Gases

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 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.

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 outline

While 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/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

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)
  
  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	P₁V₁/T₁ = P₂V₂/T₂	Combined Gas Law
P, V, T, n	P₁V₁/n₁T₁ = P₂V₂/n₂T₂	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
1. make a PVnT table
2. convert to units consistent with R
3. solve PV = nRT for the unknown property
finding how one property changes when some of the other 3 properties change
1. make a PVnT table with rows for initial and final conditions
2. convert inconsistent units (always use K, not °C!!)
3. Eliminate constant values from P₁V₁/n₁T₁ = P₂V₂/n₂T₂
4. solve the resulting equation for the unknown property
- examples
  - A steel-belted 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
  1. find moles per liter ( n/V = P/RT )
  2. convert moles/L to g/L
- finding molar mass
  1. find moles of gas ( n = PV/RT)
  2. molar mass = g/mol

Reactions involving gases

predicting volume of gas produced or consumed in a reaction
- 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 C₃H₅(ONO₂)₃ 3 CO(g) + 2 CO₂(g) + 4 H₂O(g) + 6 NO(g) + H₂CO(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 Gay-Lussac'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:
      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
- 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 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
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
  - 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
General Chemistry Online! Gases
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/print-index.shtml

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

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

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

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

	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