Transcript Document

Gas Laws, Gas Stoichiometry
Kinetic-Molecular Theory (KMT)
• The Kinetic-Molecular Theory of Matter
states that matter is composed of a large
number of small particles—individual atoms
or molecules—which are in constant motion
unless their temperature is absolute zero
( 0 K or -273 ⁰C).
• The KMT model can be used to explain
properties of solids, liquids, and gases in
terms of the energy of their particles and
the forces that act between particles.
Kinetic-Molecular Theory (KMT)
• The amount of motion is proportional to
temperature. The higher the
temperature, the more motion there is.
• Solids, liquids and gases differ in the
freedom of motion of their particles and
extent to which the particles interact.
http://preparatorychemistry.com/KMT_flash.htm
Recall the properties of solids,
liquids, gases
SOLID
Definite
volume
and shape
‘indefinite’ means ‘not
defined or specified’’
LIQUID
Definite volume,
indefinite shape
(fits shape of its
container)
GAS
Indefinite
volume,
Indefinite
shape
(expands to fill the
volume and shape
of its container)
Kinetic-Molecular Theory (KMT)
• What exactly do we mean by ‘motion’?
Motion of particles can be vibrational,
rotational, and/or translational.
academic.pgcc.edu/
~ssinex/Dance/Mol_
motion.ppt
KMT Applied to Gases
• Kinetic molecular theory can help us
understand how and why the gas laws work
and to predict when the gas laws won’t work.
• Daniel Bernoulli started kinetic molecular
theory in 1738 when he proposed a thought
model consistent with Boyle’s Law in an
attempt to explain how gases exert pressure.
Clausius refined the theory in the mid-1800s.
Kinetic-Molecular Theory of Gases
Based upon Five Assumptions
1. Gases consist of large numbers of tiny particles
that are far apart relative to their size. Gas
volumes are about 1000x greater than liquid or
solid volumes (which explains gases’
compressibility).
2. Collisions between gas particles and between
particles and container walls are elastic collisions.
Elastic collisions have no net loss of kinetic
energy (no energy lost to heat; like
billiard balls).
Kinetic-Molecular Theory of Gases
Based upon Five Assumptions
3. Gas particles are in continuous, rapid, random
motion in all directions. They therefore possess
kinetic energy, which is energy of motion. The
kinetic energy of particles is greater than the
attractive forces between them except near the
temperature at which the gas condenses and
becomes a liquid.
4. There are no forces of attraction between gas
particles. When they collide, they bounce apart.
Gases that are polar (H2O and NH3) do NOT behave
as ideal gases.
Kinetic-Molecular Theory of Gases
Based upon Five Assumptions
5. The temperature of a gas depends on the average
kinetic energy of the particles of the gas.
𝑲𝑬 =
𝟏
𝒎𝒗𝟐
𝟐
where m is the mass of the particle and v is its average
speed.
For a particular gas, the masses of the particles are the
same, so as the speed of its particles increases the
temperature of the gas increases.
All gases at the same temperature have the same kinetic
energy. At the same temperature, lighter gas particles
(such as hydrogen) have higher average speeds than do
heavier particles, such as oxygen molecules.
Kinetic-Molecular Theory of Gases
• An ideal gas is a hypothetical gas that
perfectly fits all the assumptions of the kineticmolecular theory.
• While the Kinetic-Molecular Theory of Gases
applies only to ideal gases, real gases behave
close to ideally when the temperature is high
and the pressure is low.
Critical Thinking Questions
Molecules of hydrogen escape from Earth’s
atmosphere, but molecules of oxygen and
nitrogen stay near Earth’s surface and remain in
the atmosphere. Why?
Why do gases of polar molecules not behave
ideally?
Nature of Gases - Properties
• EXPANSION
Gases do not have a definite shape or volume, and
expand to completely fill any container, taking its shape
and volume. Which two assumptions of the KMTG
explain this?
(#3 and #4)
• FLUIDITY
Gas particles glide easily or flow past each other as do
the particles of liquids. Because liquids and gases flow,
they are referred to as fluids. Which assumption of the
KMTG explains this?
