Gas Volumes and the Ideal Gas Law

SECTION 11.3 Gas Volumes and the Ideal Gas Law Section 2 presented laws that describe the relationship between the pressure, temperature, and volume of a gas. The volume of a gas is also related to the number of moles in gas. Using this relationship and the combined gas law, a single equation can be derived to describe all of the characteristics of a gas, called the ideal gas law. Key Terms Gay-Lussac s law of combining volumes of gases Avogadro s law standard molar volume of a gas ideal gas law ideal gas constant Gases react in whole number ratios. In the early nineteenth century, Joseph Gay-Lussac noticed a relationship between the volume of the reacted gases in a chemical reaction and the volume of the produced gases. For example, consider the synthesis of water vapor. hydrogen gas + oxygen gas water vapor Critical Thinking 1. Explain Which equation on this page is evidence that volume is not conserved in a chemical reaction? Why? 2 L () 1 L (1 volume) 2 L () The 2:1:2 relationship for this reaction also applied to other proportions of volume, such as 600 L, 300 L, and 600 L. Gay-Lussac noticed that simple relationships existed for many other reactions, such as the following: hydrogen gas + chlorine gas hydrogen chloride gas 1 L (1 volume) 1 L (1 volume) 2 L () Gay Lussac s law of combining volumes of gases states that the volumes of gaseous reactants and products can be expressed as ratios of small whole numbers. READING CHECK 2. In a lab, 400 ml of hydrogen gas reacts with 400 ml of chlorine gas. What product would form and how much of the product would form? 359

Equal volumes of gases under the same conditions contain equal numbers of molecules. Gay-Lussac s results contradicted Dalton s theory of the atom, which was still the leading theory at the time. Dalton believed that atoms are indivisible. He also believed that gaseous elements were isolated single atoms. The first reaction on the previous page shows that it is possible for two gases with a combined volume of 3 L to produce a gas with less volume. Therefore, either 1 L of oxygen gas has the same number of particles as 2 L of hydrogen gas, or one of the gases contained particles with multiple atoms. In 1811, Amedeo Avogadro developed a theory to explain Gay-Lussac s simple law. He rejected Dalton s idea that elements in a chemical reaction are always in the form of single atoms. He reasoned that the elements could consist of molecules that contain multiple atoms. CONNECT The inflation of automobile air bags results from a rapid series of chemical reactions producing a volume of nitrogen gas. An igniter starts a series of reactions in which two nitrogen-containing solids produce molecules of nitrogen. In as few as 40 milliseconds (0.04 s) the air bag fills to prevent impacts with the dashboard or steering wheel. Avogadro also introduced an idea known today as Avogadro s law. Avogadro s law states that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. In other words, the volume of a gas varies directly with the number of molecules in the gas. This relationship can be expressed using the equation V = kn where n is the amount of gas in moles and k is a constant. READING CHECK 3. For a gas at constant pressure and temperature, the number of moles of the gas and the volume of the gas are proportional. Hydrogen molecule Oxygen molecule Carbon dioxide molecule At the same temperature and pressure, balloons of equal volume have equal numbers of molecules, regardless of which gas they contain. 1 mol H 2 at STP = 22.4 L 1 mol O 2 at STP = 22.4 L 1 mol CO 2 at STP = 22.4 L 360 CHAPTER 11

