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Chapter 11

The Behaviour of Gases Solutions for Practice Problems Section 11.2 Student Textbook page 305 1. Problem

1. A 50 cm3 sample of nitrogen gas is collected at 101.3 kPa. If the volume is reduced to 5.0 cm3 , and the temperature remains constant, what will be the final pressure of the nitrogen? What Is Required? You must find the final pressure. What Is Given? Vi = 50.0 cm3 Vf = 50.0 cm3

Pi = 101.3 kPa

Plan Your Strategy Use either the algebraic method or the ratio method. For the algebraic method, apply the formula Pi Vi = Pf Vf . Substitute the known values into this equation and solve for Pf . Act on Your Strategy Pi Vi = Pf Vf 101.3 kPa × 50.0 cm3 = Pf × 5.0 cm3 Pf = 1.0 × 103 kPa Check Your Solution The final answer has the correct unit and number of significant figures. The volume decreases so it is expected that the pressure must increase. This answer seems to be reasonable. 2. Problem

A weather balloon has a volume of 1 000 L at a pressure of 740.0 torr. The balloon rises to a height of 1 000 m where the atmospheric pressure is measured as 450.0 torr. Assuming there is no change in temperature, what is the final volume of the weather balloon? What Is Required? You must find the final volume of the balloon. What Is given? Vi = 1 000 L

Pi = 740.0 torr Pf = 450.0 torr

Plan Your Strategy Use either the algebraic method or the ratio method.

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Using the ratio method, since the pressure is decreasing, the volume must increase, according to Boyle’s law. Therefore the initial volume must be multiplied by a pressure ratio that is greater than one. Act on Your Strategy Vf = Vi × pressure ratio torr = 1 644 L Vf = 1 000 L × 740.0 450.0 torr Check Your Solution The final answer has the correct unit and number of significant figures. The pressure decreases so it is expected that the volume must increase. This answer seems to be reasonable. 3. Problem

A 45.0 cm3 sample of nitrogen is collected at 1.0 atm. The nitrogen is compressed to a pressure of 10.0 atm. What is the final volume of the nitrogen if the temperature remains constant? What Is Required? You must find the final volume of the nitrogen. What Is Given? Vi = 1 000 L

Pi = 1.0 atm Pf = 10.0 atm

Plan Your Strategy For the algebraic method, apply the formula Pi Vi = Pf Vf . Substitute the known values into this equation and solve for Vf . Act on Your Strategy Pi Vi = Pf Vf 1.0 atm × 45.0 cm3 = 10.0 atm × Vf Vf = 4.5 cm3 Check Your Solution The final answer has the correct unit and number of significant figures. The pressure increases so it is expected that the volume must decrease. This answer seems to be reasonable. 4. Problem

A 45.6 mL sample of gas at 490 torr is compressed to a certain volume at 3 atm. What is the new volume, in litres? What Is Required? You must calculate the final volume and express the answer in litres. What Is Given? Vi = 1 000 L

Pi = 1.0 atm Pf = 10.0 atm

Plan Your Strategy Using the ratio method, since the pressure is increasing, the volume must decrease, according to Boyle’s law. Therefore the initial volume must be multiplied by a pressure fraction that is less than one. Change the 3 atm to torr using the relationship that 1 atm = 760 torr. Act on Your Strategy torr = 2 280 torr 3 atm × 760 1 atm Vf = Vi × pressure ratio Chapter 11 The Behaviour of Gases • MHR

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Vf = 45.6 mL ×

490 torr 2280 torr

= 9.80 mL = 0.00980 L or 0.01 L (1 sig. fig.)

Check Your Solution The final answer has the correct unit and number of significant figures. The pressure increases so it is expected that the volume must decrease. This answer seems to be reasonable.

