Unit 10
The Gas Laws
The Atmosphere
The Earth’s atmosphere is a layer of gases that
surrounds our planet and is retained by gravity
Gas
• A substance that is normally in the gaseous state
at normal temperature (25oC) and pressure (1 atm)
• Vapor: The gaseous form of any substance that
is a liquid or solid at normal temperature/pressure
• Atomic Gases
Ø
Noble Gases, H2, N2, O2, F2, Cl2
• Molecular Gases
Usually light molecules with weak IMFs
Ø Eg: HCl, CO2, CO, NH3, H2S, NO, NO2, SO2
Ø
Some Common Gases
• Some Common Compounds That Are Gases at Room
Temperature
Formula
Name
Characteristics
HCN
Hydrogen cyanide
Very toxic, slight odor of bitter almonds
H2S
Hydrogen sulfide
Very toxic, odor of rotten eggs
CO
Carbon monoxide
Toxic, colorless, odorless
CO2
Carbon dioxide
Colorless, odorless
CH4
Methane
Colorless, odorless, flammable
C2H4
Ethene (Ethylene)
Colorless, ripens fruit
C3H8
Propane
Colorless, odorless, bottled gas
N2O
Nitrous oxide
Colorless, sweet odor, laughing gas
NO2
Nitrogen dioxide
Toxic, red-brown, irritating odor
NH3
Ammonia
Colorless, pungent odor
SO2
Sulfur dioxide
Colorless, irritating odor
Characteristics of Gases
• Physical properties of gases are all similar
Ø
Ø
Ø
Composed of widely separated particles in constant,
random motion
No definite shape or volume and flow readily to completely
fill the container that holds them
Can be compressed to a smaller volume or expanded to
a larger volume
Ø
Composed mainly of nonmetallic elements with simple
formulas
Ø
Low molar masses and extremely low densities when
compared to liquids and solids
• Two or more gases form a homogeneous mixture
Properties that Define the State of a Gas
• Temperature - temperature of the gas is related to
the KE of the molecule
• Pressure
• Volume
• Amount of gas - usually expressed as number of
moles
Gas Pressure
• A gas exerts pressure as the molecules collide with
the surface of its container
• Pressure depends on
Ø
Ø
The number of collisions
The average force of the collisions
• Pressure is the result of these collisions (impacts)
divided by the unit area receiving the force
Gas Pressure
• In SI, force is expressed in newtons (N) and area in
square meters (m2)
• The unit of pressure in SI is the pascal (Pa) with the
units N/m2
• Kilopascals (kPa) are often used instead since the
pascal is such a small unit
• The atmosphere and mmHg (Torr) are the most
common scientific units for pressure
• Converting from one unit to another simply requires
the appropriate conversion factor(s)
Atmospheric Pressure
• Caused by air being pulled towards earth by gravity
• Atmosphere exerts pressure by the collisions of
molecules with every surface it contacts
• The weight of air per unit of area
Ø
Normal atmospheric pressure at sea
level is referred to as standard
atmospheric pressure
Ø
The average atmospheric pressure
at sea level and at 0°C is 760 mm Hg,
so one atmosphere (atm) of pressure
760 mm Hg
is
Atmospheric Pressure at Sea Level
• At sea level, atmospheric
pressure is 1 atm or 101.3
kPa or 760 mm Hg, or
14.7 psi as measured by
a barometer
• In the diagram on the left,
the pressure at sea level
forces mercury up the
tube to a height of 760
mm
• That’s why 760 mm Hg =
1 atm
Atmospheric Pressure
• Weather can change pressure
Ø
Lows and Highs on the weather map
• Altitude can change pressure
Ø
Atmospheric pressure is lower in Denver than in
Memphis because of “thinner air” due to less
collisions of gas with surface
Units of Pressure
• Where does the 1 atm come from?