(#4)
Nature of Gases
• LOW DENSITY
The density of a gaseous substance at atmospheric
pressure is about 1/1000 of the density of the same
substance in the liquid or solid state. Which assumption
of the KMTG explains this?
(#1)
• COMPRESSIBILITY
Gas particles are crowded together during
compression, so the volume is greatly decreased.
Pressurized steel cylinders can contain more than 100
times as many particles as non-pressurized containers of
the same size. Which assumption of the KMTG explains
this?
(#1)
Nature of Gases
Diffusion also happens
with liquids.
• DIFFUSION
Gases spread out and mix
with one another, even
without being stirred. This
spontaneous mixing of the
particles of two substances
caused by their random
motion is called diffusion.
Which two assumptions of the
KMTG explain this?
(#3 and #1)
Nature of Gases
• EFFUSION
In contrast, effusion is a process
by which gas particles pass through
a tiny opening. The rates of
effusion of different gases depend
on the velocities of their
particles— low mass particles
effuse faster (because of their
higher velocities) than do
molecules of higher mass (lower
velocities).
(#3 and #5) Also #1!
Phase Diagrams
A phase diagram is a
graph of pressure
versus temperature
that shows the
conditions under
which the phases of
a substance exist.
TP = triple point, the temperature and pressure conditions at which
solid, liquid, and vapor can coexist at equilibrium.
CP = critical point—the critical temperature and the critical
pressure above which water cannot exist in the liquid state
Pressure and Force
• Pressure (P) is defined as the force per unit
area on a surface.
• For the same woman, which heel exerts the
most pressure on the ground? (Hint: her
weight [the force] doesn’t change.)
Gas Pressure
• When you pump air into a
container, the pressure inside
the container will increase
because the number of
collisions of molecules of air
with the inside walls of the
container increases. These
collisions cause an outward
push or force against the
inside walls, so gas molecules
exert pressure on any surface
with which they collide.
The pressure exerted by a
gas depends on its volume,
temperature and the number
of molecules present.
Measuring Gas Pressure
• Atmospheric pressure with a
barometer, either mercury or
aneroid
• Gas pressure with manometer,
pressure gauges
Units of Pressure
Unit
Symbol
Definition/relationship
Pa
1 Pa = 1 N/m2
Millimeter of
mercury
mm Hg
Pressure that supports a
1 mm mercury column in a
barometer
Torr
torr
Atmosphere
atm
Pounds per
square inch
psi
Pascal
1 torr = 1 mm Hg
Average atmospheric
pressure at sea level and 0°C
1 atm = 760 mm Hg
= 760 torr
= 1.013 x 105 Pa
= 101.325 kPa
1 psi = 6.892 x 103 Pa
= 6.892 kPa
1 atm = 14.700 psi
Standard Temperature and Pressure
• To compare volumes of gases, one must know
the temperature and pressure at which the
volumes are measured.
• Scientists have agreed upon standard
conditions of exactly 1 atm pressure and 0°C.
These conditions are called standard
temperature and pressure and are commonly
abbreviated STP.
STP means 1.0 atm and 0°C = 273 K
Dalton’s Law of Partial Pressure
• The pressure exerted by each gas in an unreactive
mixture is independent of that exerted by other
gases present.
• The pressure of each gas in a mixture is called the
partial pressure of that gas.
• Dalton’s Law of Partial Pressures states that the
total pressure of a gas mixture is the sum of the
partial pressures of the component gases.
𝑃𝑇 = 𝑃1 + 𝑃2 + 𝑃3 + ⋯
where PT is the total pressure of the mixture and P1, P2,
P3, and so on are the partial pressures of gases 1, 2, 3,
and so on.
Demo
• Fill a syringe with air, cover the tip and try to
compress the volume. Can you completely
compress the syringe? Why or why not? What
happens to the pressure inside the syringe?
• Drop a marshmallow (which is mostly air) into a
syringe, push the plunger until it almost touches
it, then seal off the end of the barrel. Now pull on
the plunger and watch the marshmallow.
• Repeat, but drop the marshmallow into a full
syringe. Seal off the barrel and compress the
syringe. What happens to the marshmallow?
