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Learning objectives
 Understand the definition of pressure. Use the definition to predict and measure pressures experimentally.
 Describe experiments that show relationships between pressure, temperature, volume, and moles for a gas sample.
 Use empirical gas laws to predict how a change in one of the properties of a gas will affect the remaining properties.
 Use empirical gas laws to estimate gas densities and molecular weights.
 Use volumetomole relationships obtained using the empirical gas laws to solve stoichiometry problems involving gases.
 Understand the concept of partial pressure in mixtures of gases.
 Use the ideal kineticmolecular model to explain the empirical gas laws.
 List deficiencies in the ideal gas model that will cause real gases to deviate from behaviors predicted by the empirical gas laws. Explain how the model can be modified to account for these deficiencies.
Before you start...
 review the mole concept; we'll find that many of the properties of a gas depend on the number of molecules of gas present, so we will be able to simplify calculations and expressions by using moles rather than masses.
 review density; we'll find that we can predict gas densities and molecular weights using the empirical gas laws.
Lecture outlineWhile gases are less familiar in everyday life than liquids and solids, it is much easier to build models that explain the behavior of gases than it is for liquids and solids. The first lecture concentrates on describing a gas in terms of its properties: pressure, volume, temperature, and amount of substance. The second lecture shows how experimental relationships (laws) connect the properties of gases. These "gas laws" will be rationalized using a series of hypotheses about the molecular structure of gases. The final lecture examines a detailed but idealized molecular model of gases and shows how real gases depart from this ideal.
A Molecular Model of Gases
observation 
hypothesis 
Gases are easy to expand 
gas molecules don't strongly attract each other 
Gases are easy to compress 
gas molecules don't strongly repel each other 
Gases have densities that are about 1/1000 of solid or liquid densities 
molecules are much farther apart in gases than in liquids and solids 
Gases completely fill their containers 
gas molecules are in constant motion 
Hot gases leak through holes faster than cold gases 
the hotter the gas, the faster the molecules are moving 
Pressure
 definition: pressure = force/area
 units
Unit 
Symbol 
Conversions 
pascal 
Pa 
1 Pa = 1 N/m^{2} 
psi 
lb/in^{2}  
atmosphere 
atm 
1 atm = 101325 Pa = 14.7 lb/in^{2} 
bar 
bar 
1 bar = 100000 Pa 
torr 
torr 
760 torr = 1 atm 
millimeters of mercury 
mm Hg 
1 mm Hg = 1 torr 
 pressure exerted by a weight

How much pressure does an elephant with a mass of 2000 kg and a total footprint area of 5000 cm^{2} exert on the ground?
 Estimate the total footprint area of a tyrannosaur weighing 16000 kg. Assume it exerts the same pressure on its feet that the elephant does.
 measuring pressure
 strategy: relate pressure to fluid column heights
 observation: you can't draw water higher than 34 feet by suction why?
 hypothesis: atmospheric pressure supports the fluid column
pressure 
= 
force / area 

= 
mass x g / area 
(because force = weight of liquid = mass x g) 

= 
density x volume x g / area 
(because mass = density x volume) 

= 
density x g x height 
(because volume = area x height) 
 a barometer measures atmospheric pressure as a mercury column height
 a manometer measures gas pressure as a difference in mercury column heights
 two types
 closed manometer: difference in column heights gives absolute gas pressure

open manometer: difference in column heights gives difference between gas and atmospheric pressures
 kineticmolecular view of gas pressure
 gas pressure arises from force of molecular collisions
 anything that increases the number of collisions will increase pressure
The state of a gas
 most behaviors of a pure gas sample can be related to just 4 physical properties:
property 
symbol 
convenient units 
property type 
pressure 
P 
atm, torr, Pa 
intensive 
volume 
V 
L, cm^{3} 
extensive 
temperature 
T 
K 
intensive 
moles 
n 
mol 
extensive 
 any equation that relates P, V, T, and n for a material is called an
equation of state
 experiment shows PV=nRT is an approximate equation of state for gases
 gases that obey PV=nRT exactly are called
ideal gases
 R is the
gas law constant
 determine experimentally by measuring P, V, T, n, and computing R = PV/nT
 value depends on units chosen for P, V, T
 notice: 1 Joule = 1 N m = 1 (N/m^{2})(m^{3}) = 1 (Pa)(m^{3})
Experimental foundations of the ideal gas law