Section 11.3 Student Textbook page 446 5. Problem

Convert the following temperatures to the Kelvin scale. (b) 37˚C (c) 150˚C

(a) 25˚C

What Is Required? Express these temperatures in Kelvin. What Is Given? The Celsius temperatures are given. Plan Your Strategy Use the expression that TK = ˚C + 273. Act on Your Strategy

(a) TK = 25˚C + 273 = 298 K (b) TK = 37˚C + 273 = 310 K (c) TK = 150˚C + 273 = 423 K

Check Your Solution The Kelvin temperature must be a larger number than the Celsius temperature. These answers seem to be reasonable. 6. Problem

Convert the following tempratures to degrees Celsius. (a) 373 K (b) 98 K

(c) 425 K

What Is Required? Express the temperatures in Celsius. What Is Given? The Kelvin temperature is given. Plan a Strategy Use the expression ˚C = K − 273 Act on Your Strategy (a) 373K − 273 = 100˚C (b) 98K − 273 = −175˚C (c) 425K − 273 = 152˚C Check Your Solution The Celsius temperature is a lower number than the Kelvin temperature. These answers seem reasonable. 7. Problem

Give an example of something that might be at each temperature in question 5. (a) outside temperature on a pleasantly warm summer day (b) normal body temperature (c) tar that is heated to use on a road or a roof

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8. Problem

A sample of nitrogen gas surrounding a circuit board occupies a volume of 300 mL at 17˚C and 100 kPa. What volume will the nitrogen occupy at 100.0˚C if the pressure remains constant? What Is Required? You must calculate the final volume of a sample of nitrogen. What Is Given? Ti = 17˚C Tf = 100.0˚C

Vi = 300mL Vf = ?

Plan Your Strategy Using the ratio method, since the temperature is increasing, the volume must increase, according to Charles’ law. Therefore the initial volume must be multiplied by a temperature ratio that is greater than one. Change the Celsius temperatures to Kelvin using the relationship TK = ˚C + 273. Act on Your Strategy Ti = 17˚C + 273 = 290 K Vf = Vi × temperature ratio = 300 mL ×

Tf = 100.0˚C = 273 = 373 K = 386 mL

373 K 290 K

Check Your Solution The final volume has the correct unit and number of significant figures. The volume should increase when the temperature increases. This answer is reasonable. 9. Problem

A 2.5 L balloon is completely filled with helium indoors at a temperature of 24.2˚C. The balloon is taken out on a cold winter day (−17.5˚C). What will the volume of the balloon become, assuming constant pressure? What Is Required? You must determine the final volume of a balloon after a change in temperature. What Is Given? Ti = 24.2˚C Tf = −17.5˚C

Vi = 2.5 L Vf = ?

Plan Your Strategy Change the Celsius temperatures to Kelvin. Using the algebraic method, substitute V the given values into the formula TVi = Tf and solve for the final volume. i

Act on Your Strategy Ti = 24.2˚C + 273 = 297 K V Vi = Tf T i

f

f

Tf = −17.5˚C + 273 = 255.5 K or 256 K K Vf = 2.5 L × 256 = 2.2 L 297 K

Check Your Solution The final volume has the correct unit and number of significant figures. The volume should decrease when the temperature decreases. This answer is reasonable. 10. Problem

10.0 L of neon at 20.0˚C is expanded to a volume of 30.0 L. If the pressure remains constant, what must be the final temperature (in˚C)? What Is Required? Determine the final temperature of a sample of neon that has expanded. What Is Given? Ti = 20.0˚C

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Tf = ?

Vf = 30.0 L

Plan Your Strategy Change the Celsius temperature to Kelvin. Using the ratio method, since the volume is increasing, the temperature must increase, according to Charles’ law. Therefore the initial temperature must be multiplied by a volume ratio that is greater than one. Act on Your Strategy Ti = 20.0˚C + 273 = 293 K Tf = Ti × volume ratio L = 879 K T = 293 K × 30.0 10.0 L 879 K − 273 = 606˚C Check Your Solution The final temperature has the correct unit and number of significant figures. The temperature should increase for the volume to increase. This answer is reasonable. 11. Problem

A 14.5 cm3 sample of oxygen gas at 24.3˚C is drawn into a syringe with a maximum volume of 60 cm3 . What is the maximum change in temperature that the oxygen can be subjected to before the plunger pops out of the syringe? What Is Required? You must calculate the final temperature and determine the maximum change temperature that can occur before the volume of the syringe is exceeded and the plunger pops out. What Is Given? Ti = 24.3˚C Tf = ?

