CHAPTER 13 Gases 13.1 The Gas Laws The gas laws apply to ideal gases, which are described by the kinetic theory in the following five statements. Gas particles do not attract or repel each other. Gas particles are much smaller than the spaces between them. Gas particles are in constant, random motion. No kinetic energy is lost when gas particles collide with each other or with the walls of their container. All gases have the same kinetic energy at a given temperature. Boyle s Law At a constant temperature, the pressure exerted by a gas depends on the frequency of collisions between gas particles and the container. If the same number of particles is squeezed into a smaller space, the frequency of collisions increases, thereby increasing the pressure. Thus, Boyle s law states that at constant temperature, the pressure and volume of a gas are inversely related. In mathematical terms, this law is expressed as follows. Example Problem 1 Applying Boyle s Law P 1 V 2 A sample of compressed methane has a volume of 648 ml at a pressure of 503 kpa. To what pressure would the methane have to be compressed in order to have a volume of 216 ml? Examine the Boyle s law equation. You need to find, the new pressure, so solve the equation for. V P 1 1 Substitute known values and solve. 648 ml 503 kpa 216 ml 1510 kpa V 2 1
To determine whether your answer is reasonable, notice that the gas is being squeezed to a smaller volume, which requires that the pressure be increased. So, your answer is probably correct. 1. What pressure will be needed to reduce the volume of 77.4 L of helium at 98.0 kpa to a volume of 60.0 L? 2. A 250.0-mL sample of chlorine gas is collected when the barometric pressure is 105.2 kpa. What is the volume of the sample after the barometer drops to 100.3 kpa? 3. A weather balloon contains 59.5 L of helium at sea level, where the atmospheric pressure is 101.3 kpa. The balloon is released from a 4000-m mountaintop where the pressure is 61.7 kpa. What is the volume of the balloon when it is released? 4. Meteorologists want the balloon in problem 3 to float at an altitude of 35,000 m where the pressure is 575 Pa. What volume will the balloon have at that altitude? Charles s Law When the temperature of a sample of gas is increased and the volume is free to change, the pressure of the gas does not increase. Instead, the volume of the gas increases in proportion to the increase in Kelvin temperature. This observation is Charles s law, which can be stated mathematically as follows. Example Problem 2 Applying Charles s Law V 2 A weather balloon contains 5.30 kl of helium gas when the temperature is 12 C. At what temperature will the balloon s volume have increased to 6.00 kl? Start by converting the given temperature to kelvins. (12 273) K 285 K Next, solve the Charles s law equation for the new temperature,. V V 2 ; V V 2 ; 1 2 2
Then, substitute the known values and compute the result. V 6.00 kl 1 285 K 323 K 5.30 kl Finally, convert the Kelvin temperature back to Celsius. New temperature 323 273 50 C To check your answer, take note of the conditions in the problem. The volume of the gas is increasing, so you can predict that the new temperature will be higher than the original temperature. 5. A sample of SO2 gas has a volume of 1.16 L at a temperature of 23 C. At what temperature will the gas have a volume of 1.25 L? 6. A balloon is inflated with 6.22 L of helium at a temperature of 36 C. What is the volume of the balloon when the temperature is 22 C? 7. A student collects a 125.0-mL sample of hydrogen. Later, the sample is found to have a volume of 128.6 ml at a temperature of 26 C. At what temperature was the hydrogen collected? 8. A balloon has a volume of 10,500 L if the temperature is 15 C. If the temperature is 35 C, what will be the volume of the balloon? The gas laws may be combined into a single law, called the combined gas law, that relates two sets of conditions of pressure, volume, and 1temperature by the following equation. P 1 V 2 With this equation, you can find the value of any one of the variables if you know the other five. 3