Recognize three common patterns in experimental data.  
 Avogadro's experiment
 objective: relate amount of gas to volume; hold everything else constant
 sample data at 1 atm, 298 K
gas volume (mL) 
mass of gas (g) 
V/m (mL/g) 
V/n (L/mol) 
O_{2} 
100.0 
0.122 
26.2 
N_{2} 
100.0 
0.110 
25.5 
CO_{2} 
100.0 
0.176 
25.0 
 conclusions
 molar volume is almost independent of the type of gas
 volume is directly proportional to moles of gas when P, T are constant
 samples of two gases with the same V, P, T contain the same number of moles, and so, the same number of molecules!
 molecular view
 type of gas doesn't influence distance between molecules much
 example: Which is denser: dry air or moist air? Assume both air samples are at the same P, V, and T.
 Boyle's experiment
 objective: relate volume to pressure, when everything else is held constant
 sample data at 298 K for a trapped air bubble
Volume (mL) 
Pressure (Torr) 
PV (mL Torr) 
10.0 
760.0 
7.60 x 10^{3} 
20.0 
379.6 
7.59 x 10^{3} 
30.0 
253.2 
7.60 x 10^{3} 
40.0 
191.0 
7.64 x 10^{3} 
 conclusions
 volume is inversely proportional to pressure, when everything else is held constant
 molecular view
 confining molecules to a smaller space increases the number of collisions, and so increases the pressure
 Charles' experiment
 objective: relate volume to temperature, holding everything else constant

sample data for a trapped He bubble at 1 atm
Volume (mL) 
Temperature (°C) 
Temperature (K) 
V/T (mL/K) 
40.0 
0.0 
273.2 
0.146 
44.0 
25.0 
298.2 
0.148 
47.7 
50.0 
323.2 
0.148 
51.3 
75.0 
348.2 
0.147 
55.3 
100.0 
373.2 
0.148 
80 
273.2 
546.3 
0.146 
 conclusions
 volume is proportional to kelvin temperature, when everything else is held constant
 molecular view
 raising temperature increases number of collisons with container wall.
If the walls are flexible, they'll be pushed back: and the gas expands.
 summary
changing these variables in an experiment... 
while holding these variables constant... 
...reveals this relationship: 
P, V 
n, T 
P_{1} V_{1} = P_{2} V_{2} 
Boyle's Law 
V, T 
n, P 
V_{1 }/T_{1} = V_{2 }/T_{2} 
Charles' Law 
P, T 
n, V 
P_{1 }/T_{1} = P_{2 }/T_{2} 
Amonton's Law 
n, V 
P, T 
V_{1 }/n_{1} = V_{2 }/n_{2} 
Avogadro's Law 
P, V, T 
n 
P_{1 }V_{1}/T_{1} = P_{2 }V_{2}/T_{2} 
Combined Gas Law 
P, V, T, n 
P_{1 }V_{1}/n_{1}T_{1} = P_{2}V_{2}/n_{2}T_{2} 
Ideal Gas Law 
standard temperature and pressure (STP)
 convenient for reporting gas volumes
 standard temperature is 0°C (273.15 K).
 standard pressure is 1 atm in America, 1 bar elsewhere
 example

The gas in the headspace of a soda bottle has a volume of 9.0 mL at 298 K and 2 atm. What is the volume of gas at STP?
applying the ideal gas law
 estimating a property from constant values of the other 3 properties
 make a PVnT table
 convert to units consistent with R
 solve PV = nRT for the unknown property
 finding how one property changes when some of the other 3 properties change
 make a PVnT table with rows for initial and final conditions
 convert inconsistent units (always use K, not °C!!)
 Eliminate constant values from P_{1 }V_{1}/n_{1}T_{1} = P_{2}V_{2}/n_{2}T_{2}
 solve the resulting equation for the unknown property
 examples

A steelbelted car tire is inflated to a gauge pressure of 32 psi at 27°C in a garage. What will the gauge pressure be after a drive on a hot road that brings the temperature to 45°C? Note that gauge pressure is pressure above atmospheric pressure (14.7 psi).
 finding gas density given pressure, temperature, and molar mass
 find moles per liter ( n/V = P/RT )
 convert moles/L to g/L
 finding molar mass
 find moles of gas ( n = PV/RT)
 molar mass = g/mol
Reactions involving gases
 predicting volume of gas produced or consumed in a reaction
 strategy: convert between gas volumes and moles using the ideal gas law
molar volume = V/n = RT/P
 examples

Sodium metal reacts violently with water to produce aqueous sodium hydroxide and hydrogen gas. How many liters of hydrogen gas are produced for every gram of sodium that reacts, at STP?
 Nitroglycerine explodes according to
2 C_{3}H_{5}(ONO_{2})_{3}
3 CO(g) + 2 CO_{2}(g) + 4 H_{2}O(g) + 6 NO(g) + H_{2}CO(g)
If the product gases are at a temperature of 4500°C and a pressure of 1 atm, what is the total volume of gas produced from detonation of 1 mole of nitroglycerine?