Vi = 14.5 cm3 Vf = 60 cm3

Plan Your Strategy Change the Celsius temperatures to Kelvin. Using the algebraic method, substitute V the given values into the formula TVi = Tf and solve for the final temperature. i

f

Determine the temperature change that has occurred to give this final temperature. Act on Your Strategy Ti = 24.3˚C + 273 = 297 K V Vi = Tf Tf = 297 K × T i

f

60 cm3 14.5 cm3

= 1229 K

1229 K − 273 = 956˚C Therefore the maximum change in temperature is 956˚C − 24.3˚C = 932˚C Check Your Solution The temperature change has the correct unit and number of significant figures. The temperature must increase to cause the gas to expand. This answer is reasonable. 12. Problem

Methane gas can be compressed by cooling and increasing the pressure. A 600 L sample of methane gas at 25˚C and 100 kPa is cooled to −20˚C. In a second step, the gas is compressed until the pressure is quadrupled. What will be the final volume? (Hint: Use both Boyle’s law and Charles’ law to answer the question.) What Is Required? You must calculate the final volume of a sample of methane that been cooled and subjected to an increase in pressure.

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What Is Given? Vi = 600 L Vf = ?

Ti = 25˚C Tf = −20˚C

Pi = 100kPa Pf = 400 kPa

Plan Your Strategy Change the Celsius temperatures to Kelvin. The algebraic approach will be used. Using Charles’ law,

Vi Ti

Vf Tf

=

, substitute the temperature/volume data and calculate

the new volume. Then, using the Boyle’s law formula, Pi Vi = Pf Vf , substitute the known pressure/volume data and calculate the final volume resulting from the change in pressure. Act on Your Strategy Ti = 25˚C + 273 = 298 K Tf = −20˚C + 273 = 253 K V Vi = Tf Vf = 600 L × T i

f

Pi Vi = Pf Vf

Vf = 509 L ×

253 K = 509 L 298 K 100 kPa = 127 L 400 kPa

Check Your Solution The decrease in temperature should cause the volume to decrease and the increase in pressure will also decrease the volume. The final answer has the correct unit and number of significant figures. The answer seems to be reasonable.

Solutions for Practice Problems Student Textbook page 449 13. Problem

An unknown gas is collected in a 250.0 mL flask and sealed. Using electronic devices, it is found that the gas inside the flask exerts a pressure of 135.5 kPa at 15˚C. What pressure will the gas exert if the temperature (in Kelvin) is doubled? What Is Required? You must find the final pressure exerted by a gas that is subjected to a temperature change at constant volume. What Is Given? Ti = 15˚C Tf = 2 × Ti expressed in Kelvin

Pi = 135.5 kPa Pf = ?

Plan Your Strategy Change the initial temperature from Celsius to Kelvin. According to Gay-Lussac’s law, if the temperature is increasing, the pressure will increase at constant volume. Using the ratio approach, the initial pressure must be multiplied by a temperature ratio that is greater than one. Act on Your Strategy Ti = 15˚C + 273 = 288 K Tf = 2 × 288 K = 576 K Pf = Pi × temperature ratio K = 271 kPa Pf = 135.5 kPa × 576 288 K Check Your Solution The final answer has the correct unit and number of significant figures. The pressure should double if the Kelvin temperature doubles according to Gay-Lussac’s law. This answer is reasonable.

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Solutions for Practice Problems Student Textbook page 450 14. Problem

At 18˚C, a sample of helium gas stored in a metal cylinder exerts a pressure of 17.5 atm. What will the pressure become if the tank is placed in a closed room where the temperature increases to 400˚C? What Is Required? You must calculate the final pressure when the temperature of a sample of helium gas is increased. What Is Given? Ti = 18˚C Tf = 40˚C

Pi = 17.5 atm Pf = ?