Example Problem 3 Applying the Combined Gas Law A sample of nitrogen monoxide has a volume of 72.6 ml at a temperature of 16 C and a pressure of 104.1 kpa. What volume will the sample occupy at 24 C and 99.3 kpa? Start by converting the temperatures to kelvins. 16 C (16 273) K 289 K; 24 C (24 273) K 297 K Next, solve the combined gas law equation for the quantity to be determined, the new volume, V 2. P 1 V 2 ; P 1 V 2 ; V 2 P 1 Substitute the known quantities and compute V 2. V 2 P 1 104.1 kpa 72.6 ml 297 K 99.3 kpa 289 K 78.2 ml Finally, determine whether the answer makes sense. The temperature increases, which should cause the gas to expand. The decrease in pressure should cause the volume to expand further. Therefore, the final volume should be somewhat larger than the starting volume. 9. A sample of ammonia gas occupies a volume of 1.58 L at 22 C and a pressure of 0.983 atm. What volume will the sample occupy at 1.00 atm and 0 C? 10. A student collects 285 ml of O2 gas at a temperature of 15 C and a pressure of 99.3 kpa. The next day, the same sample occupies 292 ml at a temperature of 11 C. What is the new pressure of the gas? 11. A balloon is inflated with 2.42 L of helium at a temperature of 27 C. Later, the volume of the balloon has changed to 2.37 L at a temperature of 19 C and a pressure of 99.7 kpa. What was the pressure when the balloon was inflated? Avogadro s Principle In the early nineteenth century, Avogadro proposed the idea that equal volumes of all gases at the same conditions of temperature and pressure contain the same number of particles. An 4
extension of Avogadro s principle is that one mole (6.02 10 23 particles) of any gas at standard temperature and pressure (0 C and 1.00 atm pressure, STP) occupies a volume of 22.4 L. Given that the mass of a mole of any gas is the molecular mass of the gas expressed in grams, Avogadro s principle allows you to interrelate mass, moles, pressure, volume, and temperature for any sample of gas. Example Problem 4 Applying Avogadro s Principle What is the volume of 7.17 g of neon gas at 24 C and 1.05 atm? Start by converting the mass of neon to moles. The periodic table tells you that the atomic mass of neon is 20.18 amu. Therefore, the molar mass of neon is 20.18 g. 1 mol Ne moles Ne 7.17 g Ne 0.355 mol Ne 20.18 g Ne Next, determine the volume at STP of 0.355 mol Ne. 22.4 L Ne V Ne 0.355 mol Ne 1 mol Ne 7.95 L Ne If you needed only the volume at STP, you could stop here. Finally, use the combined gas law equation to determine the volume of the neon at 24 C and 1.05 atm pressure. P 1 V 2 ; P 1 V 2 ; V 2 P 1 V 2 P 1 1.00 atm 7.95 L Ne 297 K 1.05 atm 273 K 8.24 L Ne 12. How many moles of acetylene (C2H2) gas occupy a volume of 3.25 L at STP? 13. Determine the volume of 12.3 g of formaldehyde gas (CH2O) at STP. 14. What is the volume of 1.000 kg of helium gas at 36 C and a pressure of 98.7 kpa? 5
15. What is the mass of 187 L of CO2 gas? The volume is measured at STP. (Hint: Start by determining the number of moles of gas present.) 16. How many moles of silane gas (SiH4) are present in 8.68 ml measured at 18 C and 1.50 atm? (Hint: First determine the volume of the silane at STP.) 13.2 The Ideal Gas Law The pressure, volume, temperature, and number of moles of gas can be related in a simpler, more convenient way by using the ideal gas law. The following is the law s mathematical expression, PV nrt, where n represents the number of moles. The ideal gas constant, R, already contains the molar volume of a gas at STP along with the standard temperature and pressure conditions. The constant R does the job of correcting conditions to STP. You do not have to correct to STP in a separate step as in Example Problem 4. The value of R depends on the units in which the pressure of the gas is measured, as shown below. L kpa L atm L mm Hg R 8.314 0.0821 62.4 mol K mol K mol K These values are all equivalent. Use the one that matches the pressure units you are using. Example Problem 5 Applying the Ideal Gas Law 2 What pressure in atmospheres will 18.6 mol of methane exert when it is compressed in a 12.00-L tank at a temperature of 45 C? As always, change the temperature to kelvins before doing anything else. 45 C (45 273) K 318 K Next, solve the ideal gas law equation for P. PV nrt; P nrt V 6
Substitute the known quantities and calculate P. P nrt V L atm 18.6 mol 0.0821 mol K 12.00 L 318 K 40.5 atm Notice that this pressure makes sense because a large amount of gas is being squeezed into a much smaller space. 17. What is the pressure in atmospheres of 10.5 mol of acetylene in a 55.0-L cylinder at 37 C? 18. What volume does 0.056 mol of H2 gas occupy at 25 C and 1.11 atm pressure? 19. A sample of carbon monoxide has a volume of 344 ml at 85 C and a pressure of 88.4 kpa. Determine the amount in moles of CO present. 7