Boyle’s Law: Pressure-Volume
In 1662 Robert Boyle discovered that doubling the
pressure on a sample of gas at constant temperature
reduces its volume by one-half, and reducing the
pressure on a gas by one-half doubles the volume.
Therefore pressure and temperature are inversely
proportional.
𝑃𝑉 = 𝑘
𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Changes in volume and pressure at constant
temperature are given by Boyle’s Law
𝑃1 𝑉1 = 𝑃2 𝑉2
where P1 and V1 represent initial conditions, and P2
and V2 represent a different set of conditions.
Charles’ Law: Volume - Temperature
In 1787 Jacques Charles discovered the relationship between gas volume and temperature at
constant pressure. Charles found that all gases
expand to the same extent when heated by the
same amount. He also found that gas volume is
directly proportional to its temperature:
𝑉
𝑉 = 𝑘𝑇 𝑜𝑟 = 𝑘
𝑇
where the temperature T is measured using the
Kelvin scale (TK = TC + 273.15 which you may
simplify to TK = TC + 273)
Charles’ Law: Volume - Temperature
For gases at constant pressure, Charles’ Law is
given as:
𝑉1 𝑉2
=
𝑇1 𝑇2
where V1 and T1 are the initial conditions, and V2
and T2 are different conditions. Knowing three
of the four values allows one to solve for the
fourth.
Gay-Lussac’s Law: Pressure-Temperature
In 1802, Joseph Gay-Lussac recognized that
pressure and temperature of a gas at constant
volume are directly proportional to each other:
𝑃
𝑃 = 𝑘𝑇 𝑜𝑟 = 𝑘
𝑇
where T is the temperature in Kelvins and k is a
constant that depends on the volume and
quantity of gas.
Gay-Lussac’s Law: Pressure-Temperature
For a given mass of gas at constant volume, with
P1 and T1 representing the initial conditions, and
P2 and T2 representing another set of conditions,
Gay-Lussac’s Law is given as:
𝑃1 𝑃2
=
𝑇1 𝑇2
Knowing three of the four values allows one to
solve for the fourth.
Combined Gas Law
• Gases often undergo changes in pressure, volume
and temperature at the same time, so luckily it is
possible to combine Charles’, Boyle’s and GayLussac’s Laws into a single expression.
• The combined gas law expresses the relationship
between the pressure, volume, and temperature of
a fixed amount of gas:
𝑃𝑉
=𝑘
𝑇
where k is a constant that depends on the amount of
gas and T is in Kelvins.
Combined Gas Law
𝑃1 𝑉1 𝑃2 𝑉2
=
𝑇1
𝑇2
where T is in kelvins, P1, V1, and T1 are the initial
conditions, and P2, V2, and T2 are different
conditions. Knowing five of the six values allows
one to solve for the sixth.
Notice that this combined law reduces to each of
the prior laws when one of the variables is
constant. For instance, when temperature is
constant, this reduces to Boyle’s Law (𝑃1 𝑉1 = 𝑃2 𝑉2 ).
Gas Volumes and the Ideal Gas Law
Joseph Gay-Lussac in the early 1800s studied gas
volume relationships by observing the reaction of
hydrogen and oxygen gases. He found that at
constant temperature and pressure
2 𝐿 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑔𝑎𝑠 + 1 𝐿 𝑜𝑓 𝑜𝑥𝑦𝑔𝑒𝑛 𝑔𝑎𝑠
→ 2 𝐿 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑣𝑎𝑝𝑜𝑟
This showed a simple and definite 2 : 1 : 2
relationship between the volumes of the reactants
and the product. He found that this relationship
was true for any volume of reactants and products.
Gas Volumes and the Ideal Gas Law
Gay-Lussac also experimented with other gases,
such as hydrogen and chlorine, in which he
found that
1 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑔𝑎𝑠 + 1 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐ℎ𝑙𝑜𝑟𝑖𝑛𝑒 𝑔𝑎𝑠
→ 2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑐ℎ𝑙𝑜𝑟𝑖𝑑𝑒 𝑔𝑎𝑠
In 1808, he summarized his results in this
statement known as Gay-Lussac’s law of
combining volumes of gases:
At constant temperature and pressure, the
volumes of gaseous reactants and products can
be expressed as ratios of small whole numbers.