Plan Your Strategy Change the temperatures from Celsius to Kelvin. According to Gay-Lussac’s law if the temperature is increasing, the pressure will increase at constant volume. Using the ratio approach, the initial pressure must be multiplied by a temperature ratio that is greater than one. Act on Your Strategy Ti = 18˚C + 273 = 291 K Tf = 400˚C + 273 = 313 K Pf = Pi × temperature ratio K = 18.8 atm Pf = 17.5 atm × 313 291 K Check Your Solution The final answer has the correct unit and number of significant figures. The pressure should increase if the temperature increases. This answer is reasonable. 15. Problem

A gaseous refrigerant, enclosed in copper tubes, surrounds the freezer in a small refrigerator. The gas is found to exert a pressure of 110 kPa at 45˚C. The refrigerant is allowed to expand through a nozzle into an expansion chamber such that the exerted pressure decreases to 89 kPa. What is the temperature inside the freezer? What Is Required? You have to find the final temperature of a gas that has undergone a decrease in pressure at constant volume. What Is Given? Ti = 45˚C Tf = ?

Pi = 110 kPa Pf = 89 kPa

Plan Your Strategy Change the temperature from Celsius to Kelvin. Using the algebraic approach, P substitute the known values into the formula, TPi = Tf , and solve for the unknown i f Tf . Convert this temperature in Kelvin to Celsius. Act on Your Strategy Ti = 45˚C + 273 = 318 K Pi Ti

=

Pf Tf

Tf = 318K ×

89 kPa 110 kPa

= 257 K

257 K − 273 = −16˚C

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Check Your Solution The final answer has the correct unit and number of significant figures. The temperature should decrease if the pressure decreases. This answer is reasonable. 16. Problem

Before leaving on a trip to Florida, you measure the pressure inside the tires of your car at a gas station. At −7.5˚C the tire pressure is found to be 206.5 kPa. When you arrive in Florida, you stop for dinner. Before leaving, you once again measure the tire pressure at a gas station near the restaurant. Most pressure gauges in the United States are calibrated in psi. You find the tire pressure to be 34.3 psi. What is the approximate temperature in Florida? (Hint: See the MathLink on page 428 to find out how to convert psi to kPa). What Is Required? You must calculate the final pressure inside a tire after the temperature has changed at constant volume. What Is Given? Ti = −7.5˚C Tf = ?

Pi = 206.5 kPa Pf = 34.3 psi

Plan Your Strategy Change the Celsius temperature to Kelvin and change the final pressure from psi to kPa. Using the algebraic approach, substitute the known values into the equation P Pi = Tf and solve for the unknown, Tf . Convert this final temperature from Kelvin T i

f

to Celsius. Act on Your Strategy Ti = −7.5˚C + 273 = 265.5 K or 265 K kPa = 236 kPa 34.3 psi × 101.3 14.7 psi Pi Ti

=

Pf Tf

Tf = 265 K ×

236 kPa 206.5 kPa

= 303 K

303 K − 273 = 30.0˚C Check Your Solution The final answer has the correct unit and number of significant figures. The temperature should increase if the pressure increases at constant volume. This answer is reasonable.

Section 11.4 Student Textbook page 457 17. Problem

A sample of gas has a volume of 150 mL at 260 K and 92.3 kPa. What will the new volume be at 376 K and 123 kPa? What Is Required? Find the volume of a given amount of gas after it undergoes a change in temperature and pressure. What Is Given? Vi = 150 mL Vf = ?

