Chapter 10 - Gases

What the weather is gonna be like today or this week? Such knowledge not only helps us choose the clothing we wear during the day, but it also assists the farmer in planting and harvesting, and the sailor in charting a course. The gases in the atmosphere, primarily nitrogen and oxygen molecules, are heated by the Sun and move here and there due to pressure differences that create wind. It is the behavior of gases that creates our weather the gentle breeze, the violent storm, the humidity, and the rain. Tornadoes form when moist, warm air at lower elevation converges with cooler, dry air above. The resultant airflow produces winds that can approach speeds of up to 300 miles per hour. In this chapter, we examine the physical properties of gases. What is pressure? How do gases respond to changes in pressure and temperature? And what does temperature measure at the molecular level? Because gases are the simplest state of matter, they serve as an excellent starting point for probing the behavior of large collections of atoms or molecules. Indeed, it is fairly straightforward to formulate a simple model for gases that explains their behavior under most common conditions. We gain further insight by then comparing how real gases differ from this ideal model as conditions change. Thus, while we start with a description that treats all gases the same, regardless of their chemical identity, we end by gaining an understanding of important aspects of the physical behavior of molecules. By the end of this chapter, you will be able to compare the distinguishing characteristics of gases with those of liquids and solids. You will describe gas pressure, the units used to express it, and consider the air atmosphere and the pressure it exerts. You'll also be able to express the state of a gas in terms of its volume, pressure, temperature, and quantity, and examine several gas laws, which are empirical relationships among these four variables. You will examine and use the ideal gas equation, PV = nRT. You will learn how to use the ideal gas equation in additional calculations, such as the calculation of the density or molar mass of a gas. You will be able to recognize and utilize the fact that, in a mixture of gases, each gas exerts a pressure that is part of the total pressure. You will relate the macroscopic properties of gases to their behavior at the molecular level. Additionally, you will use kinetic molecular theory to predict the relative rates of effusion and diffusion. Finally, you will consider how real gases deviate from ideal behavior. Because gas molecules have a finite volume, and because attractive forces exist between molecules, the physical properties of gases are all similar. For example, a gas expands spontaneously to fill its container. Consequently, the volume of a gas equals the volume of its container. Gases are also highly compressible. When pressure is applied to a gas, its volume readily decreases. Solids and liquids, on the other hand, do not expand to fill their container and are not readily compressible. Gases also have extremely low densities. Most gases are composed mainly of nonmetallic elements with simple formulas and low molar masses. Two or more gases form a homogeneous mixture, regardless of the identities or relative proportions of the gases. The atmosphere serves as an excellent example. Two or more liquids or two or more solids may or may not form homogeneous mixtures, depending on their chemical nature. For example, when water and gasoline are mixed, the two liquids remain as separate layers. In contrast, the water vapor and gasoline vapors above the liquid form a homogeneous gas mixture. Of the few elements that exist as gases at ordinary temperatures and pressures, helium, neon, argon, krypton, and xenon are monatomic. Hydrogen, nitrogen, oxygen, fluorine, and chlorine are diatomic. Many molecular compounds are gases. Here is a list of a few of them. Notice that all of these gases are composed entirely of nonmetallic elements. Furthermore, they all have simple molecular formulas and therefore, low molar masses. Even though different gaseous substances may have different chemical properties, they behave quite similarly as far as their physical properties are concerned. For example, the nitrogen and oxygen that account for approximately 99% of our atmosphere have very different chemical properties. Oxygen supports human life, but nitrogen does not. To name just one difference, these two components of air behave physically as one gaseous material because their physical properties are essentially identical. Speaking about physical properties, these are the properties that define the state of a gas sample: the temperature, the pressure, the volume, and the amount of gas, which is usually expressed as the number of moles. Let's talk about pressure. Gases exert a pressure on any surface with which they are in contact. The gas in an inflated balloon, for example, exerts a pressure on the inside surface of the balloon. The pressure that a gas exerts is defined as