• There is a lot of “free” space There is a lot of “free” space in a gas.in a gas.

• Gases can be expanded Gases can be expanded infinitely.infinitely.

• Gases fill containers Gases fill containers uniformly and completely.uniformly and completely.

• Gases diffuse and mix rapidly.Gases diffuse and mix rapidly.

So you see there is no such thing as still air. The air molecules are constantly moving at an average of 1,000 miles per hour.

Gas properties can be modeled using math. Gas properties can be modeled using math.

Model depends on—Model depends on—• V = volume of the gas (L)V = volume of the gas (L)• T = temperature (K)T = temperature (K)– ALL temperatures in MUST be in Kelvin!!! ALL temperatures in MUST be in Kelvin!!!

No Exceptions!No Exceptions!• n = amount (moles)n = amount (moles)• P = pressureP = pressure

(atmospheres) (atmospheres)

9

0

18

9 hits9 sec

1 hitsec=

18 hits9 sec

2 hitssec=

00010203040506070809

Seconds

½ volume Pressure comes from the gas molecules hitting the side of the container. Let’s count them out loud.

Hits

So we saw that as volume decreases the pressure increases.

V=0.5 , P=2

V=0.1, P=10

V=6 , P=5

V=3, P=10

They are inversely proportional. BOYLES LAW

We can also show this by having them multiply by each other.

9

0

18

9 hits9 sec

1 hitsec=

00010203040506070809

Seconds

27 ºC = 300 K

27 ºC = 300 K

0 K

We saw that we can increase pressure by reducing the volume, but we can also do it by increasing the temperature and therefore the speed of the gas molecules. At room temperature the hits are 1 hit/sec

9

0

18

9 hits4.5 sec

2 hitssec=

00010203040506070809

Seconds

27 ºC = 300 K

327 ºC = 600 K

327 ºC = 600 K

0 K

We are going from room temperature 27 ºC = 300 K to double that temperature, which is 600 Kelvin. Let’s count the number of collisions at this higher speed. We get twice the number of collisions and therefore twice the pressure.

15 psi, 300 K30 psi

3 psi

600 K

60 K

So we just saw that when temperature goes up, so does the pressure. This makes sense because higher temperature means the gas molecules are going faster, colliding more often, and hitting harder.

Gay-Lussac’s Law

Another way to increase pressure is to increase the number of gas molecules. This is the approach the steam engine used by heating water.

Pressure is proportional to the number of gas molecules, which we count in moles.

This is also a safety problem. Any closed container that has liquid in and gets heated will likely increase pressure dramatically until the container bursts.

Let’s review what we learned. If the volume decreases the pressure will increase. Then the reverse happens if the volume increases. The pressure drops as gas molecules are farther apart.

As we also learned, we can increase pressure by introducing more molecules of the gas into the volume.

doubles

We also learned that if temperature doubles, the pressure doubles if volume is fixed. Or if the container is flexible, the volume will double with pressure staying constant. Or both can increase such that the product of the two doubles.

• P is pressure measured in atmospheres. • V is volume measured in Liters• n is moles of gas present.• R is a constant that converts the units. It's value is 0.0821

atm•L/mol•K• T is temperature measured in Kelvin.• Simple algebra can be used to solve for any of these values. • P = nRT V = nRT n = PV T = PV R = nT• V P RT nR PV

To make these quantities equal, we need a conversion constant. We call it R (the Universal Gas Constant)

• Pressure=1 atmosphere• Volume=1 Liter• n = 1 mole• R=0.0821 L atm mol-1 K-1

• What is the temperature?

Let’s find what temperature the gas must be if we have the following readings for these other properties.

Normally 1 mole of a gas at 1 atmosphere pressure takes up 22.4 liters. So it must be very cold to only have a volume of 1 liter.

Pressure of air is Pressure of air is measured with a measured with a BAROMETERBAROMETER (developed by (developed by Torricelli in 1643)Torricelli in 1643)

Hg rises in tube until force of Hg (down) Hg rises in tube until force of Hg (down) balances the force of atmosphere balances the force of atmosphere (pushing up). (Just like a straw in a (pushing up). (Just like a straw in a soft drink)soft drink)

P of Hg pushing down related to P of Hg pushing down related to • Hg densityHg density• column heightcolumn height

1608 –1647

Evangelista Torricelli

Pressure of air is Pressure of air is measured with a measured with a BAROMETERBAROMETER (developed by (developed by Torricelli in 1643)Torricelli in 1643)

Hg rises in tube until force of Hg (down) Hg rises in tube until force of Hg (down) balances the force of atmosphere balances the force of atmosphere (pushing up). (Just like a straw in a (pushing up). (Just like a straw in a soft drink)soft drink)

P of Hg pushing down related to P of Hg pushing down related to • Hg densityHg density• column heightcolumn height

Manometersfrom Greek manos meaning sparse

• 760 mm of Hg• 760 torr• 29.9 in. of Hg• 1 Atmosphere• 101.325 KPa (Kilopascals)• 14.7 lbs. per sq. in.• about 34 feet of water!about 34 feet of water!