Ø
Earth’s atmosphere is forced downwards towards the
center of the Earth by gravity. The weight of the air
column above a unit area on the surface of the Earth
causes “atmospheric pressure”
• Atmospheric pressure is defined to be 1 atm at sea
level
Ø
Up at high altitudes, the air column is shorter, so the
atmospheric pressure is lower up there (and the air gets
less dense or “thin”)
• Barometers are used to measure atmospheric
pressure
Units of Pressure
Equivalence Statements
• 1 atm = 760 mm Hg or 760 torr
• 1 atm = 101,325 Pa or 101.325 kPa
• 1 atm = 1.01325 bar
• 1 atm = 14.7 psi
• 1 torr = 1 mm Hg
• 1 bar = 1 x 105 Pa
Unit Conversion
• Convert 0.311 atm to mm Hg
236 mm Hg
• Convert 790 mm Hg to kPa
105.33 kPa
• Convert 880 torr to atm
1.16 atm
• Convert 680.5 kPa to atm
6.72 atm
Barometer
• Measures the height
of Hg in a glass tube
• Invented in 1643 by
Evangelista Torricelli
• Units are in mm Hg
• Aneroid barometer
Gas Laws and Equations
• Boyle’s Law
• Charles’ Law
• Avogadro’s Law
• Gay-Lussac’s Law
• Combined Gas Equation
• Ideal Gas Equation
Boyle’s Law
Vµ
1
P
(constant n and T)
PV = k
Boyle’s Law
PinitialVinitial = PfinalVfinal
Problems
• A balloon contains 30.0 L of helium gas at 103 kPa.
What is the volume of the helium when the balloon
rises to an altitude where the pressure is only 25.0
kPa?
• A sample of neon gas occupies a volume of 677 mL
at 134 kPa. What is the pressure of the sample if
the volume is decreased to 642 mL?
Problems
• A balloon contains 30.0 L of helium gas at 103 kPa.
What is the volume of the helium when the balloon
rises to an altitude where the pressure is only 25.0
kPa?
V = 124 L
• A sample of neon gas occupies a volume of 677 mL
at 134 kPa. What is the pressure of the sample if
the volume is decreased to 642 mL?
P = 141 kPa
Charles’ Law
VµT
(constant n and P)
T is in Kelvins
V
=k
T
• If temperature is changed when a balloon is moved from an
ice-water bath into a boiling-water bath, the gas molecules
inside it move faster due to the increased temperature
• If the external pressure remains constant, the volume will
change when the molecules expand the balloon and
collectively occupy a larger volume
Charles’ Law
Vinitial
Tinitial
=
Vfinal
Tfinal
Problems
o
• A balloon inflated in a room at 24 C has a volume of
o
4.00 L. It is then heated to a temperature of 58 C.
What is the new volume if the pressure remains
constant?
V = 4.46 L
• What is the temperature of a 2.3 L balloon if it shrinks
to a volume of 0.632 L when it is dipped into liquid
nitrogen at a temperature of 77 K?
T = 276 K
Avogadro’s Law
Vµn
(constant T and P)
V
=k
n
Vinitial Vfinal
=
ninitial nfinal
Problem
A 4.8-L sample of helium gas contains 0.22 mol of
helium. How many additional moles of helium gas
must be added to the sample to obtain a volume of
6.4 L?
.07 mols He
Gay-Lussac’s Law
At constant n and V, the pressure of an
ideal changes proportionately as its
absolute temperature changes
If absolute temperature doubles, the
pressure doubles. If the absolute
temperature is halved, the pressure is
halved
PµT
(constant n and V)
P1
T1
=
P2
T2
Problems
• The gas in a used aerosol can is at a pressure of
o
103 kPa at 25 C. If the can is thrown onto a fire,
what will the pressure be when the temperature
o
reaches 928 C?
P = 415 kPa
• A container of propane has a pressure of 108.6 kPa
at a morning temperature 15oC. By mid afternoon
the temperature has reached 32oC. What is the
pressure inside the propane tank?
P = 115 kPa
Combined Gas Equation
• Boyle’s law shows how P & V are related at constant
temperature, and Charles’s law shows how V & T are
related at constant pressure
• What if two of these variables change at once?