Ti = 260 K Tf = 376 K

Pi = 92.3 kPa Pf = 123 kPa

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Plan Your Strategy The volume will increase because the temperature is increasing (Charles’ law) and it will decrease because the pressure is increasing (Boyle’s law). Using the ratio approach, the initial volume must be multiplied by a temperature ratio that is greater than one and by a pressure ratio that is less than one. Act on Your Strategy Vf = Vi × temperature ratio × pressure ratio K kPa = 150 mL × 376 × 92.3 260 K 123 kPa = 163 mL Check Your Solution The final answer has the correct unit and number of significant figures. Given that the temperature and pressure changes have opposite effects on the volume, this answer seems reasonable. 18. Problem

A cylinder at 48 atm pressure and 290 K releases 35 mL of carbon dioxide gas into a 4.0 L container at 297 K. What is the pressure inside the container? What Is Required? Find the new pressure after a given amount of gas undergoes a change in volume and temperature. What Is Given? Vi = 35 mL Vf = 4.0 L

Ti = 290 K Tf = 297 K

Pi = 48 atm Pf = ?

Plan Your Strategy Change the initial volume from mL to L. The change in volume will make the pressure go down and the change in temperature will make the pressure go up. Using the ratio approach, the initial pressure must be multiplied by a volume fraction less than one and by a temperature fraction that is greater than one. Act on Your Plan 35 mL = 0.035 L Pf = Pi × volume ratio × temperature ratio L K × 297 = 0.43 atm Pf = 48 atm × 0.035 4.0 L 290 K Check Your Solution The final answer has the correct unit and number of significant figures. Given that the temperature and pressure changes have opposite effects on the volume, this answer seems reasonable. 19. Problem

In a large syringe, 48 mL of ammonia gas at STP is compressed to 24 mL and 110 kPa. What must be the new temperature of the gas? What Is Required? Find the new temperature after a given amount of gas undergoes a change in volume and pressure. What Is Given? Vi = 48 mL Vf = 24 mL

Ti = 273 K Tf = ?

Pi = 101.3 kPa Pf = 110 kPa

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Plan Your Strategy The change in volume will make the temperature go down (Charles’ law) and the change in pressure will make the temperature go up (Boyle’s law). Using the ratio approach, the initial temperature must be multiplied by a volume fraction less than one and by a temperature fraction that is greater than one. Change this final temperature from Kelvin to Celsius Act on Your Strategy Tf = Ti × volume ratio × pressure ratio mL 110 kPa × 101.3 = 148 K Tf = 273 K × 24 48 mL kPa 148K − 273 = −125˚C Check Your Solution The final answer has the correct unit and number of significant figures. Given that the volume and pressure changes have opposite effects on the temperature, this answer seems reasonable. 20. Problem

A 100 W light bulb has a volume of 180.0 cm3 at STP. The light bulb is turned on and the heated glass expands slightly, changing the volume of the bulb to 181.5 cm3 with an internal pressure of 214.5 kPa. What is the temperature of the light bulb (in˚C)? What Is Required? Find the new temperature after a given amount of gas undergoes a change in volume and pressure. What Is Given? Vi = 180.0 cm3 Vf = 181.5 cm3

Ti = 273 K Tf = ?

Pi = 101.3 kPa Pf = 214.5 kPa

Plan Your Strategy Using an algebraic approach, rearrange the algebraic form of the combined gas laws, Pi Vi Ti

=

Pf Vf Tf

, substitute the known values, and solve for the unknown final

temperature, Tf . Convert this final temperature from Kelvin to Celsius. Act on Your Strategy PV Pi Vi = Tf f Tf = Ti f 583 K − 273 = 310˚C

Ti Vf Pf Vi Pi

=

273 K × 181.5 cm3 × 214.5 kPa 180.0 cm3 × 101.3 kPa

= 583 K

Check Your Solution The final answer has the correct unit and number of significant figures. Given that the volume and pressure changes both are increasing the temperature, this answer seems reasonable. 21. Sulfur hexafluoride, SF6(g), is used as a chemical insulator. A 5.0 L sample of this gas

is collected at 205.0˚C and 350 kPa. What pressure must be applied to this gas sample to reduce its volume to 1.7 L at 25˚C? What Is Required? Find the new pressure after a given amount of gas undergoes a change in volume and temperature. What Is Given? Vi = 5.0 L Vf = 1.7 L

Ti = 205.0˚C Tf = 25˚C

Pi = 350 kPa Pf = ?

