1 Gases Chapter 5. 2 Gas Properties Four properties determine the physical behavior of any gas:...

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GasesGases

Chapter 5

Gas PropertiesGas Properties

Four properties determine the physical behavior of any gas:Amount of gasGas pressureGas volumeGas temperature

Gas pressureGas pressure

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

Gas 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

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 heightStandard atmospheric

pressure (1 atm) is 760 mm Hg

Units for pressureUnits for pressure

In this course we usually convert to atm

Pliquid = g•h•dPressure 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

ExampleExample

A 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 mdHg 13.6 g/cm3

hHg = 760 mm Hg

A manometer compares the pressure of a gas in a container to the atmospheric pressure

Gas Laws: BoyleGas Laws: Boyle

In 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

Gas Laws: CharlesGas Laws: Charles

In 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

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

Gas laws: CharlesGas laws: Charles

Using the Kelvin scale, Charles’ results isFor 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:

T=constant

Standard conditions for gasesStandard conditions for gases

Certain 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)

Gas laws: AvogadroGas laws: Avogadro

In 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

n=constant

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

=R or PV =nRT

Using the Ideal Gas EquationUsing the Ideal Gas Equation

Ideal gas equation may be expressed two ways:One set of conditions: ideal gas law

Two sets of conditions: general gas equation

PV =nRT

Ideal Gas Equation and molar massIdeal Gas Equation and molar mass

Solving for molar mass (M)

PV =nRTn =

PV =mRTM

M =mRTPV

Ideal Gas Equation and gas densityIdeal Gas Equation and gas density

PV =nRT =mRTM

P =mRTVM

P =dRTM

d =MPRT

Gas densityGas density

Gas density depends directly on pressure and inversely on temperature

Gas density is directly proportional to molar mass

d =MPRT

Mixtures of GasesMixtures of Gases

Ideal 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)

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.

Gas mixturesGas mixtures

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

XA =nA

ntotal

Gas MixturesGas Mixtures

For gas mixtures,

mole fractionequals

pressure fractionequals

volume fraction

Each gas occupiesthe entire container.

The volume fraction describesthe % composition by volume.

nAntotal

Ptotal

Vtotal

Gases in Chemical ReactionsGases in Chemical Reactions

To 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

=P2V2RT2

P1V1RT1

n =PVRT

A Model for Gas BehaviorA Model for Gas Behavior

Gas 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

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.

Molecular collisions and pressureMolecular collisions and pressure

Force of molecular collisions depends oncollision frequencymolecule kinetic energy, ek

ek depends on molecule mass m and molecule speed u

molecules move at various speeds in all directions

ek =12mu2

Molecular speedMolecular speed

Molecules 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

Distribution of molecule speedsDistribution of molecule speeds

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)

P =13NVmu2

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

P =13NA

PV=13NAmu

If n = 1,N = NA

andPV = RT

P =13NVmu2

PV=nRTAvogadro’s number

PV=13NAmu

NAmu2 =3RT=Mu2

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

3RT=Mu2

u2 =urms=3RTM

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

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

urms=3RTM

Interpreting temperatureInterpreting temperature

Combine the KMT and ideal gas equations again

ek =32

T=constant×T

PV=13NAmu

2 =23 ×1

2NAmu2

PV=23NA

12( mu2

)=23NAek =RT

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

Interpreting Interpreting temperaturetemperature

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

At T = 0, ek = 0

ek =32

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

Rates of diffusion/effusionRates of diffusion/effusion

The 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)

rate of effusion of Arate of effusion of B

=(urms)A

(urms)B

=3RT MA

3RT MB

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.

rate of effusion of Arate of effusion of B

=(urms)A

(urms)B

Using Graham’s Law with timesUsing Graham’s Law with times

Graham’s law can be confusing when applied to times

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

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!

Reality CheckReality Check

Ideal gas molecules Real gas molecules constant, random, same

straight-line motion point masses are NOT points – molecules

have volume; Vreal gas > Videal

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))

Real gas correctionsReal gas corrections

For 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

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

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

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

1 Gases Chapter 5. 2 Gas Properties Four properties determine the physical behavior of any gas:...

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