Chapter 12 The Behavior of gases 12.3 The Gas Laws

All gases Are less dense when warmer Are less dense when under less pressure Are less dense when there are less particles in a volume Obey laws, within limits, which allow us to predict their behavior under most conditions

Things you will learn Understand Boyle’s Law Understand Charles’s Law Understand Gay-Lussac’s Law Understand and be able to solve problems using the combined gas laws

Boyle’s Law The pressure-volume relationship of gases

Pressure and volume

Pressure and volume have an inverse relationship

Boyles Law essentials The product of a pressure and volume of any two sets of conditions at a given temperature is constant: P1 x V1 = P2 x V2 100 kPa @ 1 L = 50 kPa @ 2 L = 200 kPa @ .5L The two conditions are inversely proportional P~1/V

A high altitude balloon contains 30 L of He gas at 103 kPa A high altitude balloon contains 30 L of He gas at 103 kPa. What is the volume when it gets to an altitude where the pressure is 25 kPa?

A high altitude balloon contains 30 L of He gas at 103 kPa A high altitude balloon contains 30 L of He gas at 103 kPa. What is the volume when it gets to an altitude where the pressure is 25 kPa? Knowns: V1 = 30 L P1 = 103 kPa P2 = 25 kPa Unknown: V2

Our equation is V1 x P1 = V2 x P2 30L x 103 kPa = v2 x 25 kPa A high altitude balloon contains 30 L of He gas at 103 kPa. What is the volume when it gets to an altitude where the pressure is 25 kPa? Our equation is V1 x P1 = V2 x P2 30L x 103 kPa = v2 x 25 kPa 30 L x 103 kPa 25 kPa V2 = 124 L V2

Charles’s Law The temperature-volume relationship of gases

Charles’s Law esentials An increase in the temperature of a gas yields an increase in the volume of a gas V1/T1 = V2/T2 1 L @ 300K = 2 L @ 600k The two conditions are directly proportional V~T remember all temperature measurements are in kelvins

A balloon inflated in a room at temperature of 24°C has a volume of 4 L. The balloon is then heated to a temperature of 58°C. What is the new volume if the temperature remains constant?

Knowns: Unknown: V1 = 4 L T1 = 24°C T2 = 58°C V2 A baloon inflated in a room at temperature of 24°C has a volume of 4 L. The balloon is then heated to a temperature of 58°C. What is the new volume if the temperature remains constant? Knowns: V1 = 4 L T1 = 24°C T2 = 58°C Unknown: V2

Our equation is V1/T1= V2 /T2 4L /24°C = xL/58°C 58C x 4L / 24°C = xL A baloon inflated in a room at temperature of 24°C has a volume of 4 L. The balloon is then heated to a temperature of 58°C. What is the new volume if the temperature remains constant? Our equation is V1/T1= V2 /T2 4L /24°C = xL/58°C 58C x 4L / 24°C = xL wait, we haven’t converted to Kelvins! 331K x 4L = 4.46 L 297K

Gay-Lussac The temperature-volume relationship of gases

Gay-Lussac’s Law The pressure of a gas is directly proportional to the temperature of a gas (in Kelvins) if the volume remains constant Because these relations are directly related, they obey the formula: P1/T1 = P2/T2

Sample problem The gas in an aerosol can may be 103 kPa, meaning it won’t squirt, at 25°C, but if thrown in a fire, the pressure could be quite dangerous. How high is the pressure if the fire is 928°C ?

Sample problem The pressure in a tire is 198 kPa at the start of a trip at 27°C. At the end of the trip, it is 225 kPa. What is the internal temperature of the tire

Combining the gas laws Boyles’ Law is Charles’s Law is Gay-Lussac’s Law is

Combining the gas laws Boyles’ Law is P1 x V1 = P2 x V2 Charles’s Law is V1/T1 = V2/T2 Gay-Lussac’s Law is P1/T1 = P2/T2

The combined gas law P1 x V1 P2 x V2 T1 T2 If you hold the temperature constant, you have Boyle’s Law If you hold volume constant, you have Gay-Lussac’s Law If you hold the pressure constant, you have Charles’s Law

Sometimes you are not able to hold any of the variables constant The volume of a gas filled balloon is 30L and 153 kPa. What volume will the balloon be at STP?

