Kinetic Molecular Theory

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Kinetic Molecular Theory Gases

• An Honors/AP Chemistry Presentation

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Kinetic Molecular Theory

• Kinetic means motion
• So the K.M.T. studies the motions of molecules.
• Solids - vibrate a little
• Liquids - vibrate, rotate, and translate (a
  little)
• Gases - vibrate, rotate, and translate (a lot)!

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Basic Assumptions of KMT

• Gases consist of large numbers of molecules in
  continuous random motion.
• The volume of the molecules is negligible
  compared to the total volume.

• Intermolecular interactions are negligible.
• When collisions occur, there is a transfer of
  kinetic energy, but no loss of kinetic energy.
• The average kinetic energy is proportional to the
  absolute temperature.

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Gas Properties

• Volume - amount of space (L or mL)
• Temperature - relative amount of molecular motion
  (K)
• Pressure - the amount of force molecules exert
  over a given area (atm, Torr, Pa, psi, mm Hg)
• Moles - the number of molecules (mol)

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Temperature Conversions

• C 5/9(F-32)
• F 9/5C 32
• K C 273
• So what is the absolute temperature (K) of an
  object at -40 oF?

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Answer to Temperature Conversion

• -40 oF -40 oC
• -40 oC 233 K or 230 K

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Pressure Conversions

• 1 atm 760 mm Hg 760 Torr 101,325 Pa 14.7
  psi
• How many atmospheres is 12.0 psi?
• How many Torr is 1.25 atm?
• How many Pascals is 720 mm Hg?

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Answers to Pressure Conversions

• 12.0 psi .816 atm
• 1.25 atm 950. Torr
• 720 mm Hg 96000 Pa

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A Barometer

• A mercury barometer measures air pressure by
  allowing atmospheric pressure to press on a bath
  of mercury, forcing mercury up a long tube. The
  more pressure, the higher the column of mercury.

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More on the barometer

• Although American meteorologists will sometimes
  measure the height in inches, typically this
  pressure is measured in mm Hg.
• 1 mm Hg 1 Torr

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S.T.P.

• When making comparisons we often use benchmarks
  or standards to compare against.
• In chemistry Standard Temperature is 0 oC (273K)
  and Standard Pressure is 1 atm.

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Boyles Law

• If the amount and temperature of the gas are held
  constant, then the volume of a gas is inversely
  proportional to the pressure it exerts.
• Mathematically this means that the pressure times
  the volume is a constant.
• PV k
• P1V1P2V2

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Boyles Law in Action
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Sample Questions

• The volume of a balloon is 852 cm3 when the air
  pressure is 1.00 atm. What is the volume if the
  pressure drops to .750 atm?
• A gas is trapped in a 2.20 liter space beneath a
  piston exerting 25.0 psi. If the volume expands
  to 2.75 L, what is the new pressure?

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The Answers are

• P1V1P2V2
• (1atm)(852cm3) (.750atm)V2 V2 1140cm3
• (25.0psi)(2.20L)P2(2.75L) P2 20.0 psi

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Charles Law

• If the amount and the pressure of a gas are held
  constant, then the volume of a gas is directly
  proportional to its absolute temperature.
• Mathematically, this means that the volume
  divided by the temperature is a constant.
• V/T k
• V1/T1V2/T2

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Charles Law in Action
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Sample Questions

• The volume of a balloon is 5.00 L when the
  temperature is 20.0 oC. If the air is heated to
  40.0 oC, what is the new volume?
• 3.00 L of air are held under a piston at 0.00 oC.
  If the air is allowed to expand at constant
  pressure to 4.00 L, what is the new Celsius
  temperature of the gas?

