GasesGases

Chapter 10Chapter 10

Characteristics of GasesCharacteristics of Gases1.1. The particles in gases are far apart, The particles in gases are far apart,

therefore they are therefore they are very very compressiblecompressible..

2.2. Gases expand to fill the volume of Gases expand to fill the volume of their container their container (V (V cont.cont. = V = V gasgas))

3.3. Their motion is constant, random, Their motion is constant, random, and very fast.and very fast.

4.4. Average Kinetic NRG of gas is Average Kinetic NRG of gas is determined by Temperaturedetermined by Temperature

5.5. The volume and mass of gas The volume and mass of gas particles is negligable.particles is negligable.

6.6. For gas particles, the energy For gas particles, the energy remains constant within a system.remains constant within a system.

7.7. During collisions between gas During collisions between gas particles, the collision is perfectly particles, the collision is perfectly ellastic.ellastic.

8.8. When gas particles collide with the When gas particles collide with the walls of their container, they exert walls of their container, they exert a pressure.a pressure.

PressurePressure• Gases exert a Gases exert a pressure pressure on the walls on the walls

of their container.of their container.

• Pressure is defined as force per unit area:

P =

SI unit: 1 N/m2 = 1 Pascal (Pa)

F

A

Atmospheric PressureAtmospheric Pressure• The result of ALL the gas molecules of The result of ALL the gas molecules of

the atmosphere from sea level to the the atmosphere from sea level to the top of the exosphere.top of the exosphere.

• Gravity pulls all of them down toward Gravity pulls all of them down toward the center of the earth.the center of the earth.

• A Large Force!A Large Force!• 1 m1 m22 column air = 10,000 kg at sea column air = 10,000 kg at sea

levellevel• Air pressure changes slightly from day Air pressure changes slightly from day

to day, but changes dramatically with to day, but changes dramatically with altitude.altitude.

•Where is there higher pressure: Where is there higher pressure: up in the mountains or down at up in the mountains or down at the beach?the beach?

•Why?Why?

•What does that mean for us?What does that mean for us?

A Barometer measures atmospheric A Barometer measures atmospheric pressurepressure

The pressure of the The pressure of the atmosphere at sea level atmosphere at sea level at 25at 25ooC will hold a C will hold a column of mercury 760 column of mercury 760 mm.mm.

1 atm = 760 mm Hg1 atm = 760 mm Hg

1 atm = 760 torr1 atm = 760 torr

1 atm = 101.3 kPa1 atm = 101.3 kPa

1 atm = 29.9 inches Hg1 atm = 29.9 inches Hg

“ “Standard Atmospheric Standard Atmospheric Pressure”Pressure”

1 atm Pressure

760 mm Hg

Vacuum

A A ManometerManometer measures the pressure of measures the pressure of gas in a containergas in a container

Gas

h

• Column of mercury Column of mercury measures pressure.measures pressure.

• h is how much lower h is how much lower the pressure is than the pressure is than outside. outside.

• SUBTRACT h from SUBTRACT h from the room pressure.the room pressure.

ManometerManometer

• h is how much h is how much higher the gas higher the gas pressure is than pressure is than the atmosphere.the atmosphere.

• ADD the ADD the difference in the difference in the height to the height to the room pressureroom pressure

h

Gas

Units of pressureUnits of pressure1 atmosphere = 760 mm Hg1 atmosphere = 760 mm Hg1 mm Hg = 1 torr1 mm Hg = 1 torr1 atm = 101,325 Pascals = 101.325 1 atm = 101,325 Pascals = 101.325

kPakPa

• Use dimensional analysis to convertUse dimensional analysis to convert 1. What is 724 mm Hg in kPa?1. What is 724 mm Hg in kPa? 2. in torr?2. in torr? 3. in atm?3. in atm?

96.5 kPa

724 torr

0.952 atm

The Gas LawsThe Gas Laws

• Boyle’s Law:Boyle’s Law:

• Pressure and volume are inversely Pressure and volume are inversely related at constant temperature.related at constant temperature.

• PV= kPV= k

As one goes up, the other goes down.As one goes up, the other goes down.

• PP11VV11 = P = P22 V V22

Graphically. . .Graphically. . .

P

V (at constant T)

Boyle’s Law

P

1/V (at constant T)

Slope = k

Charles’ LawCharles’ Law• Volume of a gas varies directly Volume of a gas varies directly

with the absolute temperature at with the absolute temperature at constant pressure.constant pressure.

V = kT (if V = kT (if T is in KelvinT is in Kelvin))

VV1 1 V V22

T T11 T T22

Graphically . . .Graphically . . .