• The combined gas law applies only when the amount of
gas is constant
• The temperature must be expressed in kelvins
• A sample of gas has an initial volume of 158 mL at
a pressure of 735 mm Hg and a temperature of 34
°C. If the gas is compressed to a volume of 108 mL
and heated to a temperature of 85 °C, what is its
final pressure in mm Hg?
• The volume of a gas-filled balloon is 30.0 L at 40oC
and 153 kPa. What volume will the balloon have at
STP?
• At what temperature (in oC) does 121 mL of CO2 at
27o C and 1.05 atm occupy a volume of 293 mL at
a pressure of 1.40 atm?
• A sample of gas has an initial volume of 158 mL at
a pressure of 735 mm Hg and a temperature of 34
°C. If the gas is compressed to a volume of 108 mL
and heated to a temperature of 85 °C, what is its
final pressure in mm Hg?
P = 1253 mm Hg
• The volume of a gas-filled balloon is 30.0 L at 40oC
and 153 kPa. What volume will the balloon have at
STP? V = 39.5 L
• At what temperature (in o C) does 121 mL of CO2 at
27o C and 1.05 atm occupy a volume of 293 mL at
a pressure of 1.40 atm?
T = 696°C
Ideal Gas Equation
• So far we’ve seen that
V µ 1/P (Boyle’s law)
V µ T (Charles’s law)
V µ n (Avogadro’s law)
• Combining these, we get
nT
Vµ P
• To make this an equality, we use a constant of
proportionality (R) and reorganize to derive the
Ideal-Gas Equation
PV = nRT
The Ideal Gas Constant
• The conditions 0o C and 1 atm are called standard
temperature and pressure (STP)
• Experiments show that at STP, 1 mole of an ideal
gas occupies 22.414 L
PV = nRT
(1 atm)(22.414L)
PV
R=
=
nT
(1 mol)(273.15 K)
R = 0.082057 L • atm / (mol • K)
The Ideal Gas Constant
• If pressure is in kPa, R is 8.3145 L • kPa / (mol • K)
• If pressure is in mm Hg or torr, the value of R is
62.3637 L • mm Hg / (mol • K)
• Calculate the volume occupied by 0.845 mol of N2
gas at a pressure of 1.37 atm and a temperature of
315 K
V = 16.0 L
• An automobile tire at 23°C with an internal volume of
25.0 L is filled with air to a total pressure of 46.75 psi.
Determine the number of moles of air in the tire
n = 3.27 mols
• If I have 1.9 moles of carbon monoxide at a pressure
of 5 atm in a 50 L container, what is the temperature
of the gas?
T = 1603 K
• Calculate the volume occupied by 0.845 mol of N2
gas at a pressure of 1.37 atm and a temperature of
315 K
V = 16.0 L
• An automobile tire at 23°C with an internal volume of
25.0 L is filled with air to a total pressure of 46.75 psi.
Determine the number of moles of air in the tire
n = 3.27 mols
• If I have 1.9 moles of carbon monoxide at a pressure
of 5 atm in a 50 L container, what is the temperature
of the gas?
T = 1603 K
• A sample containing 0.35 mol argon gas at a
temperature of 13 C and a pressure of 568 torr is
heated to 56 C and a pressure of 897 torr. Calculate
the change in volume that occurs.
DV = -3 L
Density of Gases
• If we divide both sides of the ideal-gas equation by
VRT, we get
n/V = P/RT
• Moles ´ molar mass = mass
n´M=m
• If we multiply both sides by M, we get:
m/V = MP/RT
• Since m/V is density (d), the result is:
d = MP/RT
Density & Molar Mass of a Gas
• We can also use mass, # mols, pressure, and
temperature to calculate the density of a gas
d = mP/nRT
• If we know the mass, pressure, volume, and
temperature of a gas, we can find its molar mass
M = mRT/PV
• We can also solve for the molar mass if we know the
density, temperature, and pressure of the gas
M = dRT/P
• Find the molar mass of a 0.136 g sample of gas. Its
volume is 0.112 L at a temperature of 298 K and a
pressure of 1.06 atm
M = 28.0 g/mol
• Calculate the density of CO2 in grams per liter (g/L)
at 0.990 atm and 55°C
d = 1.62 g/L
• Calculate the density of hydrogen gas at STP
d = .090 g/L
• The density of a gas was measured at 1.50 atm and
27o C and found to be 1.96 g/L. Calculate the molar
mass of the gas.