Avogadro’s Law
• However, Gay-Lussac’s law seemed to violate
Dalton’s idea that gases are made up of single,
indivisible atoms! Dalton thought water was HO,
not H2O, as Gay-Lussac’s results showed:
2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑔𝑎𝑠 + 1 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑜𝑥𝑦𝑔𝑒𝑛 𝑔𝑎𝑠
→ 2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑣𝑎𝑝𝑜𝑟
• Avogadro reasoned that what was combined were
MOLECULES, and that some molecules could have
more than one atom of an element.
• Avogadro explained in 1811 that reactant elements
didn’t have to be in monatomic form, but could
contain more than one atom. Recall that oxygen,
hydrogen, nitrogen, chlorine, and iodine are all
diatomic gases.
Hydrogen
Oxygen
Carbon dioxide
• Regardless of which
gas they contain,
balloons with the
same volume at the
same temperature
and pressure have
equal number of
molecules!
AVOGADRO’S LAW
Equal volumes of gases at the same
temperature and pressure contain
equal numbers of molecules.
This means we can relate VOLUME of a gas to the
number of MOLES.
Molar Volume of a Gas
• The volume occupied by one mole of any gas at STP
(1 atm, 0°C) is known as the standard molar volume
of a gas. This volume has been found to be
22.41410 L. You may use 22.4 L.
1 𝑚𝑜𝑙
22.4 𝐿
• If you know the volume of a gas, you can use
as a conversion factor to find the number of moles,
and therefore the mass, of that gas.
• If you know the number of moles of a gas, or the
22.4 𝐿
mass, you can use
as a conversion factor to
1 𝑚𝑜𝑙
find the volume in liters.
• Remember your MOLETOWN diagram: Now you get
to go to the gated community, LITERLAND, via
Highway 22.4.
Gas Stoichiometry
For gaseous reactants or products, the
coefficients in chemical equations not only
indicate molar amounts and mole ratios but
also reveal volume ratios if the conditions stay
the same.
2𝐶𝑂 g + 𝑂2 g → 2𝐶𝑂2 g
2 molecules
2 moles
1 molecule
1 mole
2 molecules
2 moles
2 volumes
1 volume
2 volumes
Gas Stoichiometry
2𝐶𝑂 g + 𝑂2 g → 2𝐶𝑂2 g
Possible volume ratios (similar to mole ratios)
for this equation are:
2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 𝐶𝑂
1 𝑣𝑜𝑙𝑢𝑚𝑒 𝑂2
2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 𝐶𝑂
2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 𝐶𝑂2
1 𝑣𝑜𝑙𝑢𝑚𝑒 𝑂2
2 𝑣𝑜𝑙𝑢𝑚𝑒𝑠 𝐶𝑂2
Given 0.75 L of oxygen gas, how many liters of
carbon dioxide will be produced, at the same
temperature and pressure?
2 𝐿 𝐶𝑂2
0.75 𝐿 𝑂2 ×
= 1.5 𝐿 𝐶𝑂2
1 𝐿 𝑂2
Ideal Gas Law
• The gas laws we have learned so far can be
combined into a single equation. The ideal gas
law is the mathematical relationship among
pressure, volume, temperature, and number of
moles of a gas.
• The ideal gas law is the equation of state for an
ideal gas as the 4 variables define the state of a gas.
Ideal Gas Constant
The value and units of the ideal gas constant R are
found by rearranging the equation for the Ideal Gas
Law:
𝑃𝑉 = 𝑛𝑅𝑇
𝑃𝑉
𝑅=
𝑛𝑇
1 𝑎𝑡𝑚 (22.4 𝐿)
𝐿 ∙ 𝑎𝑡𝑚
𝑅=
= 0.0821
1 𝑚𝑜𝑙 (273 𝐾)
𝑚𝑜𝑙 ∙ 𝐾
This value and units for R are used when pressure is
measured in atmospheres, volume in liters
(temperature is always measured in kelvins and n is
always moles). For other units, see next slide.