This special case is sometimes known as GayLussac's Law of Combining Volumes.  
 easy special case: when all gases have identical P, T, all have identical molar volumes
 strategy: mole to mole ratios from balanced equation = gas volume to volume ratios
 example

Ammonia is synthesized by the Haber process:
N_{2}(g) + 3 H_{2}(g)
2 NH_{3}(g)
How many liters of ammonia can be synthesized from 10.0 L of N_{2}?
Temperature and molecular speeds
 molecules in a gas move at many different speeds
 the Maxwell distribution shows how many molecules are moving at a particular speed
 the distribution shifts to faster speeds at higher temperatures

A useful average speed is the root mean square velocity, v_{rms}. Maxwell related v_{rms} with temperature T and molecular weight M:  
 average molecular speed
 higher temperature = higher average molecular speed
 higher molecular weight = lower average molecular speed
 many important properties are related to average molecular velocity
 diffusion rates: gas movement due to random molecular motion
 faster molecules = faster diffusion rate
 effusion rate: gas movement through a pinhole in a container
 faster molecules = faster effusion rate
 heat conduction
 faster molecules = faster transfer of heat energy
 speed of sound
 faster molecules = faster speed of sound
 examples

Molecules traveling faster than 1.1 km/s can escape earth's atmosphere. Which is escaping into space faster: water vapor, or oxygen gas?.
 Diving chambers often use helium/oxygen mixtures instead of air. Which would conduct heat better: helium/oxygen or nitrogen/oxygen atmospheres? In which atmosphere would sound move fastest?
 average molecular kinetic energy
 kinetic energy of one molecule:
½ mv^{2}
 average kinetic energy of one mole of molecules:
½ M(v_{rms})^{2} = (3/2) RT
 implications
 average molecular kinetic energy depends only on temperature for ideal gases
Mixtures of gases
 mole fraction = moles of component per mole of mixture
 by Avogadro's law, mole fraction = volume fraction for ideal gases
 examples

2 L of He gas is mixed with 3 L of Ne gas. What is the mole fraction of each component?
 Air is approximately 79% N_{2} and 21% O_{2} by mass. What is the mole fraction of O_{2} in air?

partial pressure: pressure exerted by one gaseous component in a mixture
 partial pressure = mole fraction times total pressure
 examples

One mole of air contains 0.79 mole of nitrogen, 0.21 mole of oxygen. Compute the partial pressure of these gases at a total pressure of 1 atm and at a total pressure of 3 atm (about the pressure experienced by a diver under 66 feet of seawater).
 vapor pressure of water: partial pressure of water vapor over water in a sealed container
 examples

What is the mole fraction of water in the headspace of a soda bottle, if the gas is at 2.0 atm and 25°C? The vapor pressure of water at 25°C is 23.756 torr.
 total pressure = sum of all partial pressures
 a. k. a. Dalton's Law
 useful in correcting calculations for the effect of moisture
 examples

1.0 L of oxygen gas is collected over water at STP. How many grams of oxygen are present? The vapor pressure of water at 0°C is

Compare gas law calculations for ideal and real gases with this JavaScript equation of state calculator.  
Real gases
 differences between ideal and real gases

ideal gas 
real gas 
obey PV=nRT: 
always 
only at very low pressures 
molecular volume: 
zero 
small, but nonzero 
molecular attractions: 
zero 
small 
molecular repulsions: 
zero 
small 
 distance between molecules is related to gas concentration: n/V = P/RT
 high gas concentration = closer molecules = stronger intermolecular interactions = deviations from ideal behavior
 repulsions make pressure higher than expected by decreasing free volume
 attractions make pressure lower than expected by braking molecular collisions
 tugofwar between these two effects
 repulsions win at very high pressure
 attractions win at moderate pressure
 neither attractions nor repulsions are important at low pressure

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General Chemistry Online! GasesCopyright © 19972005 by Fred Senese Comments & questions to fsenese@frostburg.edu Last Revised 07/25/05.URL: http://antoine.frostburg.edu/chem/senese/101/gases/index.shtml
