Gases

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GasesGases

Chapter 5

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Gas PropertiesGas PropertiesFour properties determine the physical behavior of

any gas:Amount of gasGas pressureGas volumeGas temperature

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Gas pressureGas pressureGas molecules

exert a force on the walls of their container when they collide with it

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Gas pressureGas pressureGas pressure can support a column of liquidPliquid = g•h•d

g = acceleration due to the force of gravity (constant)h = height of the liquid columnd = density of the liquid

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Atmospheric Atmospheric pressurepressure

Torricelli barometer In the closed tube, the liquid

falls until the pressure exerted by the column of liquid just balances the pressure exerted by the atmosphere.

Patmosphere = Pliquid = ghd Patmosphere liquid height

Standard atmospheric pressure (1 atm) is

760 mm Hg

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Units for pressureUnits for pressure

In this course we usually convert to atm

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Gas pressureGas pressurePliquid = g•h•d

Pressure exerted by a column of liquid is proportional to the height of the column and the density of the liquid

Container shape and volume do not affect pressure

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ExampleExampleA barometer filled with perchloroethylene

(d = 1.62 g/cm3) has a liquid height of 6.38 m. What is this pressure in mm Hg (d = 13.6 g/cm3)?P = ghd = g hpce dpce = g hHg dHg hpce dpce = hHg dHg hHg = hpce d pce = (6.38 m)(1.62 g/cm3) = 0.760 m

dHg 13.6 g/cm3 hHg = 760 mm Hg

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Gas pressureGas pressureA manometer compares the pressure of a gas in a

container to the atmospheric pressure

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Gas Laws: BoyleGas Laws: BoyleIn 1662, Robert Boyle

discovered the first of the simple gas laws

PV = constant

For a fixed amount of gas at constant temperature, gas pressure

and gas volume are inversely proportional

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Gas Laws: CharlesGas Laws: CharlesIn 1787, Jacques Charles discovered a relationship

between gas volume and gas temperature:

volume (mL)

temperature (°C)

• relationship between volume and temperature is always linear

• all gases reach V = 0 at same temperature, –273.15 °C

• this temperature is ABSOLUTE ZEROABSOLUTE ZERO

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A temperature scale for gases:A temperature scale for gases:the Kelvin scalethe Kelvin scale

A new temperature scale was invented: the Kelvin or absolute temperature scale

K = °C + 273.15Zero Kelvins = absolute zero

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Gas laws: CharlesGas laws: CharlesUsing the Kelvin scale, Charles’ results is

For a fixed amount of gas at constant pressure, gas volume and gas temperature are directly proportional

A similar relationship was found for pressure and temperature:

VT

=constant

PT

=constant

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Standard conditions for gasesStandard conditions for gasesCertain conditions of pressure and temperature

have been chosen as standard conditions for gasesStandard temperature is 273.15 K (0 °C)Standard pressure is exactly 1 atm (760 mm Hg)

These conditions are referred to as STP (standard temperature and pressure)

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Gas laws: AvogadroGas laws: AvogadroIn 1811, Avogadro proposed that equal volumes of

gases at the same temperature and pressure contain equal numbers of particles.At constant temperature and pressure, gas volume is

directly proportional to the number of moles of gas

Standard molar volume: at STP, one mole of gas occupies 22.4 L

Vn

=constant

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Putting it all together:Putting it all together:Ideal Gas EquationIdeal Gas Equation

Combining Boyle’s Law, Charles’ Law, and Avogadro’s Law give one equation that includes all four gas variables:

R is the ideal or universal gas constantR = 0.08206 atm L/mol K

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PVnT

=R or PV =nRT

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Using the Ideal Gas EquationUsing the Ideal Gas EquationIdeal gas equation may be expressed two ways:

One set of conditions: ideal gas law

Two sets of conditions: general gas equation

PV =nRT

P1V1

n1T1

=P2V 2

n2T2

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Ideal Gas Equation and molar massIdeal Gas Equation and molar massSolving for molar mass (M)