CONVERSIONS

All Equal

Standard Temperature and Pressure (STP)

P = 1 atmosphereT = 0 °C

The molar volume of an ideal gas is 22.42 litersat STP.

1 mol occupies 22.42 L at STP

sphygmomanometer• sphygmometer• Greek sphygmos meaning pulse (from sphyzein to

throb)

Measures to 300mm Hg

This is the inner mechanisms of certain pressure gauges.

When a pressure cooker is used, whatWhen a pressure cooker is used, whatcauses the increased pressure?causes the increased pressure?

PV=nRTPV=nRT P=P=nRTnRT VV

Temperature goes from 25Temperature goes from 25ooC to 100C to 100ooCCTurn to Kelvin by adding 273 to CelsiusTurn to Kelvin by adding 273 to Celsius297K to 373K 75K/297K=25% increase in pressure297K to 373K 75K/297K=25% increase in pressure

P1V1=n1RT1 P2V2=n2RT2

n1T1 n1T1 n2T2 n2T2

P1V1= R P2V2= R

n1T1 n2T2

P1V1= P2V2

n1T1 n2T2

Change in Conditions Problem

We can take advantage of the fact that the R constant is the same even if the conditions of the gas changes.

1. What volume will 52.5 g of CH4 occupy at STP?

2. You heat 1.437 g NH3 in a stoppered 250 mLflask until it explodes (425 °C). What was thepressure inside the flask immediately prior to theexplosion?

The Gas Laws

a.k.a.The Old Dead Guy’s with crazy hair

Laws

Gay-Lussac’s Law

• At constant V :

Joseph-Louis Gay-Lussac

And so…. The gas laws

Boyle’s Law*• Boyle’s Law-

At constant temperature, volume is inversely proportional to pressure.

*Holds precisely only at very low pressures.

Robert Boyle

There’s more….

Charles’ Law

• At constant pressure the volume of a gas is directly proportional to temperature, and extrapolates to zero at zero Kelvin.

Avogadro’s Law

• At constant V :• For a gas at constant

temperature and pressure, the volume is directly proportional to the number of moles of gas (at low pressures).

a = proportionality constantV = volume of the gasn = number of moles of gas1 mol occupies 22.42 L at STP

Charles’ Law

Charles’ Law

• At constant pressure the volume of a gas is directly proportional to temperature, and extrapolates to zero at zero Kelvin.

Absolute Zero

Jacques Charles

There’s more….

Charles’ Law

• At constant pressure the volume of a gas is directly proportional to temperature, and extrapolates to zero at zero Kelvin.

Avogadro’s Law

• At constant V :• For a gas at constant

temperature and pressure, the volume is directly proportional to the number of moles of gas (at low pressures).

a = proportionality constantV = volume of the gasn = number of moles of gas1 mol occupies 22.42 L at STP

Avogadro’s Law

• At constant V :• For a gas at constant

temperature and pressure, the volume is directly proportional to the number of moles of gas (at low pressures).

a = proportionality constantV = volume of the gasn = number of moles of gas1 mol occupies 22.42 L at STP

Amedeo Avogadro

Avogadro developed this law after Joseph Louis Gay-Lussac had published in 1808 his law on volumes (and combining gases). The greatest problem Avogadro had to resolve was the confusion at that time regarding atoms and molecules.

The scientific community did not give great attention to his theory, so Avogadro's hypothesis was not immediately accepted. André-Marie Ampère achieved the same results three years later by another method

Molar Volume for Several Gases at STP

All of the laws can be summarized nicely…

Combined Gas Law

What temperature is required to cause the pressure of a (steel) cylinder of gas to increase from 350 to 500 mm Hg? The initial temperature was 298 K.

A gas occupies a volume of 400. mL at 500. mm Hg pressure. What will be its volume, at constant temperature, if the pressure is changed to 250 torr?

We will use Boyle’s Law: P1V1 = P2V2 (500.mm)

(400.mL)=(250.mm)(x)x= (500)(400) = 800. mL 250

Info. given: Question: what is V2?

V1 = 400. mL

P1 = 500. mm

Temperature does not changeP2 = 250 torr (760 mm=760 torr) = 250 mm

A gas occupies a volume of 410 mL at 27°C and 740 mm Hg pressure. Calculate the volume the gas would occupy at STP.

Info. given: Question: what is V2?