Hydrogen molecules combine with chlorine molecules in a 1:1 volume ratio to produce of hydrogen chloride. Hydrogen gas 1 Volume 1 Molecule + Chlorine gas 1 Volume 1 Molecule Hydrogen chloride gas Recall that 1 L of hydrogen gas combines with 1 L of chlorine gas to form 2 L of hydrogen chloride gas. According to Avogadro s law, there are twice as many particles in hydrogen chloride as there are in the other two gases. Since the compound must have at least two atoms, the reactants must contribute at least two atoms each to the reaction. The simplest equation that also satisfies Avogadro s law is shown. H 2 (g) + C l 2 (g) 2HCl(g) 1 volume 1 molecule 1 mol 1 volume 1 molecule 1 mol 2 mol Experiments eventually showed that all elements that are gases near room temperature, except for the noble gases, are diatomic molecules in their most common form. Therefore, the equation above, using Gay-Lussac s results and Avogadro s reasoning, is correct. The reaction for water is also consistent with Avogadro s law and the idea that gaseous elements are diatomic. Gay-Lussac s results imply that the simplest combination of reactants is two diatomic molecules of hydrogen and one diatomic molecule of oxygen. If the product is to satisfy a 2:1:2 ratio, then it must be a molecule with two hydrogen atoms and one oxygen atom in each molecule. 2 H 2 (g) + O 2 (g) 2 H 2 O(g) 2 mol 1 volume 1 molecule 1 mol 2 mol Critical Thinking 4. Compare Order the following samples of gas from the gas with the smallest volume to the largest volume: (a) 1.8 mol of H 2 gas, (b) 2.3 mol of As H 3 gas, (c) 4 mol of Ar gas, (d) 1.7 mol of H 2 O gas. 361

All gases have a volume of 22.4 L under standard conditions. Earlier, you learned that one mole of a molecular substance contains 6.022 10 23 molecules. For example, one mole of oxygen gas contains 6.022 10 23 molecules, each of which consists of two oxygen atoms. One mole of oxygen gas has a mass of 32.00 g. On the other hand, one mole of helium gas has 6.022 10 23 atoms and a mass of 4.003 g. According to Avogadro s law, one mole of any gas will occupy the same volume as one mole of any other gas at the same temperature and pressure. Even though oxygen gas consists of diatomic molecules that are 8 times the mass of the monatomic helium atoms, one mole of each gas at STP will occupy the same volume. The volume occupied by one mole of a gas at STP is known as the standard molar volume of a gas. It has been calculated as 22.414 10 L, which for most calculations rounds to 22.4 L. You can use the standard molar volume of a gas as a conversion factor for any gas at STP. The first conversion factor listed at the right can be used to convert moles to liters, and the second conversion factor can be used to convert from liters to moles.! Remember The number 6.022 10 23 is also known as Avogadro s number. Conversion Factors 22.4 L 1 mol 22.4 1 mol L moles-liters liters-moles PRACTICE A. At STP, what is the volume of 7.08 mol of nitrogen gas? 7.08 mol = 7.08 mol = L B. A sample of hydrogen gas occupies 14.1 L at STP. How many moles of the gas are present? 14.1 L = 14.1 L = mol C. How many molecules are in 5.60 L of oxygen gas? 5.60 L = 5.60 L = molecules 362 CHAPTER 11

In a chemical equation, the coefficients can indicate moles, molecules, or volume. Reaction stoichiometry is the study of the relationships between the substances that participate in a chemical reaction. In the chapter Stoichiometry, the simple molar ratios given by a chemical equation were used to calculate the mass relationships between the various products and reactants. For reactions involving gases, Avogadro s law can also be used to calculate the relative volumes of two or more gases, assuming the pressure and temperature conditions remain the same. Avogadro s law implies that the molar ratios established in a chemical equation also apply to the volume ratios of the gases in the equation. For example, consider the reaction of carbon monoxide gas with oxygen gas to form carbon dioxide gas. 2CO(g) + O 2 (g) 2C O 2 (g) 2 mol 1 volume 1 molecule 1 mol 2 mol This equation states that the number of moles of the gases that participate in the reaction are in a 2:1:2 ratio. The volumes of the gases are also in this same 2:1:2 ratio. That means that the relationships between any two of the gases can be expressed in the following ways. Ratio of CO to O 2 : 1 volume O 2 or C O CO 1 volume O 2 Ratio of CO to CO 2 : CO C O 2 or CO 2 CO Ratio of O 2 to CO 2 : 1 volume O 2 CO 2 or C O 2 1 volume O 2 PRACTICE D. What is the ratio of the volume of oxygen gas to the volume of water vapor in the following reaction? C 3 (g) + 5 O 2 (g) 3C O 2 (g) + 4 H 2 O(g) 363