=

V (

L)

T (ºC)-273.15ºC

Gay- Lussac LawGay- Lussac Law

• At constant volume, pressure and At constant volume, pressure and absolute temperature are directly absolute temperature are directly related.related.

• P = k TP = k T

PP1 1 P P22

T T11 T T22

Graphically . . .Graphically . . .

=

Temperature K

P

Slope = k

Avogadro's LawAvogadro's Law

• At constant temperature and At constant temperature and pressure, the volume of gas is pressure, the volume of gas is directly related to the number of directly related to the number of moles.moles.

V = k n (n is the number of moles)V = k n (n is the number of moles)

VV1 1 VV22

n n11 n n22

=

Combined Gas LawCombined Gas Law

• If the moles of gas remains If the moles of gas remains constant, use this formula and constant, use this formula and cancel out the other things that cancel out the other things that don’t change.don’t change.

•PP1 1 VV11 = P = P22 V V22

T T11

T T22

ExamplesExamples

•A spray can has a volume of 175 A spray can has a volume of 175 mL and a pressure of 3.8 atm at mL and a pressure of 3.8 atm at 22ºC. What would the pressure be 22ºC. What would the pressure be if the can was heated to 100.ºC?if the can was heated to 100.ºC?

•What volume of gas could the can What volume of gas could the can release at 22ºC and 743 torr?release at 22ºC and 743 torr?

Ideal Gas LawIdeal Gas Law•PV = nRTPV = nRT

•V = 22.41 L at 1 atm, 0ºC, n = 1 V = 22.41 L at 1 atm, 0ºC, n = 1 mole, what is R?mole, what is R?

•R is the ideal gas constant.R is the ideal gas constant.

•R = 0.0821 L atm/ mol KR = 0.0821 L atm/ mol K

•How do you get R for different units of How do you get R for different units of P?P?

•Tells you about a gas NOW.Tells you about a gas NOW.

•The other laws tell you about a gas The other laws tell you about a gas when it changes. when it changes.

Ideal Gas LawIdeal Gas Law•An An equation of stateequation of state. (a state . (a state

function)function)

• Independent of how you end up Independent of how you end up where you are at. Does not depend where you are at. Does not depend on the path.on the path.

•Given 3 you can determine the Given 3 you can determine the fourth.fourth.

•An Empirical Equation - based on An Empirical Equation - based on experimental evidence.experimental evidence.

Ideal Gas LawIdeal Gas Law

•A A hypothetical substancehypothetical substance - the - the ideal gasideal gas

• Gases only approach ideal behavior Gases only approach ideal behavior at low pressure (< 1 atm) and high at low pressure (< 1 atm) and high temperature.temperature.

•Use the laws anyway, unless told to Use the laws anyway, unless told to do otherwise.do otherwise.

•They give good estimates.They give good estimates.

ExamplesExamples•A 47.3 L container containing 1.62 A 47.3 L container containing 1.62

mol of He is heated until the mol of He is heated until the pressure reaches 1.85 atm. What is pressure reaches 1.85 atm. What is the temperature?the temperature?

•Kr gas in a 18.5 L cylinder exerts a Kr gas in a 18.5 L cylinder exerts a pressure of 8.61 atm at 24.8ºC What pressure of 8.61 atm at 24.8ºC What is the mass of Kr?is the mass of Kr?

•A sample of gas has a volume of 4.18 A sample of gas has a volume of 4.18 L at 29ºC and 732 torr. What would L at 29ºC and 732 torr. What would its volume be at 24.8ºC and 756 torr?its volume be at 24.8ºC and 756 torr?

Gas StoichiometryGas Stoichiometry•D = m/VD = m/V

•Let Let MM stand for molar mass stand for molar mass

•MM = m/n = m/n

•n= PV/RTn= PV/RT

•MM = m = m PV/RTPV/RT

• MM = mRT = m RT = DRT = mRT = m RT = DRT PV PV V P V P P P

• May be easier to convert to moles, then May be easier to convert to moles, then use stoich.use stoich.

Examples Examples •What is the density of ammonia at What is the density of ammonia at

23ºC and 735 torr?23ºC and 735 torr?

•A compound has the empirical A compound has the empirical formula CHCl. A 256 mL flask at formula CHCl. A 256 mL flask at 100.ºC and 750 torr contains .80 g 100.ºC and 750 torr contains .80 g of the gaseous compound. What is of the gaseous compound. What is the molecular formula?the molecular formula?