The volume of a gas filled balloon is 30L and 153 kPa The volume of a gas filled balloon is 30L and 153 kPa. What volume will the balloon be at STP? Knowns: V1=30 L T1=40C T2= 273K (standard temp) P1= 153 kPa P2=101.3 kPa (standard pressure

The volume of a gas filled balloon is 30L and 153 kPa The volume of a gas filled balloon is 30L and 153 kPa. What volume will the balloon be at STP? Knowns: V1=30 L T1=40C T2= 273K (standard temp) P1= 153 kPa P2=101.3 kPa (standard pressure Change all temps to Kelvins Isolate V2 and solve

Practice problems A gas at 155 kPa and 25C occupies a container with an initial volume of 1 L. By changing the volume, the pressure of the gas increases to 605 kPa as the temperature is raised to 125C. What is the new volume?

Practice problems A 5 L air sample at a temperature of -50C has a pressure of 107 kPa. What will be the new pressure if the temperature is raised to 102C and the volume expands to 7 L?

Ideal gas law Up to this point, we have left out another thing that can change pressure, volume and temperature

We have used the combined gas law to find changes in a system when we change conditions such as temperature, pressure or volume. P1 x V1 T1 P2 x V2 T2

But the amount of gas- the number of moles- can change the pressure and volume also It makes sense that the volume and pressure in a container must be proportional to the number of moles We can add the term for moles (n) to the combined gas laws that we just looked at : so P1V1/T1 becomes P1V1 / T1n1

This shows that the term P x V/ T x n is a constant This will allow us to find unknowns in a system where we are not changing conditions P1 x V1 T1 x n1 P2 x V2 T2 x n2

P1V1 = T1n1 This is called the Ideal Gas Law We need to evaluate this in order to come up with a constant which will make the equation work We do this at STP What is STP

We use STP as a starting point P= 101.3 kPa (one ATM) V= 22.4 Liters (volume of 1 mole at STP) N= 1 mole T= 273K (melting point of ice in K) The constant, R, = (P x V) / (T x n) or 8.31 (L x kPa) / (K x mol)

The ideal gas law becomes: PV=nRT Pressure x volume = # moles x temp x constant This will allow us to figure any one of the variables in any system if we know the other three

Sample problem You fill a rigid steel cylinder that has a volume of 20.0 L with nitrogen gas to a final pressure of 2 x 104 kPa at 28C. How many moles of gas are in the cylinder?

You fill a rigid steel cylinder that has a volume of 20 You fill a rigid steel cylinder that has a volume of 20.0 L with nitrogen gas to a final pressure of 2 x 104 kPa at 28C. How many moles of gas are in the cylinder? Knowns: P = 2x104 kPa V = 20 L T = 28C Unknown: N = ?

You fill a rigid steel cylinder that has a volume of 20 You fill a rigid steel cylinder that has a volume of 20.0 L with nitrogen gas to a final pressure of 2 x 104 kPa at 28C. How many moles of gas are in the cylinder? Convert 28C to K 301K Isolate n (number of moles) n = P x V R x T

n = 2x104kPa x 20L 8.31 L x kPa x 301K K x mol You fill a rigid steel cylinder that has a volume of 20.0 L with nitrogen gas to a final pressure of 2 x 104 kPa at 28C. How many moles of gas are in the cylinder? n = 2x104kPa x 20L 8.31 L x kPa x 301K K x mol

n = 2x104kPa x 20L 8.31 L x kPa x 301K K x mol You fill a rigid steel cylinder that has a volume of 20.0 L with nitrogen gas to a final pressure of 2 x 104 kPa at 28C. How many moles of gas are in the cylinder? n = 2x104kPa x 20L 8.31 L x kPa x 301K K x mol

You fill a rigid steel cylinder that has a volume of 20 You fill a rigid steel cylinder that has a volume of 20.0 L with nitrogen gas to a final pressure of 2 x 104 kPa at 28C. How many moles of gas are in the cylinder? n = 1.60 x 102 mol N2

When the pressure of a rigid hollow sphere containing 685 L of helium gas is held at 621K, the pressure of the gas is 1.89 x 103 kPa. How many moles of helium does the sphere contain?