SAMPLE PROBLEM Propane, C 3, is a gas that is sometimes used as a fuel for cooking and heating. The complete combustion reaction of propane occurs according to the following balanced equation. C 3 (g) + 5 O 2 (g) 3C O 2 (g) + 4 H 2 O(g) Assuming that all volumes are measured at the same temperature and pressure: a. What will be the volume, in liters, of oxygen required for the complete combustion of 0.350 L of propane? b. What will be the volume of carbon dioxide produced in the reaction? SOLUTION 1 ANALYZE Determine the information that is given and unknown. Given: balanced chemical equation, V of C 3 = 0.350 L Unknown: a. V of O 2 in L, b. V of CO 2 in L 2 PLAN Describe how to find the unknown values. Since the temperature and pressure conditions do not change, the volume ratios are equal to the mole ratios of the substances. These ratios can be used as conversion factors to get from the volume of C 3 to the volumes of O 2 and C O 2. 3 SOLVE Calculate the unknowns from the given information. a. The volume ratio of C 3 gas to O 2 gas is 1:5. 0.350 L C 3 = 0.350 L C 3 5 L O 2 1 L C 3 = 1.75 L O 2 b. The volume ratio of C 3 gas to C O 2 gas is 1:3. 0.350 L C 3 = 0.350 L C 3 3 L C O 2 1 L C 3 = 1.05 L C O 2 4 CHECK YOUR WORK Check the answer to see if it is reasonable. The answers are correctly expressed to three significant figures. The answers are reasonable because the chemical equation indicates that there should be a larger volume of oxygen than carbon dioxide, and a larger volume of carbon dioxide than propane. 364 CHAPTER 11

PRACTICE E. Assuming all volume measurements are made at the same temperature and pressure, what volume of H 2 is needed to react completely with 4.55 L of O 2 to produce water vapor? Balanced equation for the reaction: H 2 (g) + O 2 (g) H 2 O(g) 4.55 L O 2 = 4.55 L O 2 = F. What volume of oxygen gas is needed to react completely with 0.626 L of CO to form CO 2? Assume that all volume measurements are made at the same temperature and pressure. Balanced equation for the reaction: CO(g) + O 2 (g) CO 2 (g) 0.626 L CO = 0.626 L CO = G. Nitric acid can be produced by the reaction of N O 2 with water, according to the following balanced equation. 3N O 2 (g) + H 2 O(l) 2HN O 3 (l) + NO(g) If 708 L of N O 2 gas react with water, what volume of NO gas will be produced? Assume the gases are measured under the same conditions before and after the reaction. 708 L N O 2 = 708 L N O 2 = H. If 29.0 L of methane, C H 4, undergoes complete combustion at 0.961 atm and 140 C, how many liters of each product would be present at the same temperature and pressure? Balanced equation for the reaction: CH 4 (g) + O 2 (g) C O 2 (g) + H 2 O(g) 29.0 L CH 4 = 29.0 L CH 4 = 29.0 L CH 4 = 29.0 L CH 4 = 365

Pressure, volume, and temperature are related to the number of moles of a gas. Avogadro s law states that the number of molecules in a gas is proportional to its volume. The law can be stated as the equation n = Vk, where k is a constant. Just like Boyle s law, Charles s law, and Gay-Lussac s law, Avogadro s law is the relationship between two quantities of a gas and a constant. The other three gas laws were used to form the combined gas law, relating three different variables: pressure, temperature, and volume. When Avogadro s law is also included, the relationship can be expanded to include the number of molecules in a gas. The ideal gas law is the mathematical relationship among pressure, volume, temperature, and the number of moles in a gas, and is given by this equation, where the quantity R is a constant. PV = nrt Other gas laws can be derived from this equation by holding some variables constant. For example, if the number of moles of the gas is held constant, then the combined-gas law results. If the pressure and temperature are held constant, then Avogadro s law results. The ideal gas law shows that at least one quantity will change when the number of molecules in a gas changes. If more gas is added to a container at a fixed temperature, the pressure will increase. If more molecules are added to a flexible container at constant temperature and pressure, then the volume will increase.! Remember Boyle s law: PV = k Charles s law: V T = k Gay-Lussac s law: P T = k Avogadro s law: n V = k Critical Thinking 5. Which gas law is derived from the ideal gas law if the number of moles and the temperature are held constant? (a) When volume and temperature are constant, pressure increases as the number of molecules increases. (b) When pressure and temperature are constant, volume increases as the number of molecules increases. Temperature Pressure Temperature Pressure 0 Gas molecules added 0 Gas molecules added (a) (b) 366 CHAPTER 11