the force divided by the area of the surface on which the force is acting. The gas atoms and molecules of the atmosphere experience a gravitational acceleration due to the gravitational force. Because these particles have such tiny masses, however, their thermal energies of motion, their kinetic energy, override the gravitational forces. So the particles that make up the atmosphere don't pile up at the Earth's surface. Nevertheless, the gravitational force does operate and it causes the atmosphere as a whole to press down on the Earth's surface, creating atmospheric pressure defined as the force exerted by the atmosphere on a given surface area. Now, imagine a column of air one meter square in cross-section extending through the entire atmosphere. This column has a mass of roughly 10,000 kilograms. The downward gravitational force exerted on this column is 1 times 10 to the power of 5 Newton. The SI unit of pressure is the Pascal, named for Blaise Pascal, a French scientist who studied pressure. 1 Pascal is equal to 1 Newton divided by meter squared. A relative pressure unit is the bar. 1 bar is equal to 10 to the power of 5 Pascal, which is equal to 100 kilo Pascal. In the 17th century, many scientists and philosophers believed that the atmosphere had no weight. Evangelista Torricelli, a student of Galileo, proved this untrue. He invented the barometer, which is made from a glass tube more than 760 millimeters long, closed at one end, and completely filled with mercury. When the tube is inverted into a dish full of mercury, some of the mercury flows out, but a column of mercury remains in the tube. Torricelli argued that the mercury surface in the dish experiences the full force of the Earth's atmosphere, which pushes the mercury up the tube. The pressure exerted by the mercury column, downwards due to gravity, equals the atmospheric pressure at the base of the tube. Therefore, the height of the mercury column is a measure of the atmospheric pressure. The changes in atmospheric pressure change in millimeters of mercury or torr, are also pressure units. These units are literally the difference in the heights measured in millimeters of two connected columns of mercury, as seen in the barometer. In this figure, ATM or atmosphere is another non-SI unit for pressure. Here, we have some equivalences between the units. A manometer is used to measure the differences in pressure between atmospheric pressure and that of a gas in a vessel. The barometer, as seen on this last slide, is used to measure the pressure in the atmosphere at any given time. The normal atmospheric pressure at sea level is referred to as standard atmospheric pressure. The standard atmospheric pressure is equal to one atmosphere or 760 torr, which is equal to 760 millimeters of mercury or 101.325 kilo Pascal. Four variables are needed to define the physical conditions or state of a gas: temperature (T), pressure (P), volume (V), and the amount of gas (n), which we useful express as number of moles. As we mentioned before, the equation that expresses the relationship. Between these four paragraphs, the barrels are known as the gas laws because volume is easily measured. The first gas law to be studied expressed the effect of one variable on volume with the remaining two variables held constant. The gas volume increases as the pressure exerted on the gas decreases. Thus, an inflated weather balloon released at the Earth's surface expands as it rises because the pressure of the atmosphere decreases with increasing elevation. The British chemist, Robert Boyle, was the first person to investigate the quantitative relationship between the pressure of a gas and its volume. He found, for example, that a decrease in the pressure of a gas to half its original value causes the volume to double. Conversely, doubling the pressure causes the volume to decrease to half its original volume. Boyle's law, which summarizes this observation, states that the volume of a fixed quantity of gas maintained at a constant temperature is inversely proportional to the pressure. When two measurements are inversely proportional, one gets smaller as the other gets larger. Boyle's law can be expressed mathematically as volume equals a constant times 1 over the pressure. Alternatively, we can rearrange this equation to have the pressure times the volume equal to a constant. The value of the constant depends on the temperature and the amount of gas in the sample. This means that if we compare two conditions, with an initial pressure and volume equal to another condition with a different pressure and volume, the graph of volume versus pressure results in a curve obtained for a given quantity of gas at a fixed temperature. However, a linear relationship is obtained when the volume is plotted versus 1 over P. This figure illustrates that the volume of an inflated balloon increases when the temperature of the gas inside the balloon increases, and decreases when the temperature of the gas decreases. This relationship between gas volume and temperature was discovered in 1787 