What pressure will be exerted by What pressure will be exerted by .450 mol of a gas at 25C if it is contained in a .650 L vessel?

A deep underground cavern contains 2 A deep underground cavern contains 2.24 x 106 L of methane (CH4) at a pressure of 1.50 x 103 kPa and a temperature of 42C. How many kilograms of CH4 does this deposit contain?

What volume will 12 g of oxygen gas (O2) occupy at 25C and a pressure of 52.7 kPa?

Ideal gas?? The ideal gas law assumes that gas particles occupy no volume and that there are no attractions between particles. This is never true. Gases under high pressure and low temperatures turn to liquids. Gases do occupy volume (obviously), and there are attractions (intermolecular forces) as we learned in chapter 10. Boyle’s law implies that gases are infinitely compressible, but this is not the case

Partial pressures (Dalton’s Law) The pressure exerted by a mixture of gases (our atmosphere, for example) equals the sum of the pressures of each individual gas in the mixture Ptotal = Pa + Pb + Pc

Partial pressures Ptotal =ntotalRT / V Each gas in a mixture acts independently of each other The total pressure exerted by the gases in a mixture is directly related to the number of moles in the mixture Ptotal ~ ntotal Ptotal =ntotalRT / V

The total pressure of a mixture of hydrogen, argon and nitrogen is 120 kPa. The partial pressure of hydrogen is 32 kPa and the partial pressure of argon is 58 kPa. What is the partial pressure of the nitrogen?

The total pressure of a mixture of hydrogen, argon and nitrogen is 120 kPa. The partial pressure of hydrogen is 32 kPa and the partial pressure of argon is 58 kPa. What is the partial pressure of the nitrogen? Using Ptotal = Pa + Pb + Pc 120 kPa = 32 kPa + 58 kPa + Pnitrogen Pnitrogen = 30 kPa

A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2?

The total pressure depends on the total number of moles (.55) A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2? The total pressure depends on the total number of moles (.55) Ptotal =ntotalRT / V

The total pressure depends on the total number of moles (.55) A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2? The total pressure depends on the total number of moles (.55) Ptotal =ntotalRT / V Ptotal = .55 mole x 8.3 x 298K / 10L = 136kPa

A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2?

A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2? The partial pressure of each gas depends on the number of moles of each gas (.25 mole CO2) PCO2 =nCO2RT / V

A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2? The partial pressure of each gas depends on the number of moles of each gas (.25 mole CO2) PCO2 =nCO2RT / V PCO2 =.25 mol x 8.3 x 298 / 10 = 62 kPa

A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2?

A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2? Ptotal = Pa + Pb + Pc

Ptotal = Pa + Pb + Pc 136 kPa = 62 kPa + PO2 PO2 = 74 kPa A mixture of .25 mole CO2 and .30 mole of O2 is in a 10 L container at 298K. What is the total pressure of the mixture? What is the partial pressure of the CO2? What is the partial pressure of the O2? Ptotal = Pa + Pb + Pc 136 kPa = 62 kPa + PO2 PO2 = 74 kPa

The gas laws Boyle’s Law Charle’s Law Gay-Lussac’s Law Combined gas law Ideal gas law Dalton’s Law

Boyle’s Law P1V1 = P2V2 Pressure and volume change Temperature is held constant

Charle’s Law V1/T1 = V2/T2 Volume and temperature change Pressure remains constant

Gay-Lussac’s Law P1/T1 = P2/T2 Pressure and temperature change Volume remains constant

Combined gas law P1V1/T1 = P2V2/T2 Pressure, volume and temperature change # moles remains constant

Ideal gas law PV = nRT Used for finding conditions where nothing is changing. One variable can be missing Units must be: P (kPa) V (L) N (moles) T (Kelvins) R (8.3)

Dalton’s Law Ptotal = Pa + Pb + Pc The total pressure of a mixture of gases is the sum of the pressures of each different gas (partial pressures)