Gas Laws and KMT

Chapter 5

Pressure• Barometer – first

pressure measuring device– Torricelli, 1643– A glass tube filled with

mercury, inverted into a dish of mercury.

– At sea level, height of mercury in the tube is 760 mm

• Why does the Hg stay in the tube, defying gravity?

Air Pressure

• Why does a barometer measure lower air pressure when a storm is approaching?– Lower air pressure means the weight of air

being pulled toward the earth is lower• Air is being pulled UP, so air is rushing into a

low (wind)

• Air pressure is also lower at higher elevation– At 9600 ft, air pressure is only 560 mm– Less air pushing down on earth’s surface

ManometerThe principle of a manometer measurement depends on the fact that given the same fluid, pressure is the same at equal heights. Pgas= Patm – h OR Pgas= Patm + h

Manometer is a substitute for a Barometer and both measure mm HgMm Hg =Torr

Pressure

• Standard Atmosphere = 760 torr = 760 mm Hg

• Pressure = Force/area

• SI units of measure– Force = Newtons– Area = m2

– SI unit of measure for pressure is Pascal (Pa)

• 1 Standard atmosphere – 101,325 Pa

Gas Laws

• Boyles Law

• Charles Law

• Guy-Lussac’s Law (Not used much)

• Avogadro’s Law

• Ideal Gas Law

• All Lead To:– Gas Stoichiometry

Boyles Law

• Boyle studied pressure and volume

• PV = k– Temperature constant– Amount of gas constant

• Variation:– V=k/P– P=k/V

• Boyles Law is also frequently written and used as:– P1V1 = P2V2

Charles Law• Studied relationship between pressure and

temperature• Determined that plots of volume vs. temperature

are linear• V = bT

– Constant pressure– Constant amount of gas– NOTE: gas cannot have a negative volume, so

temperature cannot be negative. Thus we MUST use Kelvin scale for temperature at all times.

– More on this later

• Variations:– V/T=b

Charles Law

• Charles Law is also frequently written and used as:– V1/T1 = V2/T2

Avogadro’s Law

• Postulated that equal volumes of gases at the same temperature and pressure contain the same number of ‘particles’.

• Avogadro’s Law– V = an

• a = proportionality constant

• N = number of moles of gas

• Variations:– V/n = a (constant)

Combined Gas Law

• Assumes constant amount of gas– PV/T = k– Or

– P1V1/T1 = P2V2/T2

Ideal Gas Law

• Boyles Law: V = k/P (constant T & n)• Charles Law: V = bT (constant P & n)• Avogadro’s Law: V=an (constant P & T)• Combined: V = R(Tn/P)

– Or PV=nRT– R is universal gas constant (0.08206

L*Atm/mole*K)– MAKE SURE ALL TEMPS ARE IN KELVIN

• This is the IDEAL GAS LAW– Real gasses behave somewhat differently

Gas Stoichiometry

• Molar volume of a gas = 22.42 L at standard temperature and pressure

• STP (standard temperature and pressure)– 0ºC (273K)– 1 atm (760 torr or 760 mm Hg

• Using gas density:– Density = mass/volume– PV=nRT P = nRT/V

Gas Stoichiometry Using Density

• Density = mass/volume– PV=nRT P = nRT/V– n = mass/molar mass =m/molar mass– P = (m/molar mass)RT/V– P = mRT/V(molar mass)– m/V = density (d)– P = dRT/molar mass– Molar mass = dRT/P

Dalton’s Law of Partial Pressures

• Applies to Gasses

• For a mixture of gasses in a container, the total pressure is the sum of the pressures that each gas will exert if it were alone– PTOTAL = P1 + P2 + P3+…..

– PTOTAL = n1RT/V + n2RT/V + n3RT/V….

– Equals (n1 + n2 + n3 + …)RT/V

– Equals NtotalRT/V

Collecting Gas Over Water

• Whenever you collect gas over water, water vapor is present:– Water molecules escape from surface of

water– Pressure due to water, depends on

temperature, and is the vapor pressure of water.

– Total pressure of gas collected is Pressure of gas + pressure of water vapor

Kinetic Molecular Theory

• A theory summarizes observed behavior

• A model allows you to use theories to predict behavior– Also can be viewed as a way of

understanding, a way of thinking, a mental construct

• KMT is a model based on gas laws

Kinetic Molecular Theory• Particles are so small compared to distances

between the particles that the volume of the particles can be assumed to be negligible (zero).

