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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.
- Solids - vibrate a little
- Liquids - vibrate, rotate, and translate (a

little) - Gases - vibrate, rotate, and translate (a lot)!

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.

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)

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?

Answer to Temperature Conversion

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

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?

Answers to Pressure Conversions

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

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.

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

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.

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

Boyles Law in Action

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?

The Answers are

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

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

Charles Law in Action

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?

The Answers Are

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

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

The Gay-Lussac Law in Action

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?

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

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

Avogadros Law in Action

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?

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

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

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

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?

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

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

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

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.

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

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.

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.

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

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

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?

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.

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

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

Sample Water Vapor Pressures

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?

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

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?

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

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?

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.

A Thought Question

- Which of the following ideal gases would have the

largest average kinetic energy at 25oC? He, N2,

CO, or H2

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

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.

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.

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)

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

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.

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

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.

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?

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.

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.

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.

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.

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.

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.

Phase Diagrams

- A phase diagram shows how the different states of

matter exist based on the pressure and

temperature.

A Typical Phase Diagram

Water is not Typical

and Helium is weird!