V1 = 410 mL

T1 = 27°C T2 = 0 °C

P1 = 740 mm P2 = 760 mm

We will use the combined gas law: 2

22

1

11

TVP

TVP

Oops…use Kelvin 27°C=300K; 0°C=273K

mL363mm760K273

)K300()mL410)(mm740(

PT

TVP

V2

2

1

112

?0mm760C0

)C27()mL410)(mm740(

PT

TVP

V2

2

1

112

Suppose you have 856 mL of a gas. A weather front comes through, and the barometric pressure changes from 780 mm Hg to 720 mmHg. Along with this, the temperature changesfrom 86 °F (30. °C) to 72 °F (22 °C). What isthe new volume of your gas?

Gas Stoichiometry

• When gases are involved in a reaction, das properties must be combined with stoichiometric relationships.

E.g. Determine the volume of gas evolved at 273.15 K and 1.00 atm if 1.00 kg of each reactant were used. Assume complete reaction (i.e. 100% yield)

CaO(s) + 3C(s) CaC2(s) + CO(g).• Strategy:

– Determine the number of moles of each reactant to which this mass corresponds.

– Use stoichiometry to tell us the corresponding number of moles of CO produced.

– Determine the volume of the gas from the ideal gas law.

Kinetic Molecular Theory• The first kinetic interpretation of gases was Robert Hooke

in 1676.• At the time, Isaac Newton’s picture of a gas was the

accepted view. Newton suggested that gas particles exert pressure on the walls of a container because of repulsive forces between the molecules.

Isaac Newton

Enter Maxwell and Boltzmann• James Clerk Maxwell in 1859 and Ludwig

Boltzmann in the 1870s finally got people to listen to a kinetic theory of gasses.

Postulates of the Theory

• Gases are composed of molecules which are small compared to the average distance between them.

d(N2,g) = 0.00125 g/L (273°C)d(N2,liq) = 0.808 g/mL (-195.8°C)

• Molecules move randomly, but in straight lines until they collide with other molecules, or the walls of the container.

• Forces of attraction and repulsion between gas molecules are negligible (except during collisions!)

• All collisions between gas molecules are elastic.• The average kinetic energy of a molecule in a gas sample

is proportional to the absolute temperature.

Kinetic Molecular Theory and the Ideal Gas Law

• Consider that pressure is due to the large number of collisions of gas molecules with the walls of the container.

p (frequency of collisions)·(average force)

muNV

1u p

u: average speedm: massN: number of molecs.

2Nmu pV

Kinetic Molecular Theory and the Ideal Gas Law

muNV

1u p

½mu2 is the kinetic energy and T

NT pV N is proportional tothe number of moles.

nT pV Insert a constantof proportionality.nRT pV

Root Mean Square Speed

• The root mean square speed of gas molecules depends on the temperature and the molar mass.

M

RTu

3

What is the rms speed of O2 molecules at 21 oC and15.7 atm?

223100.32

294314.83

sm

xu

M

TRu

3

1489 smu

13

1122

100.32

294314.83

molkgx

KKmolsmkgu

Graham’s Law of Effusion

• The rate of effusion of gas from a system is proportional to the rms speed of the molecules.

M

RTu

3

At constant temperature:

M1rate

An Example• What is the ratio of rates of effusion of CO2

and SO2 from the same container at the same temperature and pressure?

2

2

2

2

SOfor effusion of rate

COfor effusion of rate

CO

SO

M

M

molg

molg

/0.44

/1.64

SOfor effusion of rate

COfor effusion of rate

2

2

21.1SOfor effusion of rate

COfor effusion of rate

2

2 This is because CO2 molecules move1.21 times faster than SO2 molecules!

Another Example

• If it takes 4.69 times as long for a particular gas to effuse as it takes hydrogen under the same conditions, what is the molecular weight of the gas?

The time of effusion is inversely proportional to the rate of effusion.

g/mol 2.0

M 4.69

g/mol 2.0

M 22.0 M g/mol 44.0

Dalton’s Law of Partial Pressures

• John Dalton, in 1801, suggested that each gas in a mixture exerts a pressure and that the total pressure is the sum of these partial pressures.

CBAtot pppp

More on partial pressures

• Each component gas has a partial pressure which can be found using the ideal gas law.

RTnVp AA Where nA is the number of moles of component A.

And the mole fraction is given by:

tot

A

tot

AA p

p

n

nX

An example on partial pressures

• A 1.00 L sample of dry air at 786 Torr and 25 oC contains 0.925 g N2 plus other gasses (such as O2, Ar and CO2.) a) What is the partial pressure of N2? b) What is the mole fraction of N2?

molg

molg 0330.0

0.28925.0

atm

L

KKmolLatmmol807.0

00.1

2980821.00330.0 11

Torratm

Torratm 613

760807.0 780.0

786

6132

Torr

TorrX N