M = 32.2 g/mol
• A sample of phosphorous that weighs 3.243 x 10-2 g
exerts a pressure of 31.89 kPa in a 56.0 mL bulb at
550o C. What is the molar mass and molecular
formula of the phosphorous vapor? M = 124 g/mol - P4
• An unknown diatomic gas has a density of 3.164 g/L
at STP. What is the identity of the gas?
M = 70.95 g/mol - Cl2
Dalton’s Law of Partial Pressures
• Gas laws tell us that the total pressure of a mixture
depends solely on the number of moles of gas and
not the kinds of molecules
• Partial pressure – the pressure of an individual gas
in a mixture of gases
• The total pressure exerted by the mixture of nonreactive gases is equal to the sum of the partial
pressures of each individual gas
Ptotal = p1 + p2 + p3 + …+ pn
Mole Fraction
• Because each gas in a mixture acts as if it is alone,
we can relate amount in a mixture to partial
pressures:
• That ratio of moles of a substance to total moles is
called the mole fraction, χ.
Pressure and Mole Fraction
• The end result is
• The partial pressure of an individual gas is equal to
the total pressure multiplied by the mole fraction of
that gas
• 2.00 mol He is mixed with 1.00 mol Ar. Find the
partial pressure of each at 1.75 atm pressure
PHe = 1.17 atm
PAr = .583 atm
• A mixture of helium, neon, and argon has a total
pressure of 558 mm Hg. The partial pressure of
helium is 341 mm Hg and the partial pressure of
neon is 112 mm Hg. What is the partial pressure of
argon?
105 mm Hg
Consider the following apparatus containing He gas
in both sides at 45o C. Initially the valve is closed.
After the valve is opened, what is the pressure of
the helium gas?
2.00 atm
9.00 L
3.00 atm
3.00 L
PHe = 2.25 atm
27.4 L of O2 gas at 25.0o C and 1.30 atm, and 8.50 L
of He gas at 25.0o C and 2.00 atm were pumped into
a tank with a volume of 5.81 L at 25oC
Calculate the new partial pressure of O2
6.13 atm
Calculate the new partial pressure of He
2.93 atm
Calculate the total pressure in the tank
9.06 atm
Collecting a Gas over Water
• Since gases have such small densities, it can be
difficult to measure their mass
• A common way to determine the amount of gas
present is by collecting it over water and measuring
the height of displaced water
• This arrangement is called
a pneumatic trough, and it
was widely used in the early
days of chemistry
Ø
As the gas enters the bottle, it
displaces the water and becomes
trapped in the closed, upper part of the bottle
Collecting a Gas over Water
When a gas from a
chemical reaction is
collected through water,
water molecules become
mixed with the gas
molecules
The pressure of water
vapor in the final mixture
is the vapor pressure of
water at the temperature
at which the gas is
collected
Zn(s)+ 2HCl
ZnCl2 + H2(g)
PTotal = PH2 + PH 2O
Vapor of Water & Temperature
52
H2 gas is collected over water at 22.5°C. Find the
pressure of the dry gas if the atmospheric pressure
is 94.4 kPa
PH2 = 91.7 kPa
A gas is collected over water at a temp of 35.0°C
when the barometric pressure is 742.0 torr. What is
the partial pressure of the dry gas?
Pgas = 699.8 torr
Gas Stoichiometry
• Mass-Mass
• Mass-Volume
• Volume-Volume
Ø
Volume is proportional to moles, so….
Ø
Mole relationship from a balanced chemical reaction can
be used directly
Ø
No conversions needed!
Gas Stoichiometry
@ STP
• For volume-volume problems, if the gases are all at
STP, then you only need to use the mole ratio to
determine the volume of reactant or product
• For mass-volume problems, you can only use 22.4
L/mol if the gas you are using is at STP or the gas
you are producing will be at STP conditions!!