by two French scientists, Jacques Charles and Joseph Louis Gay-Lussac. Charles' law states that the volume of a fixed amount of gas maintained at constant pressure is directly proportional to its absolute temperature. Some typical volume-temperature data are shown in this figure. Notice that the extrapolated line passes through -273 degrees Celsius. Also, note that the gas is predicted to have a zero volume at this temperature. However, this condition is never realized because all gases liquefy or solidify before reaching this temperature. In 1848, William Thomson, a British physicist known as Lord Kelvin, proposed an absolute temperature scale now known as the Kelvin scale. On this scale, zero Kelvin is the absolute zero and equals -273.15 degrees Celsius. Mathematically, Charles' law takes the form of V = k times the temperature. Alternatively, we can rearrange this equation to have the volume divided by the temperature equal to a constant. The value of the constant depends on the pressure and the amount of gas. The relationship between the quantity of a gas and its volume was obtained through the work of Joseph Louis Gay-Lussac and Amedeo Avogadro. Gay-Lussac observed in 1808 that the volume ratios of gases reacting with one another are in the ratio of small whole numbers. This is very similar to the concept of the number of moles in a chemical equation. For example, two volumes of hydrogen gas reacts with one volume of oxygen gas to form two volumes of water vapor. This is stoichiometry based and this led to three years later, Amedeo Avogadro interpreted the Gay-Lussac's observation by proposing what is known as Avogadro's Law. The volume of a gas, maintained at constant temperature and pressure, is directly proportional to the number of moles of the gas. So, that means that at standard temperature and pressure, one mole of any gas occupies 22.4 liters, which is 6.02 x 10^23 gas molecules. That means that basically the volume equals a constant times N, or the number of moles. We can rearrange this equation by saying that the initial volume divided by N is going to be equal to a constant. And if we have two different conditions, the initial volume divided by the number of moles in the initial state will be equal to the final volume divided by the final number of moles. Here we have three different gases with the same volume, all of them 22.4 liters, the same pressure (one atmosphere), the same temperature (0 degrees), and even though they have different molar mass, we still have the same number of molecules. That means that for the same number of molecules, you're going to have basically the same volume for that gas. The three laws we just examined were obtained by holding two of the four variables (pressure, volume, temperature, and n) constant and seeing how the remaining two variables affect each other. We can express each law as a proportionality relationship, using a symbol that we can read as "is proportional to." So, here the volume is proportional to 1/P (Boyle's Law). Additionally, the volume is directly proportional to the temperature (Charles's Law), and the volume is proportional to N (Avogadro's Law). We can combine these relationships into a general gas law where the volume will be proportional to the number of moles times the temperature divided by P. If we call the proportionality constant R, we can organize this equation and obtain the equation PV = NRT, which is the ideal gas equation, also called the ideal gas law. An ideal gas is a hypothetical gas whose pressure, volume, and temperature relationships are described completely by the ideal gas equation. The term R in the ideal gas equation is the gas constant, which makes the equation an equality, not only a proportion. The value of R and its units depend on the units of pressure, volume, the number of moles, and temperature. The temperature in the ideal gas equation must always be the absolute temperature in kelvins, instead of Celsius. The quantity of gas N is normally expressed in moles. The units chosen for pressure and volume are most often atmosphere and liters, respectively. However, other units can be used in countries other than the United States. The Pascal is the most commonly used unit for pressure. In this table, we can show the numerical value for R in various units. When working with the ideal gas equation, you must choose the form of R in which the units agree with the units of pressure, volume, number of moles, and temperature given in the problem. Now, we will use the ideal gas equation first to define the relationship between the density of a gas and its molar mass, and then to calculate the volumes of gases formed or consumed in chemical reactions. Remember that the density units are mass divided by volume, or mass per unit of volume. We can rearrange the ideal gas equation to obtain similar units. We can rearrange the equation to have mass over volume. So, if we multiply both sides of this equation by the molar mass (remember that the molar mass is mass over moles), we can obtain an equation for density.