•The molar mass of the The molar mass of the atmosphere at the surface of atmosphere at the surface of Titan, Saturn’s largest moon, is Titan, Saturn’s largest moon, is 28.6 g/mol. The surface 28.6 g/mol. The surface temperature is 95 K, and the temperature is 95 K, and the pressure is 1.6 atm. Assuming pressure is 1.6 atm. Assuming ideal behavior, calculate the ideal behavior, calculate the density of Titan’s atmosphere.density of Titan’s atmosphere.

•5.9 g/L5.9 g/L

2NaN2NaN33 (s)(s) 2Na 2Na (s)(s) + 3 N + 3 N22 (g)(g)

•The safety air bags in automobiles The safety air bags in automobiles are inflated by nitrogen gas (see the are inflated by nitrogen gas (see the equation above). If an air bag has a equation above). If an air bag has a volume of 36 L and is to be filled volume of 36 L and is to be filled with nitrogen gas at a pressure of with nitrogen gas at a pressure of 1.15 atm at a temperature of 1.15 atm at a temperature of 26.026.0ooC, how many grams of NaNC, how many grams of NaN33 must be decomposed?must be decomposed?

•72 g NaN72 g NaN33

Gases and StoichiometryGases and Stoichiometry

•Reactions happen in molesReactions happen in moles

•At Standard Temperature and At Standard Temperature and Pressure (STP, 0ºC and 1 atm) 1 Pressure (STP, 0ºC and 1 atm) 1 mole of gas occuppies 22.42 L.mole of gas occuppies 22.42 L.

• If not at STP, use the ideal gas If not at STP, use the ideal gas law to calculate moles of law to calculate moles of reactant or volume of product.reactant or volume of product.

ExamplesExamples

•Mercury can be achieved by the Mercury can be achieved by the following reaction above. What following reaction above. What volume of oxygen gas can be volume of oxygen gas can be produced from 4.10 g of mercury produced from 4.10 g of mercury (II) oxide at STP?(II) oxide at STP?

•At 400.ºC and 740 torr?At 400.ºC and 740 torr?

(g)O + Hg(l) HgO 2heat

ExamplesExamples• Using the following reactionUsing the following reaction

• calculate the mass of sodium hydrogen calculate the mass of sodium hydrogen carbonate necessary to produce 2.87 L carbonate necessary to produce 2.87 L of carbon dioxide at 25ºC and 2.00 atm.of carbon dioxide at 25ºC and 2.00 atm.

• If 27 L of gas are produced at 26ºC and If 27 L of gas are produced at 26ºC and 745 torr when 2.6 L of HCl are added 745 torr when 2.6 L of HCl are added what is the concentration of HCl?what is the concentration of HCl?

NaCl(aq) + CO (g) +H O(l)2 2

NaHCO (s) + HCl 3

ExamplesExamples

• Consider the following reactionConsider the following reaction What volume of NO at What volume of NO at

1.0 atm and 1000ºC can be produced 1.0 atm and 1000ºC can be produced from 10.0 L of NHfrom 10.0 L of NH33 and excess O and excess O22 at the at the

same temperature and pressure?same temperature and pressure?

• What volume of OWhat volume of O22 measured at STP will measured at STP will

be consumed when 10.0 kg NHbe consumed when 10.0 kg NH33 is is

reacted?reacted?

4NH (g) + 5 O 4 NO(g) + 6H O(g)3 22 ( )g

The Same reactionThe Same reaction

• What mass of HWhat mass of H22O will be produced from O will be produced from

65.0 L of O65.0 L of O22 and 75.0 L of NH and 75.0 L of NH33 both both

measured at STP? measured at STP?

• What volume Of NO would be produced?What volume Of NO would be produced?

• What mass of NO is produced from 500. What mass of NO is produced from 500. L of NH3 at 250.0ºC and 3.00 atm?L of NH3 at 250.0ºC and 3.00 atm?

4NH (g) + 5 O 4 NO(g) + 6H O(g)3 22 ( )g

Dalton’s Law of Partial Dalton’s Law of Partial PressuresPressures

•The total pressure in a container is The total pressure in a container is the sum of the pressure each gas the sum of the pressure each gas would exert if it were alone in the would exert if it were alone in the container.container.

•The total pressure is the sum of The total pressure is the sum of the partial pressures.the partial pressures.

•PPTotalTotal = P = P11 + P + P22 + P + P33 + P + P44 + P + P55 ... ...

•For each P = nRT/VFor each P = nRT/V

Dalton's LawDalton's Law

• PPTotalTotal = n = n11RT + nRT + n22RT + nRT + n33RT +...RT +...

V V V V V V

• In the same container R, T and V are the In the same container R, T and V are the same.same.