• The particles are in constant motion. Collisions of the particles with the walls of the container are the cause of pressure exerted by the gas.

• The particles are assumed to exert no forces on each other; they are assumed to neither attract nor repel each other.

• The average kinetic energy of a collection of gas particles is assumed to be directly proportional to the Kelvin temperature of the gas.

Boyles Law

• If volume decreases, pressure increases.

• KMT says a decrease in volume means the particles will hit the wall more often

Pressure and Temperature

• Ideal Gas Law: Pressure is directly proportional to temperature

• KMT: as temperature increases;– Speeds of particles increases– Particles hit wall with greater force– Particles hit walls with greater frequency– Result: increased pressure

Charles Law

• Ideal Gas Law: at constant pressure, volume of gas is directly proportional to temperature Kelvin

• KMT: When heated;– Speed of molecules increases– Hit walls with greater force– Hit walls with greater frequency– Only way to keep pressure constant is to

increase volume

Avogadro’s Law

• Ideal Gas Law: Volume is directly proportional to number of particles present– Constant temperature & pressure

• KMT: If you add more particles to a container;– Pressure would increase– Only way to maintain pressure is to

increase volume

Dalton’s Law

• Dalton: Total pressure is the sum of the partial pressures

• KMT: Assumes;– all gas particles are independent of each

other– Volumes of individual particles are

unimportant– Identities of particles do not matter

Deriving Ideal Gas Law

• Apply particle physics to assumptions of KMT:– Use definitions of velocity, momentum, force,

pressure– See Appendix 2 for details– KE = 1/2 mv2 where v is root mean squared

speed.

– Total kinetic energy is KE = NA(1/2 mv2) where N is Avogadro's number

– Final derivation is: P = 2/3 (n NA(1/2 mv2) / V)

What is Temperature?

• KMT bases temperature on Kelvin– Because it is based on average kinetic

energy of the particles– Requires an absolute energy scale– Hence: Kelvin

Problem:

• Calculate the average kinetic energy of the CH4 particles in a sample of CH4 gas at 273K and at 546K

Thursday

• Effusion and Graham’s Law

• Diffusion

• Real Gases and van der Waal’s equation.

Effusion & Diffusion

• Diffusion – the mixing of gases without agitation

• Effusion – passage of a gas through a tiny orifice (hole)

Effusion

• Graham’s Law of Effusion

Rate of Effusion for gas 1Rate of Effusion for gas 2

=√M2

√M1

M1 and M2 are molar masses for the gases.

Diffusion

• Diffusion takes a long time– Even though molecules are travleing 450

and 660 m/s– Why?– Tube is filled with air– Lots of collisions with air that don’t lead to

a reaction– Difficult to describe theoretically

Next

• Real Gases– Corrections for pressure– Corrections for volume

Real Gases

• Ideal gas behavior is best thought of as the behavior approached by real gases under certain conditions.

• Ideal gas behavior fails at:– Low temperatures– High pressures

• Real gases behave most like ideal gases at:– High temperature– Low pressure

Real Gases

• Ideal gas assumption of volume is incorrect:– Molecules always take up some space.– Correct for volume by subtracting volume

for the molecules

– VREAL = VIDEAL-nb

• n is number of moles

• b is an empirical correction constant

• So:– P’ = nRT/(V-nb)

Real Gases

• But we still have to correct for the fact that real gases DO have attraction forces.– Effect is to make observed pressure POBS smaller

than it normally would if there were no attractions:

– POBS = (P’ – correction factor)(nRT/(V-nb) – correction factor

– Size of correction factor depends on concentration of the gas molecules in particles per liter (n/V)

• Higher concentration, more likely particles are close enough to attract. Depends on square of number of particles because 2 particles have to get close enough.

Real Gases

• So:– POBS = P’ – s(n/V)2

• Inserting correction factors for both volume and attractions gives the equation:– POBS = (nRT/(V-nb)) – a(v/V)2

Volume of container

Volume correction factor

Pressure correction factor

Van Der Waals Equation

• POBS + a(n/V)2 x (V-nB) = nRT