• 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
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
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
• 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
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?
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
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