PV =nRTn =

mM

PV =mRTM

M =mRTPV

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Ideal Gas Equation and gas densityIdeal Gas Equation and gas density

PV =nRT =mRTM

P =mRTVM

d =mV

P =dRTM

d =MPRT

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Gas densityGas densityGas density depends directly

on pressure and inversely on temperature

Gas density is directly proportional to molar mass

d =MPRT

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Mixtures of GasesMixtures of GasesIdeal gas law applies to pure gases and to mixturesIn a gas mixture, each gas occupies the entire

container volume, at its own pressureThe pressure contributed by a gas in a mixture is

the partial pressure of that gasPtotal = PA + PB (Dalton’s Law of Partial

Pressures)

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Mixtures of Mixtures of GasesGases

When a gas is collected over water, it is always “wet” (mixed with water vapor).Ptotal = Pbarometric = Pgas + Pwater vapor Example: If 35.5 mL of H2 are collected over water at

26 °C and a barometric pressure of 755 mm Hg, what is the pressure of the H2 gas? The water vapor pressure at 26 °C is 25.2 mm Hg.

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Gas mixturesGas mixturesThe mole fraction represents the contribution of

each gas to the total number of moles.XA = mole fraction of A

XA =nA

ntotal

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Gas MixturesGas MixturesFor gas mixtures,

mole fractionequals

pressure fractionequals

volume fraction

Each gas occupiesthe entire container.

The volume fraction describesthe % composition by volume.

nAntotal

=PA

Ptotal

=V A

V total

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Gases in Chemical ReactionsGases in Chemical ReactionsTo convert gas volume into moles for

stoichiometry, use the ideal gas equation: If both substances in the problem are gases, at the

same T and P, gas volume ratios = mole ratios.

P2 = P1 and T2 = T1

n2

n1

=P2V 2RT2

P1V1RT1

=V 2

V1

n =PVRT

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A Model for Gas BehaviorA Model for Gas BehaviorGas laws describe what gases do, but not why.Kinetic Molecular Theory of Gases (KMT) is the

model that explains gas behavior.developed by Maxwell & Boltzmann in the mid-1800sbased on the concept of an ideal or perfect gas

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Ideal gasIdeal gas Composed of tiny particles in constant, random, straight-line motion Gas molecules are point masses, so gas volume is just the empty

space between the molecules Molecules collide with each other and with the walls of their

container The molecules are completely independent of each other, with no

attractive or repulsive forces between them. Individual molecules may gain or lose energy during collisions, but

the total energy of the gas sample depends only on the absolute temperature.

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Molecular collisions and pressureMolecular collisions and pressureForce of molecular collisions depends on

collision frequencymolecule kinetic energy, ek

ek depends on molecule mass m and molecule speed u molecules move at various speeds in all directions

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ek =12mu2

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Molecular speedMolecular speedMolecules move at various speedsImagine 3 cars going 40 mph, 50 mph, and 60 mph

Mean speed = u = (40 + 50 + 60) ÷ 3 = 50 mphMean square speed (average of speeds squared)

u2 = (402 + 502 + 602) ÷ 3 = 2567 m2/hr2 Root mean square speed

urms = √2567 m2/hr2 = 50.7 mph

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Distribution of molecule speedsDistribution of molecule speeds

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The basic equation The basic equation of KMTof KMT

Combining collision frequency, molecule kinetic energy, and the distribution of molecule speeds gives the basic equation of KMTP = gas pressure and V = gas volumeN = number of moleculesm = molecule massu2 = mean square molecule speed (average of speeds squared)

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P =13NVmu2

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Combine the Equations ofCombine the Equations ofKMT and Ideal GasKMT and Ideal Gas

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P =13NA

Vmu2

PV=13NAmu2 =RT

If n = 1,N = NA

andPV = RT

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P =13NVmu2

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PV=nRTAvogadro’s number

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Combine the Equations ofCombine the Equations ofKMT and Ideal GasKMT and Ideal Gas

