Gases
Variable Density
Solids and liquids have a definite volume, thus they have a fixed density at a given temperature
Gases have a variable density at a given temperature
 Pressure and temperature and molar mass determine density for a gas
 Density is directly proportional to pressure and to molar mass
 Density and temperature are inversely proportional
Gas Laws Overview
Describes macroscopic behavior of gases – volume, temperature, pressure (and mol number)
Combined Gas Law
 P and V inversely proportional
 V and T directly proportional
 P and T directly proportional
PV / T = constant
Avogadro’s Law - V & n
 V and n directly proportional

P1V1 / T1 = P2V2 / T2
[n = mol number]
Applicable to stoichiometry of gas phase reactions
Ideal Gas Law
PV/T=nR
PV=nRT
PV/nT=R
R = 0.08206 L·atm·mol–1·K–1 = 62.37 L·mmHg·mol–1·K–1
Gas Law Calculations
Solving for dynamic conditions
 Initial and final values given
 Usually for two variables
 Can use combined gas law
Solving for static conditions
 Three values given
 Can use ideal gas law
Examples:
Calculate the new volume of a 5.0 L balloon when the pressure drops from 1.0 atm to 0.90 atm.
P1 V1 = P2 V2
[T is constant]
(1.0 atm)(5.0 L) = (0.90 atm) V2
V2 = 5.6 L
Calculate the initial pressure of a piston containing 2.5 mL of air if the final volume and pressure are 7.5
mL and 0.30 atm, respectively.
P1 V1 = P2 V2
[T is constant]
P1 (2.5 mL) = (0.30 atm)(7.5 mL)
P1 = (0.30 atm)(7.5 mL) / (2.5 mL) = 0.90 atm
Calculate the final volume of a 5.0 L balloon if the temperature is raised from 300 K to 600 K.
V1 / T1 = V2 / T2
[P is constant]
5.0 L / 300 K = V2 / 600 K
V2 = (5.0 L)(600 K) / (300 K) = 10 L
Calculate the final temperature if the pressure in a tank drops from 20.0 atm to 8.0 atm. Initial
temperature is 300 K.
P1 / T1 = P2 / T2
[V is constant]
20.0 atm / 300 K = 8.0 atm / T2
T2 = (8.0 atm)(300 K) / (20.0 atm) = 120 K
Calculate the volume of a balloon containing 2.00 mol of gas with P = 1.00 atm and T = 300 K.
PV=nRT
(1.00 atm)(V) = (2.00 mol)( 0.08206 L·atm·mol–1·K–1)(300 K)
V = (2.00 mol)( 0.08206 L·atm·mol–1·K–1)(300 K) / (1.00 atm) = 49.2 L
Calculate the number of moles in a cylinder containing 20.0 L of gas at 298 K and a pressure of 3.00 atm.
PV=nRT
(3.00 atm)(20.0 L) = n (0.08206 L·atm·mol–1·K–1)(298 K)
n = (3.00 atm)(20.0 L) / (0.08206 L·atm·mol–1·K–1)(298 K) = 2.45 mol
Calculate the temperature of a cylinder holding 5.00 mol of gas with P = 6.00 atm and V = 22.5 L.
PV=nRT
T = (6.00 atm)(22.5 L) / [(5.00 mol)( 0.08206 L·atm·mol –1·K–1)] = 329 K
Calculate the molar volume of a gas with P = 750 mmHg and T = 295 K.
PV=nRT
(750 mmHg / 760 mmHg·atm–1)(V) = (1.00 mol)( 0.08206 L·atm·mol–1·K–1)(295 K)
V = (1.00 mol)( 0.08206 L·atm·mol–1·K–1)(295 K) / (0.95 atm) = 24.5 L
Calculate the molar volume of a gas at STP. Standard T = 273 K and standard P = 1 atm.
PV=nRT
(1 atm)(V) = (1.00 mol)( 0.08206 L·atm·mol–1·K–1)(273 K)
V = (1.00 mol)( 0.08206 L·atm·mol–1·K–1)(273 K) / (1 atm) = 22.4 L
Molar volume at STP is 22.4 L for an ideal gas.
This knowledge generally allows calculations that use only the simple gas laws (Boyle's and Charles' Laws).
Other Applications of the Ideal Gas Law
The ideal gas law can be rearranged to show the relationship between gas density and molar mass.
n/V=P/RT
n=m/M
d=m/V=PM/RT
M =dRT/P
M =mRT/PV
Of course, at STP these become: d = M / 22.4 Lmol –1
Examples:
Calculate the density of butane (C4H10) at 0ºC and 1.00 atm.
M = 58.14 gmol –1
d = P M / R T = (1.00 atm)(58.14 gmol –1) / (0.08206 Latmmol –1K –1)(273.15 K)
d = 2.59 gL–1
Calculate the molar mass of a gas with a density of 1.18 g L –1 at 25.0ºC and 1.00 atm.
M =dRT/P
M = (1.18 gL–1) (0.08206 Latmmol –1K –1)(298.15 K) / (1.00 atm)
M = 28.9 gmol –1
Dalton’s Law of Partial Pressures
PTotal = P1 + P2 + P3 + …
P1 = X1 × PTotal
Total pressure is sum of partial pressures in a mixture of gases
The partial pressure of a component in a gas mixture is the product of mole fraction and total pressure
Vapor Pressure
Small amount of liquid evaporates to form vapor over a liquid and exerts a vapor pressure.
Thus, for gases collected by bubbling through water
Vapor pressure increases with temperature.
PTotal = Pgas + Pwater
Water
T(ºC)
P(mmHg)
T(ºC)
P(mmHg)
0
4.6
60
149.4
10
9.2
70
233.7
20
17.5
80
355.1
30
31.8
90
525.8
40
55.3
100
760.0
50
92.5
Examples:
Calculate the partial pressure of oxygen in a scuba tank filled with heliox (helium and oxygen mixture).
The partial pressure of helium is 600 mmHg and the total pressure is 760 mmHg.
P = P1 + P2
760 mmHg = 600 mmHg + P2
P2 = 160 mmHg
A sample of acetylene is collected over water at a temperature of 20ºC. Total pressure is 500.0 mmHg.
Calculate the partial pressure of acetylene.
P = P1 + P2
500.0 mmHg = 17.5 mmHg + P2
P2 = 482.5 mmHg
Calculate the partial pressure (in mmHg) of O2 in air that has a total pressure of 0.985 atm and contains
20.95% O2.
PO2 = (0.2095)(0.985 atm)
PO2 = 0.206 atm
PO2 = (0.206 atm)(760 mmHgatm–1) = 157 mmHg
Calculate the partial pressure (in mmHg) of He in air that has a total pressure of 0.995 atm and contains
5.24 ppm He.
PHe = (5.24×10–6)(0.995 atm)(760 mmHgatm–1)
PHe = 3.96×10–3 mmHg
Kinetic Theory of Gases
Explains macroscopic behavior of gases – volume, temperature, and pressure.
[Quantum mechanical and relativistic effects are negligible.]
1. Gases consist of a statistically large number of particles with finite mass.
2. Particles are in constant, random motion.
[Average kinetic energy of particles directly proportional to absolute temperature].
3. Elastic collisions between particles and walls.
[Conservation of energy]
4. Intermolecular forces between particles are negligible.
5. Total particle volume negligible relative to container.
[Large intermolecular spacing]
For real gases at different temperatures and pressures:
 Intermolecular forces may not be negligible
 Particle volume may not be negligible
Thus, real gases may deviate from ideal behavior
 High pressure and/or low temperature lead to non-ideal behavior

