Section 11.9: The Gas Laws—Boyle’s Law, Gay Lussac's Law, and the Combined Gas Law Mini Investigation: Modelling a Lung, page 555 Answers may vary. Sample answers: A. Gas pressure is directly proportional to the number of collisions between gas entities and the container. If the number of gas entities is constant and we expand the volume of the container, there will be fewer collisions with the walls of the container. This means the pressure inside the balloon will be reduced as the volume increases. B. When we pull down on the balloon cover, the balloon inside inflates. The balloon cover acts like the diaphragm in our chest cavity and the balloon inside the plastic cup acts like our lungs. When we pull downward on the balloon, it creates a region of low pressure and outside air moves into the balloon inside the cup. The air moves into the balloon because the air pressure outside is greater than the pressure inside. This is similar to inhalation. In our chest cavity, our diaphragm lowers creating low pressure in our lungs due to increased volume and air moves into our lungs. C. The model demonstrates inhalation showing the relationship between changes in volume and inhalation but the demonstration does not show exhalation. The single balloon inside the cup simulates the functioning two lungs. The details of lung anatomy are not shown. Mini Investigation: Cartesian Diver, page 556 Answers may vary. Sample answers: A. The eye dropper contains some water and a bubble of air. The density of the dropper with the water and the air bubble is the same as the water in the pop bottle, so the diver initially floats. B. As we apply pressure to the pop bottle we compress the air in the eye dropper. This causes the air entities to compress, and more water moves into the diver. Water moving into the eyedropper causes the eyedropper to become heavier and it sinks. C. Submarines are composed of an inner and outer steel shell. In between is a region called the hull. When floating on the surface, the hull is filled mostly with air. This is similar to the Cartesian diver floating. To allow the submarine to dive the hull is filled with water, which makes the submarine heavier and it sinks. The Cartesian diver sinks when more water is introduced into the eyedropper by applying pressure. Tutorial 1 Practice, page 5591. Given: initial pressure, P1 = 446 kPa initial volume, V1 = 8.25 L final volume, V2 = 12 L The amount of gas and the temperature remain constant. Required: final pressure, P2 Analysis: Use Boyle’s law.
P
1V
1= P
2V
2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
Step 2. Substitute given values (including units) into the equations and solve.
P2=
446 kPa ! 8.25 L
12 L
P2= 307 kPa
Statement: A pressure of 307 kPa must be applied to the gas when it occupies 12 L. 2. Given: final pressure, P2 = 2.84 kPa initial volume, V1 = 1.75 L final volume, V2 = 6.50 L The amount of gas and the temperature remain constant. Required: initial pressure, P1 Analysis: Use Boyle’s law.
P
1V
1= P
2V
2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
P1=
P2V
2
V1
Step 2. Substitute given values (including units) into the equations and solve.
P1=
2.84 kPa ! 6.50 L
1.75 L
P1= 10.5 kPa
Statement: The original pressure of the gas was 10.5 kPa. 3. Given: initial pressure, P1 = 85 kPa initial temperature, t1 = 25 °C final temperature, t2 = –11.4 °C The volume and amount of gas remain constant. Required: final pressure, P2 Analysis: Use Gay-Lussac's law.
P1
T1
=P
2
T2
Solution: Step 1. Convert temperature values to kelvins.
Step 2. Rearrange the equation to isolate the unknown variable.
P2=
P1T
2
T1
Step 3. Solve the equation (including units).
P2=
85 kPa ! 261.6 K
298 K
P2= 75 kPa
Statement: The pressure of the gas inside the soccer ball is 75 kPa. Tutorial 2 Practice, page 5601. Given: initial pressure, P1 = 92.5 kPa initial volume, V1 = 775 mL initial temperature, t1 = –28 °C final volume, V2 = 825 mL final temperature, t2 = 15 °C The amount of gas remains constant. Required: final pressure, P2 Analysis: Use the combined gas law.
P1V
1
T1
=P
2V
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= –28 + 273
T1= 245 K
T2= t
2+ 273
= 15+ 273
T2= 288 K
Step 2. Rearrange the equation to isolate the unknown variable.
P2=
P1V
1T
2
V2T
1
Step 3. Substitute given values (including units) into the equation and solve.
P2=
92.5 kPa ! 775 mL ! 288 K
825 mL ! 245 K
P2= 102 kPa
Statement: The pressure at the bottom of the mountain is 102 kPa.
