mass measurements to calculate the molar mass of an unknown gas. Usually, the mass and volume of an unknown gas are measured under conditions of known pressure and temperature. Given this information, we can determine the amount of the gas in moles from the ideal gas law. Then we can calculate the molar mass by dividing the mass (in grams) by the amount (in moles) as shown in Example 5.8. Because the particles in an ideal gas do not interact (as we discuss in more detail in Section 5.8), each component in an ideal gas mixture acts independently of the other components. For example, the nitrogen molecules in air exert a certain pressure—78% of the total pressure—that is independent of the presence of the other gases in the mixture. Likewise, the oxygen molecules in air exert a certain pressure—21% of the total pressure—that is also independent of the presence of the other gases in the mixture. The pressure due to any individual component in a gas mixture is the partial pressure (Pn) of that component. We can calculate partial pressure from the ideal gas law by assuming that each gas component acts independently Pₙ=nₙ RT/V Notice that this equation is the ideal gas law applied to one individual component gas. For a multicomponent gas mixture, we calculate the partial pressure of each component from the ideal gas law as follows: The partial pressure of a component in a gaseous mixture is its mole fraction multiplied by the total pressure. For gases, the mole fraction of a component is equivalent to its per- cent by volume divided by 100%. Therefore, based on Table 5.2, we calculate the partial pressure of nitrogen (PN2) in air at 1.00 atm as follows: PN2 = 0.78 * 1.00 atm = 0.78 atm Similarly, the partial pressure of oxygen in air at 1.00 atm is 0.21 atm and the partial pressure of argon in air is 0.01 atm. Applying Dalton's law of partial pressures to air at 1.00 atm: Ptotal = PN2 + PO2 + PAr Ptotal = 0.78 atm + 0.21 atm + 0.01 atm = 1.00 atm
Pressure Units
We measure pressure in a number of different units. A common unit of pressure, the millimeter of mercury (mmHg), originates from how pressure is measured with a barometer (Figure 5.4▶). A barometer is an evacuated glass tube, the open end of which is submerged in a pool of mercury. Atmospheric pressure on the the liquid mercury's surface forces the liquid upward into the evacuated tube. Because mercury is so dense (13.5 times more dense than water), atmospheric pressure can support a column of Hg that is only about 0.760 m or 760 mm (about 30 in) tall. (By contrast, atmospheric pressure can support a column of water that is about 10.3 m tall.) The mercury column rises with increasing atmospheric pressure or falls with decreasing atmospheric pressure. The unit millimeter of mercury is often called a torr, after the Italian physicist Evangelista Torricelli (1608-1647) who invented the barometer 1 mmHg = 1 torr A second unit of pressure is the atmosphere (atm), the average pressure at sea level. One atmosphere of pressure pushes a column of mercury to a height of 760 mm, so 1 atm and 760 mmHg are equal: 1 atm = 760 mmHg A fully inflated mountain bike tire has a pressure of about 6 atm, and the pressure at the top of Mount Everest is about 0.31 atm. The SI unit of pressure is the pascal (Pa), defined as 1 newton (N) per square meter. 1 Pa = 1 N/m2 The pascal is a much smaller unit of pressure than the atmosphere. 1 atm = 101,325 Pa Other common units of pressure include inches of mercury (in Hg) and pounds per square inch (psi). 1 atm = 29.92 in Hg 1 atm = 14.7 psi Table 5.1 summarizes these units.
Collecting Gases over water
When the product of a chemical reaction is gaseous, we often collect the gas by the dis- placement of water. For example, suppose we use the following reaction as a source of hydrogen gas: Zn(s) + 2 HCl(aq) ---> ZnCl2(aq) + H2(g) To collect the gas, we set up an apparatus like the one in Figure 5.14▼. As the hydrogen gas forms, it bubbles through the water and gathers in the collection flask. The hydrogen gas collected in this way is not pure, however. It is mixed with water vapor because some water molecules evaporate and mix with the hydrogen molecules. The partial pressure of water in the resulting mixture is its vapor pressure, which depends on the temperature (Table 5.3). Vapor pressure increases with increasing temperature because higher temperatures cause more water molecules to evaporate. Suppose we collect hydrogen gas over water at a total pressure of 758.2 mmHg at 25 °C. What is the partial pressure of the hydrogen gas? We know that the total pressure is 758.2 mmHg and that the partial pressure of water is 23.78 mmHg (its vapor pressure at 25 °C):