At extremes, gas may condense into liquid
Nitrogen boils at 77 K
[Liquid air at about T = –200 ºC]
Van der Waals Equation
Accounts for non-ideality using two constants that depend on the identity of the gas
Graham’s Laws of Effusion and Diffusion
Effusion – process by which individual gas molecules flow through holes in a container without
intermolecular collisions
Diffusion – particles moving from an area of high concentration to an area of low concentration

Rate of effusion (diffusion) proportional to inverse square root of molar mass
Denser gases (higher molar mass) diffuse and effuse more slowly
Henry’s Law and Gas Solubility
 Gas solubility is directly proportional to the partial pressure of the gas
 Henry’s law constant – depends on solute, solvent, and temperature
[Not universal like ideal gas constant.]
Background and Reference Information
DRY AIR
Pressure = Force / Area


Pressure is omni-directional
Atmospheric pressure – referenced at sea level
Common Units
1 atm
78.08% N2
0.93% Ar
20.95% O2
0.03% CO2
= 760 mmHg
= 14.7 psi
= 101.325 kPa
= 29.92 inHg
= 1013.25 mb (millibar)
Temperature – Average Kinetic Energy
Absolute temperature is directly proportional to average kinetic energy of the gas particles
TK = TC + 273.15
generally, can use: TK = TC + 273
Hemoglobin and Gas Transport
 Partial pressure of O2 in air ~160 mmHg and in alveoli ~100 mmHg

Partial pressure of CO2 in air ~40 mmHg and in alveoli ~45 mmHg

CO2 present in blood is primarily in form of bicarbonate ion, HCO 3–
Historical development of gas laws and kinetic molecular theory
NB: many discoveries were made independently by two scientists; attribution can be controversial
Barometer invented
Air pump invented
Manometer invented
Mercury thermometer invented
[1643, Torricelli]
[1652, Otto van Guericke]
[1661, Christian Huygens]
[1714, Daniel Fahrenheit]
Boyle’s Law
[1662, Pressure and Volume]
 Volume of a gas is inversely proportional to its pressure (at constant temperature)
Amonton’s Law
[1702, Pressure and Temperature]
 Pressure of a gas is directly proportional to its absolute temperature
Charles’ Law
[1787-1802, Volume and Temperature]
 Volume of a gas is directly proportional to its absolute temperature

Applicable to physical and chemical processes
Dalton’s Law
[1801, Partial Pressures]
 Total pressure of a mixture of gases is the sum of the partial pressures of the components
Henry’s Law
[1803, Solubility of Gases]
 At constant temperature, the solubility of a gas in some liquid is directly proportional to the partial
pressure of that gas in equilibrium with that liquid
Gay-Lussac’s Law

Avogadro’s Law

[1809, Volumes in Reactions]
Law of combining volumes – volumes of reactants and products are ratio of small integers
[1811, Volume and Mol Number]
Two gas samples with equal volume contain equal number of particles (same mol number)
Graham’s Laws
[1831, Diffusion or Effusion]
 Rate of effusion inversely proportional to the square root of molar mass
Ideal gas law


[1834, Clapeyron; 1857, Clausius (from kinetic theory)]
Pressure, volume, temperature, and mol number (number of particles) related by constant
Combined gas law (for constant mol number) useful variant for closed systems
Van der Waals’ equation

Correction to pressure term and volume term
[1873, corrections for real gases]