2. Given: initial pressure, P1 = 99.0 kPa initial volume, V1 = 2.75 L initial temperature, t1 = 21.0 °C final pressure, P2 = 105 kPa final temperature, t2 = 71.0 °C Amount of gas remains constant. Required: final volume, V2 Analysis: Use the combined gas law.
P1V
1
T1
=P
2V
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 21.0 + 273
T1= 294 K
T2= t
2+ 273
= 71.0 + 273
T2= 344 K
Step 2. Rearrange the equation to isolate the unknown variable.
V2=
P1V
1T
2
P2T
1
Step 3. Substitute given values (including units) into the equation and solve.
V2=
99.0 kPa ! 2.75 L ! 344 K
105 kPa ! 294 K
V2= 3.03 L
Statement: The final volume of the gas is 3.03 L. 3. Given: initial pressure, P1 = 253 kPa initial volume, V1 = 450 mL initial temperature, t1 = 15 °C final pressure, P2 = 405 kPa final volume, V2 = 310 mL Amount of gas remains constant. Required: final temperature, t2 Analysis: Use the combined gas law.
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 15+ 273
T1= 288 K
Step 2. Rearrange the equation to isolate the unknown variable.
T2=
P2V
2T
1
P1V
1
Step 3. Substitute given values (including units) into the equation and solve.
T2=
405 kPa ! 310 mL ! 288 K
253 kPa ! 450 mL
T2= 317.6 K [two extra digits carried]
Step 4. Convert temperature values back to Celsius.
T2= t
2+ 273
t2= T
2! 273
= 318 ! 273
t2= 45 ºC
Statement: The final temperature is 45 °C. Research This: Investigating Gas Law Simulations, page 551 Answers may vary. Sample answers: A. Three important criteria for the gas law simulations are ease-of-use, efficient display of data, and accuracy of the data in describing the gas law. These criteria were chosen because these reflect some of the advantages of using a computer simulation over a textbook or other resource. B. I chose the “Gas Properties” simulation hosted by the University of Colorado at Boulder because it simulated a wide range of gas phenomena as well as scoring the highest out of all the simulations for our three chosen criteria. C. I would improve the simulations’ instructions to make them clearer, more easily found, and have more information. I would also improve the visuals of the simulation. D. Using simulations helps to visualize how different quantities are related and how they change with time. Using a simulation allows you to see how changing one variable affects other variables instantly, and having a corresponding visual aid (such as molecules being pumped into a box representing the pressure variable) makes it easier to understand why the affected variables change in the way they do.
2. Given: initial volume, V1 = 45.2 L initial temperature, T1 = 280 K final temperature, T2 = 560 K The initial pressure is equal to final pressure. The amount of gas remains constant. Required: final volume, V2 Analysis: Use Charles’ law. Use 1 2
1 2
V V
T T= to simplify the equation.
V1
T1
=V
2
T2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
V2=
V1T
2
T1
Step 2. Substitute given values (including units) into the equation and solve.
V2=
45.2 L ! 560 K
280 K
V2= 9.0 !10 L
Statement: The final volume of the nitrogen gas is 9.0 × 10 L.
3. Given: initial pressure, P1 = 340 kPa initial temperature, t1 = 15 °C final temperature, t2 = –15 °C The volume and amount of gas remain constant. Required: final pressure, P2 Analysis: Use Gay-Lussac’s law.
P1
T1
=P
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 15+ 273
T1= 288 K
T2= t
2+ 273
= –15+ 273
T2= 258 K
Step 2. Rearrange the equation to isolate the unknown variable.
P2=
P1T
2
T1
Step 3. Solve the equation (including units).
P2=
340 kPa ! 258 K
288 K
P2= 3.0 !10
2 kPa
Statement: The new pressure will be 3.0 × 102 kPa. 4. (a) Given: initial pressure, P1 = 101.3 kPa initial volume, V1 = 3.50 L initial temperature, T1 = 320 K final volume, V2 = 8.40 L final temperature, T2 = 320 K The amount of gas and the temperature remain constant. Required: final pressure, P2 Analysis: Use Boyle’s law.
P
1V
1= P
2V
2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
Step 2. Substitute given values (including units) into the equations and solve.
P2=
101.3 kPa ! 3.50 L
8.40 L
P2= 42.2 kPa
Statement: The final pressure is 42.2 kPa. (b) Given: initial pressure, P1 = 210 kPa initial temperature, T1 = 415 K final pressure, P2 = 420 kPa final volume, V2 = 120 mL final temperature, T2 = 415 K The amount of gas and the temperature remain constant. Required: initial volume, V1 Analysis: Use Boyle’s law.