The ideal gas law relates pressure to volume to temperature. One major difference between the ideal gas law and the combined gas law is the nature of the constant. The value of k in the combined gas law stays the same when the conditions of a particular sample of gas change. However, its value changes for different samples. The combined gas law can only be used to compare two sets of measurements for the same sample. The constant R in the ideal gas law is known as the ideal gas constant. Its value is the same for every sample of gas. Therefore, the ideal gas law can be used to determine an unknown value from just one set of measurements. The value of R can be calculated by solving for R in the ideal gas equation, and then substituting values for one mole of an ideal gas at STP. The result is the following equation. R = nt PV (1 atm)(22.414 10 L) = = 0.082 058 L atm (1 mol)(273.15 K) mol K For the calculations in this book, use the rounded value 0.0821 L atm/(mol K) TIP Other values of R for different units are shown in the table. Numerical Values of Gas Constant, R Unit of R Numerical value of R Unit of P Unit of V Unit of T Unit of n L mmhg 62.4 mm Hg L K mol mol K L atm 0.0821 atm L K mol mol K J mol K 8.314 Pa m 3 K mol mol K L kpa 8.314 kpa L K mol Note: 1 L atm = 101.325 J; 1 J = 1 Pa m 3 Finding P, V, T, or n from the Ideal Gas Law The ideal gas law can be applied to determine the current conditions of a gas sample if three of the four variables are known. It can also be used to calculate the molar mass or density of a gas sample. Be sure to match the units in all of the given values to the correct value for R before applying the ideal gas law. READING CHECK 6. To use the ideal gas law to determine the number of moles in a gas sample, what three quantities must be measured? 367

SAMPLE PROBLEM What is the pressure in atmospheres exerted by a 0.500 mol sample of nitrogen gas in a 10.0 L container at 298 K? SOLUTION 1 ANALYZE Determine the information that is given and unknown. Given: V = 10.0 L, n = 0.500 mol, T = 298 K Unknown: P in atm 2 PLAN Write the equation that can be used to determine the unknown. The ideal gas law can be used because three quantities are given. Rearranging to solve for P gives this equation. P = nrt V 3 SOLVE Substitute known values to find the value of the unknown. Because the given values are in moles, liters, and kelvins, use R = 0.0821 L atm/(mol K). No conversions are necessary. P = nrt (0.500 0.0821 L atm mol) ( V = mol K ) (298 K) 10.0 L = 1.22 atm 4 CHECK YOUR WORK Check the answer to see if it is reasonable. The units cancel as desired, and the final answer is expressed in the correct number of significant figures. The answer is also close to an estimated value of (0.5 0.1 300) = 1.5 10 PRACTICE I. What pressure, in atmospheres, is exerted by 0.325 mol of hydrogen gas in a 4.08 L container at 35 C? P = nrt V = = 368 CHAPTER 11

SECTION 11.3 REVIEW VOCABULARY 1. State Avogadro s law and explain its significance. REVIEW 2. What volume (in ml) at STP will be occupied by 0.0035 mol of CH 4? 3. What would be the units for R if P is in pascals, T is in kelvins, V is in liters, and n is in moles? 4. A 4.44 L container holds 15.4 g of oxygen at 22.55 C. What is the pressure? Critical Thinking 5. ANALYZING DATA Nitrous oxide is sometimes used as a source of oxygen gas: 2 N 2 O(g) 2 N 2 (g) + O 2 (g) a. What volume of each product will be formed from 2.22 L N 2 O? b. At STP, what is the density of the product gases when they are mixed? 369