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Plan Your Strategy Convert the Celsius temperature to Kelvin. Using an algebraic approach, rearrange the algebraic form of the combined gas laws,

Pi Vi Ti

=

Pf Vf Tf

, substitute the known

values, and solve for the unknown final pressure Pf . Act on Your Strategy PV Pi Vi = Tf f T i

f

Pf =

Pi Vi Tf Vf Ti

=

350 kPa × 5.0 L × 298 K 1.7 L × 478 K

= 6.4 × 102 kPa

Check Your Solution The final answer has the correct unit and number of significant figures. Given that the volume and temperature changes have opposite effects on the initial pressure, this answer seems reasonable.

Solutions for Practice Problems Student Textbook page 461 22. Problem

To speed up a reaction in a vessel pressurized at 98.0 kPa, a chemist added 202.65 kPa of hydrogen gas. What was the resulting pressure? What Is Required? You must find the total pressure in a mixture of two gases. What Is Given? The initial pressure and the pressure contributed by the additional gas are known. Plan Your Strategy According to Dalton’s law, the total pressure will be the sum of the pressures contributed by each component gas. Mathematically, Ptotal = P1 + P2 . Act on Your Strategy P1 = 98.0 kPa P2 = 202.65 kPa Ptotal = 98.0 kPa + 202.65 kPa = 300.6 kPa Check Your Solution This is a reasonable answer that makes sense from Dalton’s law. 23. Problem

A gas mixture contains 12 % Ne, 23 % He, and 65 % Rn. If the total pressure is 116 kPa, what is the partial pressure of each gas? What Is Required? You must find how much each gas in a mixture of gases contributes to the total pressure. What Is Given? The percentage of each gas in a mixture is given. Plan Your Strategy According to Dalton’s law, each gas will contribute a portion of the total pressure that is proportional its percentage in the mixture. Act on Your Strategy Ptotal = PNe + PHe + PRn PNe = 12 % of Ptotal = 0.12 × 116 kPa = 14 kPa PHe = 23 % of Ptotal = 0.23 × 116 kPa = 27 kPa PRn = 65 % of Ptotal = 0.65 × 116 kPa = 75 kPa Chapter 11 The Behaviour of Gases • MHR

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Check Your Solution The sum of the partial pressures is equal to the total pressure. The answers are reasonable. 24. Problem

The partial pressure of argon gas, making up 40 % of a mixture, is 325 torr. What is the total pressure of the mixture in kPa? What Is Required? You must find the total pressure in this mixture of gases. What Is Given? The partial pressure of the argon component and the percentage argon in the mixture are given. Plan Your Strategy The pressure that a component contributes to the total pressure is in proportion to its percentage of the total amount of gas. Knowing that 40 % of the gas in the mixture accounts for 325 torr, calculate the pressure corresponding to 100 % of the mixture. Let X = the total pressure in the mixture. Act on Your Strategy 40% = 100% 325 torr X

X=

100% 40%

× 325 torr = 8.1 × 102 torr

Check Your Solution The calculated value has the correct unit and number of significant figures. It is a little more than double the partial pressure of the argon. This seems to be a reasonable answer. 25. Problem

A mixture of nitrogen and carbon dioxide gas is at a pressure of 1.00 atm and a temperature of 278 K. If 30 % of the mixture is nitrogen, what is the partial pressure of the carbon dioxide? What Is Required? You must find the how much carbon dioxide gas contributes to the total pressure. What Is Given? The total pressure and the percentage that nitrogen makes up of the total mixture are given. Plan Your Strategy The pressure that a component contributes to the total pressure is in proportion to its percentage of the total amount of gas. The carbon dioxide should contribute its percentage of the mixture to the total pressure. Act on Your Strategy partial pressure of CO2 = 70 % of Ptotal = 0.70 × 1.00 atm = 0.70 atm Check Your Solution This seems like a reasonable contribution for 70 % of the mixture.

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