P
1V
1= P
2V
2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
V1=
P2V
2
P1
Step 2. Substitute given values (including units) into the equations and solve.
V1=
420 kPa ! 120 mL
210 kPa
V1= 240 mL
Statement: The initial volume is 240 mL. (c) Given: initial pressure, P1 = 720 mm Hg initial volume, V1 = 345 mL initial temperature, T1 = 420 K final pressure, P2 = 620 mm Hg final temperature, T2 = 640 K The amount of gas remains constant. Required: final volume, V2 Analysis: Use the combined gas law.
P1V
1
T1
=P
2V
2
T2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
Step 2. Substitute given values (including units) into the equation and solve.
V2=
720 mm Hg ! 345 mL ! 640 K
620 mm Hg ! 420 K
V2= 610 mL
Statement: The final volume, V2, is 610 mL. 5. Given: initial pressure, P1 = 130 kPa initial temperature, t1 = 6.50 °C final temperature, t2 = 670.0 °C The volume and amount of gas remain constant. Required: final pressure, P2 Analysis: Use Gay-Lussac’s law.
P1
T1
=P
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 6.50 + 273
T1= 279.50 K
T2= t
2+ 273
= 670.0 + 273
T2= 943.0 K
Step 2. Rearrange the equation to isolate the unknown variable.
P2=
P1T
2
T1
Step 3. Solve the equation (including units).
P2=
130 kPa ! 943.0 K
279.50 K
P2= 4.4 !10
2 kPa
Statement: The new pressure of the neon gas is 4.4 × 102 kPa. 6. Given: initial pressure, P1 = 95 kPa initial temperature, t1 = 22 °C final pressure, P2 = 350 kPa The volume and amount of gas remain constant. Required: final temperature, t2 Analysis: Use Gay-Lussac’s law.
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 22 + 273
T1= 295 K
Step 2. Rearrange the equation to isolate the unknown variable.
T1=
P1T
2
P2
Step 3. Solve the equation (including units).
T2=
350 kPa ! 295 K
95 kPa
T2= 1.087 ! 10
3 K
Step 4. Convert temperatures values back to Celsius.
T2= t
2+ 273
t2= T
2! 273
= 1.087 "103! 273
t2= 8.1"10
2 °C
Statement: The tank can sustain a maximum temperature of 8.1 × 102 °C. 7. Given: initial pressure, P1 = 101.3 kPa initial temperature, t1 = 20 °C final temperature, t2 = –15 °C The volume and amount of gas remain constant. Required: final pressure, P2 Analysis: Use Gay-Lussac’s law.
P1
T1
=P
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 20 + 273
T1= 293 K
T2= t
2+ 273
= –15+ 273
T2= 258 K
Step 2. Rearrange the equation to isolate the unknown variable.
Statement: The final pressure of the nitrogen gas is 89.2 kPa. 8. Given: initial pressure, P1 = 152 kPa final temperature, t2 = 200 °C The initial temperature, t1, is standard ambient temperature, which is 25 °C. The volume and amount of gas remain constant. Required: final pressure, P2 Analysis: Use Gay-Lussac’s law.
P1
T1
=P
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 25+ 273
T1= 298 K
T2= t
2+ 273
= 200 + 273
T2= 473 K
Step 2. Rearrange the equation to isolate the unknown variable.
P2=
P1T
2
T1
Step 3. Solve the equation (including units) to determine if the final pressure is greater than 253 kPa.
P2=
152 kPa ! 473 K
298 K
P2= 241 kPa
Statement: The pressure of the gas at 200 °C is 241 kPa. This is not greater than the 253 kPa limit, so yes, the glassware can withstand the gas being heated to this temperature. 9. (a) I would expect the tire pressure to decrease. As the temperature decreases the motion of entities will decrease. This will reduce the volume the entities occupy and reduce the number of collisions with the tire surface, thereby reducing the pressure. (b) Given: initial pressure, P1 = 227.5 kPa initial temperature, t1 = 10 °C final temperature, t2 = –5 °C The volume and amount of gas remain constant. Required: final pressure, P2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 10 + 273
T1= 283 K
T2= t
2+ 273
= –5+ 273
T2= 268 K
Step 2. Rearrange the equation to isolate the unknown variable.
P2=
P1T
2
T1
Step 3. Solve the equation (including units).