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The Answers Are

• V1/T1V2/T2
• 5.00L/293K V2/313K V25.34L
• 273K/3.00L T2/4.00L T2364K91oC

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The Gay-Lussac Law

• If the amount and volume of the gas are held
  constant, then the pressure exterted by the gas
  is directly proportional to its absolute
  temperature.
• Mathematically this means that the pressure
  divided by the temperature is a constant.
• P/T k
• P1/T1P2/T2

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The Gay-Lussac Law in Action
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Sample Questions

• A tank of oxygen is stored at 3.00 atm and -20
  oC. If the tank is accidentally heated to 80 oC,
  what is the new pressure in the tank?
• A piston is trapped in place at a temperature of
  25 oC and a pressure of 112 kPa. At what celcius
  temperature is the pressure 102 kPa?

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The Answers are

• P1/T1P2/T2
• (3atm)/(253 K) P2/ (353 K) P2 4.19 atm
• (298 K)/(112 kPa)T2/(102kPa) T2 271 K -2
  oC

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Avogadros Law

• If the temperature and the pressure of a gas are
  held constant, then the volume of a gas is
  directly proportional to the amount of gas.
• Mathematically, this means that the volume
  divided by the of moles is a constant.
• V/n k or V/m k
• V1/n1V2/n2 or V1/m1V2/m2

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Avogadros Law in Action
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Sample Questions

• The volume of a balloon is 5.00 L when there is
  .250 mol of air. If 1.25 mol of air is added to
  the balloon, what is the new volume?
• 3.00 L of air has a mass of about 4.00 grams. If
  more air is added so that the volume is now 24.0
  L, what is the mass of the air now?

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The Answers Are

• V1/n1V2/n2 or V1/m1V2/m2
• 5.00L/.250 mol V2/1.50 mol V230.0 L
• 4.00g/3.00L m2/24.0L m232.0 g

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The Combined Gas Law

• This law combines the inverse proportion of
  Boyles Law with the direct proportions of
  Charles, Gay-Lussacs, and Avogadros Laws.
• P1V1/(n1T1) P2V2/(n2T2)
• or
• P1V1/T1 P2V2/T2

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Four Gas Laws in One

• The combined gas law could be used in place of
  any of the previous 4 gas laws.
• For example, in Boyles Law, we assume that the
  amount and temperature are constant. So if we
  cross them off of the combined gas law
• P1V1/(n1T1) P2V2/(n2T2)
• P1V1 P2V2

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Another Example

• A sample of hydrogen has a volume of 12.8 liters
  at 104 oF and 2.40 atm. What is the volume at
  STP?

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The answer is

• P1V1/(n1T1) P2V2/(n2T2)
• P12.40atm,V112.8L, T1104oF40oC313K, T2273K,
  P21atm, n1n2
• (2.4atm)(12.8L)/(313K) (1atm)V2/(273K)
• V2 26.8 L

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The Ideal Gas Law

• If, P1V1/(n1T1) P2V2/(n2T2)
• Then PV/(nT) constant
• That constant is R, the ideal gas law constant.
• R .0821 Latm/(molK)
• R 8.314 J/(molK)
• So, PVnRT

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But what about

• Since n m/M, we can substitute into PV nRT
  and get
• PVM mRT
• Since D m/V, we can substitute in again and get
• PM DRT

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So which one is it?

• Like a good carpenter, it is good to have many
  tools so that you can choose the right tool for
  the right job.
• If I am solving a gas problem with density, I use
  PM DRT.
• If I am solving a gas problem with moles, I use
  PV nRT.
• If I am solving a gas problem with mass, I use
  PVM mRT.

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Such as.

• Under what pressure would oxygen have a density
  of 8.00 g/L at 300 K?
• PM DRT
• P(32 g/mol) (8 g/L)(.0821 latm/molK)(300 K)
• P 6.16 atm

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An Important Number

• What is the volume of 1 mole of a gas at STP?
• PV nRT V nRT/P
• V (1mol)(.0821Latm/molK)(273K)/ (1atm)
• V 22.4 L
• This is called the standard molar volume of an
  ideal gas.

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Gas Stoichiometry

• We had said that stoichiometry implied a ratio of
  molecules, or moles. Up until now we only used
  mole ratios.
• However Avogadro said that the volume is directly
  proportional to the number of molecules.
• This means that we can do stoichiometry with
  volume or moles.

