CHAPTER 10. GASES
Student LEARNING OUTCOMES
• Recognize always use temperature Kelvin in gas problems & to use & understand standard conditions, STP
• Know to always use units in gas-law problems to keep track of required conversions
• Be able to convert between pressure units
• Be able to calculate P, V, n, & T using various gas law equations & relations, & calculate partial pressures
• Realize that Ideal gas behavior do not behave ideally; high & low temperatures
V  constant  T or
V
 constant
T
• Be able to calculate density & molecular weight of a gas
• Be able to understand & perform calculations based on collecting gas over water
• Be able to describe Kinetic Molecular Theory & how it explains P & T, gas laws, & rates
• Be able to understand, describe, & calculations based on effusion & diffusion
CHAPTER 10 OUTLINE
10.1 Characteristics of Gases
• Liquids or solids under ordinary conditions also exist as gases
•referred to as vapors
• Many properties of gases differ from solids and liquids:
•gases highly compressible & occupy full volume of containers
•gas subjected to pressure, volume decreases
•gases always form homogeneous mixtures with other gases
• Gases occupy small fraction of volume of their containers
•each molecule of gas behaves as though other molecules were absent
10.2 Pressure
•Pressure is force acting on object per unit area: P = F/A
Atmospheric Pressure and the Barometer
• SI unit of force is newton (N); 1 N = 1 kg-m/s2
• SI unit of pressure is pascal (Pa); 1 Pa = 1 N/m2; related unit bar, equal to 105 Pa
• Atmospheric pressure measured with barometer
•standard atmospheric pressure is 760 mm of Hg
•non-SI units used include: atmospheres (atm), millimeter of mercury (mm Hg) or torr
•1 atm = 760 mm Hg = 760 torr = 1.01325 x 105 Pa = 101.325 kPa
10.3 The Gas Laws
• Equations express relationships of T (temperature), P (pressure), V (volume), and n (number moles gas)
The Pressure–Volume Relationship: Boyle’s Law
• Boyle’s law: volume of fixed quantity of gas, at constant T, inversely proportional to pressure
• Mathematically: plot of V versus P is a hyperbola
•plot V versus 1/P a straight line passing through origin
•According to Boyle’s law, volume increases, pressure decreases
V  constant 
1
or PV  constant
P
The Temperature–Volume Relationship: Charles’s Law
• Hot-air expands when heated
•Charles’s law: volume of fixed quantity of gas, at constant P, directly proportional to absolute temperature
• Mathematically: plot of V versus T is a straight line
•Note, value of constant depends on P & n
•T measured in ˚C, intercept on temperature axis is –273.15 ˚C
• Define absolute zero, 0 K = –273.15 ˚C
The Quantity–Volume Relationship: Avogadro’s Law
• Gay-Lussac’s law of combining volumes: At given T & P, volumes of gases react are ratios small whole numbers
• Avogadro’s hypothesis: Equal volumes of gases at same T & P contain same number of molecules
• Avogadro’s law: volume of gas at given T & P directly proportional to number of moles of gas
•mathematically: V = constant  n
•22.4 L of any gas at 0 ˚C & 1 atmosphere contains 6.02 x 1023 gas molecules
10.4 The Ideal-Gas Equation
• Summarizing the gas laws:
•Boyle: V  1/P (constant n, T)
•Charles: V  T (constant n, P)
•Avogadro: V  n (constant P, T)
•Combined: V  nT/P
• Ideal gas equation: PV = nRT
•ideal gas is hypothetical gas whose P, V, and T behavior completely described by ideal-gas equation
•gas constant, R = 0.08206 L-atm/mol-K; other R values of various units given Table 10.2
• STP (standard temperature and pressure) = 0 ˚C, 273.15 K, 1 atm
• Molar volume of 1 mol of ideal gas at STP is 22.41 L
Relating the Ideal-Gas Equation and the Gas Laws
• PV = nRT and n and T are constant, then PV is constant and Boyle’s law
•other laws generated similarly
• In general, have gas under two sets of conditions, then
P1 V P2 V2

n 1T n 2 T2
1
1
• When P, V, and T all change for fixed number of moles of gas
•for this set of circumstances, (PV/T) =nR = constant which gives
P1V1 P2V2

T1
T2
10.5 Further Applications of the Ideal-Gas Equation
Gas Densities and Molar Mass
• Density units of mass over volume
• Rearranging ideal-gas equation with M as molar mass
n
P
nM PM


