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Kinetic Molecular Theory

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Kinetic Molecular Theory & Gases An Honors/AP Chemistry Presentation Kinetic Molecular Theory Kinetic means motion So the K.M.T. studies the motions of molecules. – PowerPoint PPT presentation

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Title: Kinetic Molecular Theory


1
Kinetic Molecular Theory Gases
  • An Honors/AP Chemistry Presentation

2
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)!

3
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.

4
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)

5
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?

6
Answer to Temperature Conversion
  • -40 oF -40 oC
  • -40 oC 233 K or 230 K

7
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?

8
Answers to Pressure Conversions
  • 12.0 psi .816 atm
  • 1.25 atm 950. Torr
  • 720 mm Hg 96000 Pa

9
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.

10
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

11
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.

12
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

13
Boyles Law in Action
14
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?

15
The Answers are
  • P1V1P2V2
  • (1atm)(852cm3) (.750atm)V2 V2 1140cm3
  • (25.0psi)(2.20L)P2(2.75L) P2 20.0 psi

16
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

17
Charles Law in Action
18
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?

19
The Answers Are
  • V1/T1V2/T2
  • 5.00L/293K V2/313K V25.34L
  • 273K/3.00L T2/4.00L T2364K91oC

20
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

21
The Gay-Lussac Law in Action
22
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?

23
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

24
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

25
Avogadros Law in Action
26
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?

27
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

28
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

29
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

30
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?

31
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

32
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

33
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

34
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.

35
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

36
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.

37
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.

38
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

39
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

40
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?

41
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.

42
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

43
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

44
Sample Water Vapor Pressures
45
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?

46
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

47
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?

48
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

49
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?

50
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.

51
A Thought Question
  • Which of the following ideal gases would have the
    largest average kinetic energy at 25oC? He, N2,
    CO, or H2

52
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

53
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.

54
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.

55
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)

56
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

57
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.

58
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
59
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.

60
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?

61
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.

62
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.

63
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.

64
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.

65
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.

66
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.

67
Phase Diagrams
  • A phase diagram shows how the different states of
    matter exist based on the pressure and
    temperature.

68
A Typical Phase Diagram
69
Water is not Typical
70
and Helium is weird!
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