Ideal Gas Law
• R is the ideal gas constant and its value depends
on the units chosen for pressure, volume and
temperature.
Unit of R
Numerical
value of R
Unit of P
Unit of V
Unit of T
Unit of n
𝐿 ∙ 𝑚𝑚 𝐻𝑔
𝑚𝑜𝑙 ∙ 𝐾
62.4
mm Hg
L
K
mol
𝐿 ∙ 𝑎𝑡𝑚
𝑚𝑜𝑙 ∙ 𝐾
0.0821 or
8.21 x 10-2
atm
L
K
mol
8.314
Pa
𝑚3
K
mol
8.314
kPa
L
K
mol
𝐽
𝑚𝑜𝑙 ∙ 𝐾
𝐿 ∙ 𝑘𝑃𝑎
𝑚𝑜𝑙 ∙ 𝐾
Note: 1 L  atm = 101.325 J and 1 J = 1 Pa  m3
Sample Problem – Boyle’s Law
A sample of nitrogen gas has a volume of 478 cm3
and a pressure of 104.1 kPa. What volume would
the gas occupy at 88.2 kPa if the temperature
remains constant?
(Answer is 564 cm3)
P1 = _________
104.1 kPa
V1 = _________
478 cm3
P2 = _________
88.2 kPa
V2 = _________
?
P1V1 = P2V2
𝑃1 𝑉1
𝑉2 =
𝑃2
(104.1𝑘𝑃𝑎)(478 𝑐𝑚3 )
=
88.2 𝑘𝑃𝑎
Sample Problem – Charles’ Law
Air in a closed cylinder is heated from 25°C to
36°C. If the initial pressure is 3.80 atm, what is
the final pressure?
(Answer = 3.94 atm)
V1 = _________
3.80 atm
T1 = _________
25 ⁰C → 298 K
V2 = _________
?
T2 = _________
36 ⁰C → 309 K
𝑉1 𝑉2
=
𝑇1 𝑇2
𝑉1 𝑇2
𝑉2 =
𝑇1
(3.80 𝑎𝑡𝑚)(309 𝐾)
=
298 𝐾
Sample Problem – Gay-Lussac’s Law
A 20 L cylinder contains 6 atm of gas at 27 °C.
What would the pressure of the gas be if the gas
was heated to 77 °C?
(Answer = 7 atm)
Sample problem – Combined Gas
A gas is heated from 263.0 K to 298.0 K and the
volume is increased from 24.0 liters to 35.0 liters
by moving a large piston within a cylinder. If the
original pressure was 1.00 atm, what would the
final pressure be?
(Answer = 0.78 atm)
Sample Problem – Ideal Gas Law
What is the pressure in atmospheres exerted by a
0.500 mol sample of nitrogen gas in a 10.0 L
container at 298 K?
Givens: n = 0.500 mol, V = 10.0 L, T = 298 K
Unknown: P of N2 gas in atm
Plan: Use ideal gas law and rearrange for P:
𝑃𝑉 = 𝑛𝑅𝑇
→
𝑃=
𝑛𝑅𝑇
𝑉
𝐿 ∙ 𝑎𝑡𝑚
0.500 𝑚𝑜𝑙 (0.0821
)(298 𝐾)
𝑚𝑜𝑙
∙
𝐾
𝑃=
10.0 𝐿
= 1.22 𝑎𝑡𝑚
Sample Problem – Avogadro’s Law
At STP, 100.0 g of N2 has what volume?
(Recall that at STP, 1 mol of any gas has a volume
of 22.4L.)
(Answer=79.97 L)
1 𝑚𝑜𝑙 𝑁2
28.02 𝑔 𝑁2
= 3.57 𝑚𝑜𝑙 𝑁2
V1 = 22.4L
________ 100.0 g N2 ×
n1 = 1.0
________
mol
𝑉1 𝑉2
=
V2 = ________
?
𝑛1 𝑛2
n2 = Convert
________
from mass
= 3.57 mol
𝑉1 𝑛2 22.4𝐿 𝑥 3.57 𝑚𝑜𝑙
𝑉2 =
=
𝑛1
1 𝑚𝑜𝑙