• PPTotalTotal = (n = (n11+ n+ n22 + n + n33+...)RT+...)RT V V

• PPTotalTotal = (n = (nTotalTotal)RT)RT V V

The mole fractionThe mole fraction

• Ratio of moles of the substance to Ratio of moles of the substance to the total moles.the total moles.

• symbol is Greek letter chi symbol is Greek letter chi

= n= n11 = P= P1 1

n nTotal Total PPTotalTotal

ExamplesExamples

• The partial pressure of nitrogen in air The partial pressure of nitrogen in air is 592 torr. Air pressure is 752 torr, is 592 torr. Air pressure is 752 torr, what is the mole fraction of nitrogen?what is the mole fraction of nitrogen?

• What is the partial pressure of What is the partial pressure of nitrogen if the container holding the nitrogen if the container holding the air is compressed to 5.25 atm?air is compressed to 5.25 atm?

0.787

4.13 atm

ExamplesExamples

3.50 L

O2

1.50 L

N2

2.70 atm• When these valves are opened, what is When these valves are opened, what is

each partial pressure and the total each partial pressure and the total pressure?pressure?

4.00 L

CH4

4.58 atm 0.752 atm

• When valve is opened, the gases mix When valve is opened, the gases mix and fill the total volume of all vessels: and fill the total volume of all vessels:

4.0 L + 1.5 L + 3.5 L = 9.0 L4.0 L + 1.5 L + 3.5 L = 9.0 L• Since the #moles and temp. is Since the #moles and temp. is

constant, this is a simple Boyle’s Law constant, this is a simple Boyle’s Law problem:problem: 4.0 L4.0 LPPCH4CH4 = 2.7 atm = 1.2 atm = 2.7 atm = 1.2 atm

9.0 L9.0 L 1.5 L1.5 L

PPN2N2 = 4.58 atm = 0.76 atm = 4.58 atm = 0.76 atm

9.0 L9.0 L 3.5 L3.5 L

PPO2O2 = .752 atm = 0.29 atm = .752 atm = 0.29 atm

9.0 L9.0 L

So, the totalpressure of allthe gases is1.2 + 0.76 + 0.29 = 2.25 atm

Kinetic Molecular Theory Kinetic Molecular Theory (KMT)(KMT)•Theory tells why the things Theory tells why the things

happen.happen.

•Explains why ideal gases behave Explains why ideal gases behave the way they do.the way they do.

•Assumptions that simplify the Assumptions that simplify the theory, but don’t work in real theory, but don’t work in real gases.gases.

Kinetic Molecular Theory…Kinetic Molecular Theory…

1.1. The particles are so small we can The particles are so small we can ignore their volume.ignore their volume.

2.2. The particles are in constant motion The particles are in constant motion and their collisions cause pressure.and their collisions cause pressure.

3.3. The particles do not exert forces on The particles do not exert forces on each either (attractive or repulsive)each either (attractive or repulsive)

4.4. The average Kinetic NRG is The average Kinetic NRG is proportional to the Kelvin proportional to the Kelvin temperature (KE = 1/2 mvtemperature (KE = 1/2 mv2 2 ))

What it tells usWhat it tells us

•(KE)(KE)avgavg = 3/2 RT = 3/2 RT

•This the meaning of temperature.This the meaning of temperature.

• Molecules move about at a random average KE.

• It is not a true average, but the root-mean-square speed (RMS speed).

• RMS is the square root of the average squared speeds. It is close to the average.

Combine these two Combine these two equationsequations

• (KE)(KE)avgavg = N = NAA(1/2 mu (1/2 mu 22 ) )

• (KE)(KE)avgavg = 3/2 RT = 3/2 RT

Combine these two Combine these two equationsequations

• (KE)(KE)avgavg = N = NAA(1/2 mu (1/2 mu 22 ) )

• (KE)(KE)avgavg = 3/2 RT = 3/2 RT

Where M is the molar mass in kg/mole, Where M is the molar mass in kg/mole, and R has the units 8.3145 J/Kmol.and R has the units 8.3145 J/Kmol.

• The velocity will be in m/sThe velocity will be in m/s

urms = 3RT

M

Range of velocitiesRange of velocities

• The average distance a molecule travels The average distance a molecule travels before colliding with another is called the before colliding with another is called the mean free path and is small (near 10mean free path and is small (near 10-7-7))

• Temperature is an average. There are Temperature is an average. There are molecules of many speeds in the molecules of many speeds in the average.average.