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PV=13NAmu2 =RT

NAmu2 =3RT=Mu2

NA x m (Avogadro’s number x mass of one molecule)= mass of one mole of molecules (molar mass M)

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Combine the Equations ofCombine the Equations ofKMT and Ideal GasKMT and Ideal Gas

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3RT=Mu2

u2 =urms=3RTM

We can calculate the root mean square speedfrom temperature and molar mass

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Calculating root mean square speedCalculating root mean square speedTo calculate root mean square speed from

temperature and molar mass:Units must agree!Speed is in m/s, so

R must be 8.3145 J/mol K M must be in kg per mole, because Joule = kg m2 / s2

Speed is inversely related to molar mass: light molecules are faster, heavy molecules are slower

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urms=3RTM

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Interpreting temperatureInterpreting temperatureCombine the KMT and ideal gas equations again

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ek =32RNA

T=constant×T

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PV=13NAmu

2 =23 ×1

2NAmu2

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PV=23NA

12( mu2)=2

3NAek =RT

Again assume n=1, so N = NA and PV = RT

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Interpreting Interpreting temperaturetemperature

Absolute (Kelvin) temperature is directly proportional to average molecular kinetic energy

At T = 0, ek = 0

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ek =32RNA

T

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Diffusion and EffusionDiffusion and EffusionDiffusion (a) is

migration or mixing due to random molecular motion

Effusion (b) is escape of gas molecules through a tiny hole

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Rates of diffusion/effusionRates of diffusion/effusionThe rate of diffusion or effusion is directly

proportional to molecular speed:

The rates of diffusion/effusion of two different gases are inversely proportional to the square roots of their molar masses (Graham’s Law)

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rate of effusion of Arate of effusion of B=(urms)A

(urms)B= 3RT MA

3RT MB= MB

MA

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Using Graham’s LawUsing Graham’s Law

Graham’s Law applies to relative rates, speeds, amounts of gas effused in a given time, or distances traveled in a given time.

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rate of effusion of Arate of effusion of B=(urms)A

(urms)B= MB

MA

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Using Graham’s Law with timesUsing Graham’s Law with timesGraham’s law can be confusing when applied to

times

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rate Arate B=

nA/tAnB/tB

= MB

MA

For same amounts of A and B,tBtA

= MB

MA

rate = amount of gas (n) time (t)

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Use common senseUse common sensewith Graham’s Lawwith Graham’s Law

When you compare two gases, the lighter gasescapes at a greater rate has a greater root mean square speedcan effuse a larger amount in a given timecan travel farther in a given timeneeds less time for a given amount to escape or travel

Make sure your answer reflects this reality!

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Reality CheckReality CheckIdeal gas molecules Real gas molecules

constant, random, same straight-line motion point masses are NOT points – molecules have volume; Vreal gas

> Videal gas

independent of each other are NOT independent – molecules are attracted to each other, so Preal gas < Pideal gas

gain / lose energy during same (some energy may becollisions, but total energy absorbed in moleculardepends only on T ek))

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Real gas correctionsReal gas correctionsFor a real gas,

a corrects for attractions between gas molecules, which tend to decrease the force and/or frequency of collisions (so Preal < Pideal)

b corrects for the actual volume of each gas molecule, which increases the amount of space the gas occupies (so Vreal > Videal)

The values of a and b depend on the type of gas

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An equation for real gases:An equation for real gases:the van der Waals equationthe van der Waals equation

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Preal+n2aV 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ Vreal– nb( ) =nRT=PidealVideal

Add correction to Preal to make it equal to Pideal,because intermolecular attractions decrease real pressure

Subtract correction to Vreal to make it equal to Videal,because molecular volume increases real volume

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When do I needWhen do I needthe van der Waals equation?the van der Waals equation?

Deviations from ideality become significant whenmolecules are close together (high pressure)molecules are slow (low temperature)

At low pressure and high temperature, real gases tend to behave ideally

At high pressure and low temperature, real gases do not tend to behave ideally

}non-idealconditions