• 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 volume-to-mole 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 kinetic-molecular 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.

• 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.

A Molecular Model of Gases

Pressure

• definition: pressure = force/area
• units

  Unit	Symbol	Conversions
  pascal	Pa	1 Pa = 1 N/m2
  psi	lb/in2
  atmosphere	atm	1 atm = 101325 Pa = 14.7 lb/in2
  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 cm² 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
• kinetic-molecular 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, cm3	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/m²)(m³) = 1 (Pa)(m³)

Experimental foundations of the ideal gas law

• 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)
    O2	100.0	0.122	26.2
    N2	100.0	0.110	25.5
    CO2	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 103
    20.0	379.6	7.59 x 103
    30.0	253.2	7.60 x 103
    40.0	191.0	7.64 x 103

  • 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	P1 V1 = P2 V2	Boyle's Law
  V, T	n, P	V1 /T1 = V2 /T2	Charles' Law
  P, T	n, V	P1 /T1 = P2 /T2	Amonton's Law
  n, V	P, T	V1 /n1 = V2 /n2	Avogadro's Law
  P, V, T	n	P1 V1/T1 = P2 V2/T2	Combined Gas Law
  P, V, T, n	P1 V1/n1T1 = P2V2/n2T2	Ideal Gas Law

• 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:
      N₂(g) + 3 H₂(g) 2 NH₃(g)
      How many liters of ammonia can be synthesized from 10.0 L of N₂?

Temperature and molecular speeds

• 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 mv²
  • 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% N₂ and 21% O₂ by mass. What is the mole fraction of O₂ 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

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
  • tug-of-war between these two effects
    • repulsions win at very high pressure
    • attractions win at moderate pressure
    • neither attractions nor repulsions are important at low pressure

Copyright © 1997-2005 by Fred Senese
Comments & questions to fsenese@frostburg.edu
Last Revised 02/23/18.URL: http://antoine.frostburg.edu/chem/senese/101/gases/index.shtml