13.3 Gas Stoichiometry In Chapter 11, you learned how to use moles and molar mass along with a balanced chemical equation to calculate the masses of reactants and products in a chemical reaction. Now that you know how to relate volumes, masses, and moles for a gas, you can do stoichiometric calculations for reactions involving gases. 8
Example Problem 7 Gas Stoichiometry Using Mass Ammonium sulfate can be prepared by a reaction between ammonia gas and sulfuric acid as follows. 2NH 3(g) H 2SO 4(aq) (NH 4) 2SO 4(aq) What volume of NH 3 gas, measured at 78 C and a pressure of 1.66 atm, will be needed to produce 5.00 10 3 g of (NH 4) 2SO 4? First, you need to compute the number of moles represented by 5.00 10 3 g of (NH 4) 2SO 4. Using atomic mass values from the periodic table, you can compute the molar mass of (NH 4) 2SO 4 to be 132.14 g/mol. n (NH4 ) 2 SO 4 5.00 103 g (NH 4 ) 2 SO 4 132.14 g /mol (NH 4 ) 2 SO 4 37.84 mol (NH 4 ) 2 SO 4 Next, determine the number of moles of NH 3 that must react to produce 37.84 mol (NH 4) 2SO 4. n NH3 37.84 mol (NH 4 ) 2 SO 4 2 mol NH 3 1 mol (NH 4 ) 2 SO 4 75.68 mol NH 3 Finally, use the ideal gas law equation to calculate the volume of 75.68 mol NH 3 under the stated conditions. Solve the equation for V, the volume to be calculated. PV nrt; V nrt P Convert the temperature to kelvins, substitute known quantities into the equation, and compute the volume. L atm 75.68 mol NH3 0.0821 nrt mol K 351 K V 1310 L P 1.66 atm Notice that the values for the molar mass of (NH 4) 2SO 4 and the number of moles of NH 3 have more than three significant figures, whereas the calculated volume has only three. When you do a problem in a stepwise way, you should maintain at least one extra significant figure in the intermediate values you calculate. Then, round off values only at the end of the problem. 9
24. A piece of aluminum with a mass of 4.25 g is used to generate hydrogen gas by the following method. 2Al(s) 6HCl(aq) 2AlCl3(aq) 3H2(g) The hydrogen is collected at a temperature of 15 C and a pressure of 94.4 kpa. What volume of hydrogen is produced? 25. Ammonium nitrate decomposes when heated to produce dinitrogen monoxide and water. NH4NO3(s) N2O(g) 2H2O(l) If 12.2 g of NH4NO3 reacts, what volume of NO, measured at 98.5 kpa and 14 C, is produced? 26. Carbon disulfide burns in air to produce carbon dioxide gas and sulfur dioxide gas. CS2(l) 3O2(g) CO2(g) 2SO2(g) Determine the mass of CS2 required to produce 15.7 L of SO2 gas at 25 C and 99.4 kpa. (Hint: This problem is the reverse of Example Problem 7. Start by calculating the moles of SO2 present.) 27. When potassium chlorate is heated, it decomposes to produce potassium chloride and oxygen gas. Write a balanced equation for this reaction, and calculate the mass of potassium chlorate needed to produce 5.00 10 2 ml of oxygen gas at 1.108 atm and 39 C. Chapter 13 Review 28. Describe the spacing of the particles of an ideal gas and the nature of the collisions of these particles. 29. Why does the pressure of a gas increase when the gas is squeezed into a smaller volume? 30. A sample of gas is squeezed until its pressure triples. How is the volume of the gas changed by the squeezing? 31. A sample of gas is heated until its Kelvin temperature doubles. How is the sample s volume changed by the heating? 32. How many moles of gas are there in 1000.0 L of nitrogen gas measured at STP? 33. In the reaction N2(g) 3H2(g) 2NH3(g), what volume of ammonia is formed when 360 ml of H2 gas reacts? Assume that both gases are at STP. 10
Answer Key Chapter 13 1. 126 kpa 2. 262.2 ml 3. 97.7 L 4. 10,500 L 5. 46 C (319 K) 6. 5.94 L 7. 18 C (291 K) 8. 8680 L 9. 1.44 L 10. 95.6 kpa 11. 1.00 10 2 kpa 12. 0.145 mol 13. 9.18 L 14. 6420 L 15. 367 g 16. 5.45 10 4 mol 17. 4.86 atm 18. 1.2 L 19. 0.0102 mol or 1.02 10 2 mol 24. 5.99 L 25. 3.69 L 26. 24.0 g 27. 2KClO3(s) 2KCl2(s) 3O2(g); 1.77 g 11
Answer Key (continued) Chapter 13 Review 28. The spacing is much greater than the size of the particles. No kinetic energy is lost in the collisions. 29. Squeezing concentrates the gas particles in a smaller space. As a result, they collide with each other and the walls of their container more often. 30. The volume is reduced to one-third the original volume. 31. The volume doubles. 32. 44.6 mol 33. 240 ml 12