Nitrogen monoxide and oxygen gas combine to form
the brown gas nitrogen dioxide, which contributes to
smog. How many liters of nitrogen dioxide are
produced when 34 L of oxygen react with an excess
of nitrogen monoxide at STP?
2NO(g) + O2(g) ® 2NO2(g)
V = 68 L NO2
Assuming STP, how many milliliters of oxygen are
needed to produce 20.4 mL SO3 according to this
unbalanced equation?
SO2(g) +
O2(g) ®
V = 10.2 mL O2
SO3(g)
2H2(g) + O2(g) à 2H2O(g)
• If 3.25 L of oxygen react, how many liters of water
vapor are formed?
V = 6.50 L H2O
• Volume-Volume is just Avogadro’s Law!
Mass-Volume Problems
• Key step – get to moles!
• Mass conversion – use molar mass
• Volume conversion – use gas equation
• Need to know temperature and pressure
2Na(s) + 2H2O(l) à 2NaOH(aq) + H2(g)
25.0 g of sodium react with excess water at STP. How
many liters of hydrogen are produced?
V = 12.2 L
Potassium chlorate decomposes into potassium
chloride and oxygen gas. How many grams of KClO3
are needed to produce 5.00 L of oxygen at 0.750 atm
and 18oC?
m = 12.82 g
What volume (in L) of H2 at 355 K and 738 mm Hg is
required to synthesize 35.7 g of methanol, given:
CO(g) + 2H2(g) → CH3OH(g)
V = 66.9 L H2
Kinetic-Molecular Theory (KMT)
• Laws tell us what happens in
nature. Each of the gas laws
we have discussed tell us what
is observed under certain
conditions
• Why are these laws observed?
We will discuss a theory to
explain our observations
Main Tenets of KMT
a. Gases consist of large numbers of molecules that are
in continuous, random motion
b. The combined volume of all the molecules of the gas
is negligible relative to the total volume in which the
gas is contained
c. Attractive and repulsive forces between gas
molecules are negligible
d. Energy can be transferred between molecules during
collisions, but the KEavg of the molecules does not
change with time, as long as the temperature of the
gas remains constant
Main Tenets of Kinetic-Molecular Theory
e. The KEavg of the molecules is proportional to the
absolute temperature
Effusion & Diffusion
Effusion is the escape
of gas molecules
through a tiny hole into
an evacuated space
Diffusion is the spread of
one substance throughout
a space or a second
substance
Graham’s Law of Effusion
• Graham’s Law of Effusion states that the rate of
effusion is inversely proportional to the square root
of the molar mass of the gas
• The “lighter” gas always has a faster rate of speed
• The ratio of effusion rates of two different gases is
given by the equation
A mixture of helium and methane is placed in an
effusion apparatus. Calculate the ratio of their
effusion rates
M of CH4 = 16.04 g/mol
M of He = 4.003 g/mol
rate
rate
He
CH4
=
√ 4.003
16.04
= 2.002
A molecule of oxygen gas has an average speed of
12.3 m/s at a given temp and pressure. What is the
average speed of hydrogen molecules at the same
conditions?
VH2 = 49.0 m/s
An unknown gas diffuses 4.0 times faster than O2.
Find its molar mass
M = 2.0 g/mol
Real Gases
• In the real world, the behavior of gases only conforms
to the ideal-gas equation at relatively high temperature
and low pressure
• Even the same gas will show wildly different behavior
under high pressure at different temperatures
Differences Between Ideal and Real
Gases
Ideal Gas
Obey PV=nRT
Always
Real Gas
Molecular volume
Zero
Only at very low P
and high T
Small but nonzero
Molecular attractions
Zero
Small
Molecular repulsions
Zero
Small
Deviations from Ideal Behavior
The assumptions made in the kinetic-molecular model
(negligible volume of gas molecules themselves, no
attractive forces between gas molecules, etc.) break
down at high pressure and/or low temperature
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