V RT
V RT
• Mass of a gas be determined as follows:
M
dRT
PM
d 
P
RT
Volumes of Gases in Chemical Reactions
• Ideal-gas equation relates P, V, and T to number of moles of gas
• n used in stoichiometric calculations
10.6 Gas Mixtures and Partial Pressures
• Gas molecules o far apart, assume behave independently
• Dalton observed: total pressure of mixture of gases equals sum of pressures each gas exert if alone
•partial pressure: pressure exerted by a particular component of a gas mixture
• Dalton’s law of partial pressures
•gas mixture the total pressure given by sum of partial pressures each component: Pt = P1 + P2 + P3 + . . .
• Each gas obeys ideal gas equation, thus,
RT
RT
Pt  (n1  n2  n3  )
 nt
V
V
Partial Pressures and Mole Fractions
• Let n1 be the number of moles of gas 1 exerting a partial pressure P1, then P1 = Pt
• is the mole fraction (n1/nt); mole fraction is a dimensionless number
Collecting Gases over Water
• Common to synthesize gases & collect by displacing a volume of water
• To calculate amount of gas produced, need to correct for partial pressure of water: Ptotal = Pgas + Pwater
• Vapor pressure of water varies with temperature; values found in Appendix B
10.7 Kinetic-Molecular Theory
• Kinetic-molecular theory explains gas behavior; a theory of moving molecules
• Summary: gases consist of large number of molecules in constant random motion
•combined volume all molecules is negligible compared with volume of container
•intermolecular forces (between gas molecules) are negligible
•energy transferred between molecules during collisions, the average KE constant at constant temp
•collisions are perfectly elastic
•average KE of gas molecules is proportional to absolute temp
• Kinetic-molecular theory gives understanding of P & T on the molecular level
•pressure of gas results from collisions with walls of container
•magnitude of P determined by how often and how hard molecules strike
• Absolute temperature of gas is measure of average KE
•some molecules have less KE or more KE than the average (distribution)
•as temperature increases, average KE of gas molecules increases
•as KE increases, velocity of gas molecules increases
• Root-mean-square (rms) speed, u, is speed of gas molecule having average kinetic energy
•average KE, , related to rms speed:  = ½ mu2
m = mass molecule
Application to the Gas Laws
• Can understand empirical observations of gas properties within framework of kinetic-molecular theory
• Effect of increase volume (at constant temperature):
•average KE of gas remains constant, therefore, u is constant
•volume increases, gas molecules travel further to hit walls of the container, therefore, pressure decreases
• Effect of increase in temperature (at constant volume):
•average KE of gas molecules increases, there more collisions with container walls, therefore, u increases
•change in momentum in each collision increases (molecules strike harder), therefore, pressure increases
10.8 Molecular Effusion and Diffusion
• Average KE of gas related to its mass:  = ½ mu2
• Two gases at same temp: lighter gas has higher rms speed than heavier gas; then mathematically:
u
3RT
M
•lower molar mass, M, higher rms speed for gas at constant temp
• Two consequences of dependence of molecular speeds on mass are:
•effusion: escape of gas molecules through tiny hole into evacuated space
•diffusion: spread of one substance throughout a space or throughout a second substance
Graham’s Law of Effusion
• Two gases with molar masses, M1 and M2, and with effusion rates, r1 and r2:
•relative rate of effusion given by Graham’s law:
r1

r2
M2
M1
•Only molecules which hit small hole will escape through it, therefore, higher rms speed more likely
gas molecule will hit hole
r1 u1


r2 u 2
M2
M1
Diffusion and Mean Free Path
• Diffusion faster for light gas molecules & significantly slower than rms speed
•slowed by collisions of gas molecules with one another
• Average distance traveled by gas molecule between collisions called mean free path
10.9 Real Gases: Deviations from Ideal Behavior
• Ideal gas equation: PV =nRT
•1 mol of ideal gas, PV/RT = 1 for all pressures
•real gas, PV/RT varies from 1 significantly
•higher pressure, more deviation from ideal behavior
•1 mol of ideal gas, PV/RT = 1 for all temperatures
•temp increases, gases behave more ideally
• Assumptions in kinetic-molecular theory show where ideal gas behavior breaks down:
•molecules of gas have finite volume & do attract each other
• Pressure on gas increases, molecules forced closer together
•as molecules get closer, free space which molecules move gets smaller
•smaller the container, more total space gas molecules occupy, therefore, higher P, less gas resembles ideal
•gas molecules get closer together, intermolecular distances decrease
•smaller distance between molecules,more attractive forces develop between molecules,therefore, less ideal
• Temperature increases, gas molecules move faster and further apart
•means more energy available to break intermolecular forces
•the negative departure from ideal-gas behavior disappears
The van der Waals Equation
• Add two terms to ideal gas equation to correct for
•volume of molecules: (V – nb)
•for molecular attractions:
 n2a 
 2 
V 


• Correction terms generate the van der Waals equation:

n2a 
 P  2 V  nb  nRT

V 

•a and b are empirical constants that differ for each gas
•van der Waals constants for some common gases found in Table 10.3
• Understand effect of intermolecular forces on P, consider a molecule about to strike wall of container
•striking molecule attracted by neighboring molecules, therefore, impact on wall is lessened