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Example 1 of Gas Stoichiometry

• What volume of hydrogen is needed to synthesize
  6.00 liters of ammonia?
• N2 (g) 3 H2 (g) --gt 2 NH3 (g)
• 6.00 L H2 x (2 NH3/3 H2) 4.00 L NH3

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Example 2 of Gas Stoichiometry

• What mass of nitrogen is needed to synthesize
  20.0 L of ammonia at 1.50 atm and 25 oC?
• N2 (g) 3 H2 (g) --gt 2 NH3 (g)
• 20.0 L NH3 x (1 N2/2 NH3) 10.0 L N2
• PVM mRT
• (1.5 atm)(10 L)(28 g/mol) m(.0821Latm/molK)(298K
  )
• m 17.2 g N2

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Daltons Law

• When we talk about air pressure, we need to
  understand that air is not oxygen.
• Air is a solution of nitrogen (78.09), oxygen
  (20.95), argon (.93), and CO2 (.03).
• So when we talk about air pressure, which gas are
  we talking about?

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ALL OF THEM!

• Daltons Law of Partial Pressures states that the
  total pressure of a system is equal to the sum of
  the partial (or individual) pressures of each
  component.
• Ptotal P1 P2 Px
• So if air pressure is 1 atm, then we can assume
  that the N2 is .78 atm, the O2 is .21 atm, and
  the Ar is about .01 atm.

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A Corollary

• If we extend Boyles Law and Avogadros Law, we
  could infer that, at constant temperature and
  volume, the pressure of a gas is directly
  proportional to its pressure.
• P1/Ptotal n1/ntotal

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An important example

• A sample of CaCO3 is heated, releasing CO2, which
  is collected over water (a typical practice).

• The pressure in the collection bottle is the sum
  of the pressure of the CO2 plus the pressure of
  the water vapor (since some water always
  evaporates).
• Ptotal PCO2 PH2O

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Sample Water Vapor Pressures
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So in our example

• If a total pressure of 365 Torr is collected at
  25 oC in a 100 ml collection bottle
• What is the partial pressure of CO2?
• What mass of CaCO3 decomposed?

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Heres how it works

• Ptotal PCO2 PH2O
• 365 Torr PCO2 23.8 Torr
• PCO2 341.2 Torr .449 atm
• PVM mRT
• (.449 atm)(.100 L)(44.0 g/mol)
  m(.0821Latm/molK)(298K)
• m .0807 g CO2

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Corollary Problem

• A gas collection bottle contains .25 mol of He,
  .50 mol Ar, and .75 mol of Ne. If the partial
  pressure of Helium is 200 Torr
• What is the total pressure in the system?
• What are the partial pressures of Ne and Ar?

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The answers are

• nHe .25 mol, nAr .50 mol, nNe .75 mol, PHe
  200 Torr.
• ntotal 1.50 mol
• Ptotal/Phe ntotal/nHe
• Ptotal/200Torr 1.50 mol/.25 mol
• Ptotal 1200 Torr
• PAr/Ptotal nAr/ntotal
• Par/1200 .50 mol/1.50 mol
• PAr 400 Torr
• PNe 1200 Torr - 400 Torr - 200 Torr
• Pne 600 Torr

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Temperature and Kinetic Energy

• Earlier, I stated that temperature is a relative
  measure of molecular motion.
• By definition, Kinetic energy is a measure of the
  energy of motion.
• Pretty similar right?

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Yes they are

• KEav 3/2RT
• The average kinetic energy depends only on the
  absolute temperature.
• R, the Ideal Gas Law Constant, should be 8.314
  J/molK, since we will want the energy in the
  proper SI unit of Joules.

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A Thought Question

• Which of the following ideal gases would have the
  largest average kinetic energy at 25oC? He, N2,
  CO, or H2

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They are all the same!