P2=
227.5 kPa ! 268 K
283 K
P2= 2.2 !10
2 kPa
Statement: The new tire pressure is 2.2 × 102 kPa. 10. Given: initial pressure, P1 = 125 kPa initial volume, V1 = 1250 cm3 final pressure, P2 = 97 kPa The amount of gas and the temperature remain constant. Required: final volume, V2 Analysis: Use Boyle’s law.
P
1V
1= P
2V
2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
V2=
P1V
1
P2
Step 2. Substitute given values (including units) into the equations and solve.
V2=
125 kPa ! 1250 cm3
97 kPa
V2= 1.6 !10
3 cm
3
Statement: The volume of the balloon at the new altitude would be 1.6 × 103 cm3.
11. Given: initial pressure, P1 = 2.7574 × 104 kPa initial volume, V1 = 4.50 × 104 L final volume, V2 = 6.00 × 104 L The amount of gas and the temperature remain constant. Required: final pressure, P2 Analysis: Use Boyle’s law.
P
1V
1= P
2V
2
Solution: Step 1. Rearrange the equation to isolate the unknown variable.
P2=
P1V
1
V2
Step 2. Substitute given values (including units) into the equations and solve.
P2=
2.7574 !104 kPa ! 4.50 !10
4 L
6.00 !104 L
P2= 2.07 !10
4 kPa
Statement: The new pressure would be 2.07 × 104 kPa. 12. (a) Answers may vary. Some examples of gases stored under pressure are: oxygen for patients, helium for balloons, and acetylene torches for welding. (b) These gases need to be stored safely and away from incompatible materials. The cylinders must be well maintained and container seals must be in good condition. The containers must be transported carefully. (c) These gases need to be stored away from incompatible materials, since they could react if any gas leaks. They need to be stored safely to minimize the risk of fires, spills, or leaks. In the case of fire, as the temperature increases, the gases would expand and the pressure inside the cylinder would increase. If the pressure inside the storage tanks exceeds the limits of the cylinder an explosion could occur. They should be stored away from busy areas. If the tanks are damaged they could leak or explode. The gas cylinders must be checked regularly to ensure that all valves are working properly. The cylinders must be inspected for leaks or any dents that could release gas. Gases that escape could react with entities in the environment. These Gases must be transported carefully so the containers are not damaged, otherwise release of gas or a container explosion could occur. 13. Given: initial pressure, P1 = 685 kPa initial volume, V1 = 8.0 mL initial temperature, t1 = 12 °C final pressure, P2 = 99 kPa final temperature, t2 = 26 °C The amount of gas remains constant. Required: final volume, V2 Analysis: Use the combined gas law.
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 12 + 273
T1= 285 K
T2= t
2+ 273
= 26 + 273
T2= 299 K
Step 2. Rearrange the equation to isolate the unknown variable.
V2=
P1V
1T
2
P2T
1
Step 3. Substitute given values (including units) into the equation and solve.
V2=
685 kPa ! 8.0 mL ! 299 K
99 kPa ! 285 K
V2= 58 mL
Statement: The volume of the gas bubble when it reaches the surface is 58 mL. 14. Given: initial pressure, P1 = 112 kPa initial volume, V1 = 5.50 L initial temperature, t1 = 25 °C final pressure, P2 = 32.0 kPa final temperature, t2 = –34 °C The amount of gas remains constant. Required: final volume, V2 Analysis: Use the combined gas law.
P1V
1
T1
=P
2V
2
T2
Solution: Step 1. Convert temperature values to kelvins.
T1= t
1+ 273
= 25+ 273
T1= 298 K
T2= t
2+ 273
= !34 + 273
T2= 239 K
Step 2. Rearrange the equation to isolate the unknown variable.
Step 3. Substitute given values (including units) into the equation and solve.
V2=
112 kPa ! 5.50 L ! 239 K
32.0 kPa ! 298 K
V2= 15.4 L
Statement: The final volume of the balloon is 15.4 L. 15. Given: initial pressure, P1 = 116 kPa initial volume, V1 = 893 mL initial temperature, t1 = 44 °C final pressure, P2 = 102 kPa final volume, V2 = 1.03 L The amount of gas remains constant. Required: final temperature, t2 Analysis: Use the combined gas law.
P1V
1
T1
=P
2V
2
T2
Solution: Step 1. Convert temperature values to kelvins and volume values to litres.
T1= t
1+ 273
= 44 + 273
T1= 317 K
V2= 893 mL
= 893 mL !1 L
1000 mL
V2= 0.893 L
Step 2. Rearrange the equation to isolate the unknown variable.
T2=
P2V
2T
1
P1V
1
Step 3. Substitute given values (including units) into the equation and solve.