• Shown on a graph called a velocity Shown on a graph called a velocity distributiondistribution

num

ber

of p

arti

cles

Molecular Velocity

273 K

num

ber

of p

arti

cles

Molecular Velocity

273 K

1273 K

num

ber

of p

arti

cles

Molecular Velocity

273 K

1273 K

2273 K

VelocityVelocity

•Average increases as Average increases as temperature increases.temperature increases.

•Spread increases as temperature Spread increases as temperature increases.increases.

Molecular Effusion and Molecular Effusion and DiffusionDiffusion

•Effusion:Effusion:

Movement of gas through a small Movement of gas through a small opening into an evacuated container opening into an evacuated container (vacuum)(vacuum)

•The effusion rate measures how fast The effusion rate measures how fast this happens.this happens.

•Graham’s LawGraham’s Law -the rate of effusion is -the rate of effusion is inversely proportional to the square inversely proportional to the square root of the mass of its particles.root of the mass of its particles.

Graham’s LawGraham’s Law

Rate of effusion for gas 1

Rate of effusion for gas 2

M2

M1

ExamplesExamples• A compound effuses through a A compound effuses through a

porous cylinder 3.20 times faster porous cylinder 3.20 times faster than helium. What is it’s molar mass?than helium. What is it’s molar mass?

• If 0.00251 mol of NHIf 0.00251 mol of NH33 effuse through effuse through

a hole in 2.47 min, how much HCl a hole in 2.47 min, how much HCl would effuse in the same time?would effuse in the same time?

• A sample of NA sample of N22 effuses through a effuses through a

hole in 38 seconds. what must be the hole in 38 seconds. what must be the molecular weight of gas that effuses molecular weight of gas that effuses in 55 seconds under identical in 55 seconds under identical conditions?conditions?

DiffusionDiffusion

•The mixing of gases (the The mixing of gases (the spreading of gas through a room)spreading of gas through a room)

•Slow considering that gases Slow considering that gases move at 100’s of meters per move at 100’s of meters per secondsecond

•Why is it so slow?Why is it so slow?

•Collisions with other molecules Collisions with other molecules slow it down.slow it down.

•Best estimate is Graham’s LawBest estimate is Graham’s Law

Real GasesReal Gases

•Real molecules do take up space Real molecules do take up space and they do interact with each and they do interact with each other (especially polar other (especially polar molecules).molecules).

•Need to add correction factors to Need to add correction factors to the ideal gas law to account for the ideal gas law to account for these.these.

Volume CorrectionVolume Correction

•The actual volume free to move in is The actual volume free to move in is less because of particle size.less because of particle size.

•More molecules will have more More molecules will have more effect.effect.

•Corrected volume V’ = V - nbCorrected volume V’ = V - nb

•b is a constant that differs for each b is a constant that differs for each gas.gas.

•P’ = nRTP’ = nRT (V-nb) (V-nb)

Pressure correctionPressure correction

•Because the molecules are attracted Because the molecules are attracted to each other, the pressure on the to each other, the pressure on the container will be less than idealcontainer will be less than ideal

•depends on the number of depends on the number of molecules per liter.molecules per liter.

•since two molecules interact, the since two molecules interact, the effect must be squared.effect must be squared.

Pressure correctionPressure correction Because the molecules are attracted Because the molecules are attracted

to each other, to each other, the pressure on the the pressure on the container will be less than idealcontainer will be less than ideal

depends on the number of molecules depends on the number of molecules per liter.per liter.

since two molecules interact, the since two molecules interact, the effect must be squared.effect must be squared.

Pobserved = P’ - a

2

( )Vn

AltogetherAltogether• PPobsobs= nRT - a n = nRT - a n 22

V-nb VV-nb V

• Called the Van der Waal’s equation Called the Van der Waal’s equation

if rearrangedif rearranged

Corrected Corrected Corrected Corrected Pressure Pressure Volume Volume

( )

Pobs + an

V

2

x V - nb nRT

Where does it come fromWhere does it come from

•a and b are determined by a and b are determined by experiment.experiment.

•Different for each gas.Different for each gas.

•Bigger molecules have larger b.Bigger molecules have larger b.

•a depends on both size and a depends on both size and polarity.polarity.

•once given, plug and chug.once given, plug and chug.

ExampleExample

•Calculate the pressure exerted by Calculate the pressure exerted by 0.5000 mol Cl0.5000 mol Cl22 in a 1.000 L in a 1.000 L container at 25.0ºCcontainer at 25.0ºC

•Using the ideal gas law.Using the ideal gas law.

•Van der Waal’s equationVan der Waal’s equation

a = 6.49 atm La = 6.49 atm L22 /mol /mol22

b = 0.0562 L/molb = 0.0562 L/mol