• Since Keav 3/2RT, the mass does not make a
  difference (ideally).
• KE 3/2(8.314J/molK)(298K)
• KE 3716 J/mol

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Speed vs Kinetic Energy

• In physics, you learned that KE 1/2mv2. The
  velocity, v, describes the speed of an object in
  a specific direction. If the mass, m, is
  measured in kg and the velocity is measured in
  m/s, then the kinetic energy would be measured in
  Joules.

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Physics to Chemistry

• Rewriting the physics version, we could say that
  vv(2KE/m).
• In chemistry, the Kinetic energy is measured in
  J/mol, so the mass would have to be measured in
  Kg/mol which is essentially molar mass.

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Root Mean Square Speed

• In Chemistry, we are not worried about velocities
  in multiple directions. We want an average speed
  independent of direction.
• We call this Vrms - the root mean square speed.
• Vrms v(3RT/M)

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A Thought Question Revisited

• Which of the following ideal gases would have the
  largest root mean square speed at 25oC? He, N2,
  CO, or H2

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This Time They Are Different

• Vrms v(3RT/M)
• For He, Vrms v(3RT/M) v(38.314J/molK298K)/4g
  /mol 1363 m/s
• For N2, Vrms v(3RT/M) v(38.314J/molK298K)/28
  g/mol 515 m/s
• Since the molar mass is the same for N2 and CO,
  their Vrms would be the same, 515 m/s.
• For H2, Vrms v(3RT/M) v(38.314J/molK298K)/2g
  /mol 1928 m/s
• Because H2 is the lightest, it moves the fastest.

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And this leads us to

• Grahams Law
• The rate of effusion (or diffusion) is inversely
  proportional to the square root of the molar
  mass.
• Effusion is the process of a gas escaping from
  one container through a small opening.
• Diffusion is the process of a gas spreading out
  in a large container.

Rate1vM1 Rate2vM2
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Rate vs. Speed

• When we say rate, we are talking about an amount
  of gas (moles, grams, or even liters) per unit of
  time.
• This is not the same as speed which is distance
  over time.
• However, the main idea is the same lighter gases
  move/effuse faster.

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For example

• Under a given set of conditions, oxygen diffuses
  at 10 L/hr. A different gas diffuses at 20 L/hr
  under the same conditions. What is the molar
  mass of this gas?

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2 ways to solve this

• By the equation
• Rate1vM1 Rate2vM2
• 10 L/hr(v32g/mol) 40 L/hr vM2
• M2 2 g/mol
• By Logic
• If the rate of the unknown gas is 4 times faster,
  it must be 42, or 16, times lighter.
• 32 g/mol divided by 16 is 2 g/mol.

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Real vs. Ideal

• At the start of the presentation, we talked about
  the major assumptions of the Kinetic Molecular
  Theory.
• If a gas obeys the KMT, it is ideal.
• If it doesnt obey the KMT, it is real.

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So What does that Mean?

• The molecules of an ideal gas do not interact
  with one another, except to collide elastically.
• The molecules of a real gas will interact, to
  some degree.
• Since no gases are always ideal, the trick is to
  make a real gas behave ideally.

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Real Gases Behaving Ideally

• If we dont want the molecules attracting or
  repelling one another, the first issue is to use
  a nonpolar gas.
• If we use smaller amounts of the gas, there are
  less chances of them interacting.

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Real Gases Behaving Ideally

• If we put the gas in larger volumes, the
  molecules will not interact as much.
• Likewise, if we keep the gas under low pressure ,
  the molecules will not interact as much.
• This could also be stated by having molecules
  that have low densities.

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Real Gases Behaving Ideally

• The smaller the molecules, the less likely they
  are to interact.
• Lastly, at higher temperatures the molecules are
  moving too fast to actually interact with one
  another - they are more likely to collide
  elastically.

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Phase Diagrams

• A phase diagram shows how the different states of
  matter exist based on the pressure and
  temperature.

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A Typical Phase Diagram
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Water is not Typical
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and Helium is weird!