Lesson Objectives
The student will:
•
•
•
apply the relationship that of any gas at standard
conditions will occupy .
convert gas volume at STP to moles and to molecules, and vice
versa.
apply Dalton’s law of partial pressures to describe the composition
of a mixture of gases.
Vocabulary
•
•
•
•
•
Dalton's law of partial pressure
diffusion
Graham's law
molar volume
partial pressure
Introduction
When molecules are at the same temperature, they have the same kinetic
energy regardless of mass. In this sense, all molecules are created equal
(as long as they are at the same temperature). They all exert the same
force when they strike a wall. When trapped in identical containers, they
will exert the exact same pressure. This is the logic of Avogadro's Law.
If we alter this situation a little and say we have the same number of
molecules at the same temperature exerting the same pressure, then
similar logic allows us to conclude that these groups of molecules must
be in equal-sized containers.
Molar Volume of Gases at STP
As you know, of any substance contains molecules. We also know that Avogadro's law states that equal volumes
of gases under the same temperature and pressure will contain equal
number of molecules. With a sort of reverse use of Avogadro’s Law, we
know then that molecules of any gas will occupy the same
volume under the same temperature and pressure. Therefore, we can say
that of any gas under the same conditions will occupy the
same volume. You can find the volume of of any gas under
any conditions by plugging the values into the universal gas law,
, and solving for the volume. We are particularly interested
in the molar volume, or the volume occupied by , of any gas
under standard conditions. At STP, we find that:
This is the volume that of any gas will occupy at STP. You
will be using this number to convert gas volumes at STP to moles, and
vice versa. You will use this value often enough to make it worth
memorizing.
Example:
How many moles are present in Solution:
of hydrogen gas at STP?
Example:
What volume will of methane gas
occupy at STP?
Solution:
Dalton’s Law of Partial Pressures
The English chemist, John Dalton, in addition to giving us the atomic
theory, also studied the pressures of gases when gases are mixed
together but do not react chemically. His conclusion is known as
Dalton’s law of partial pressures:
For a mixture of gases in a container, the total pressure exerted is the
sum of the pressures that each gas would exert if it were alone.
This can be expressed mathematically as:
In the last section, molecules were described as robots. The robots were
identical in their ability to exert force. When a group of diverse
molecules are at the same temperature, this analogy works well. In the
case of molecules, they don’t all look alike, but they have the same
average kinetic energy and therefore exert the same force when they
collide. If we have gaseous molecules at in a container
that exert a pressure of , then adding another gaseous
molecules of any substance at the same temperature to the container
would increase the pressure to . If there are in a container (and at the same temperature), each single molecule is
responsible for of the total pressure. In terms of the force of
collision, it doesn’t make any difference if the molecules have different
masses. At the same temperature, the smaller molecules are moving
faster than the larger ones, so the striking force is the same. It is this
ability to exert force that is measured by temperature. We can
demonstrate that different sized objects can have the same kinetic energy
by calculating the kinetic energy of a golf ball and the kinetic energy of
a bowling ball at appropriate velocities. Suppose a golf ball
is traveling at and a bowling ball is traveling at
.
The kinetic energies are the same. If these balls were to strike a pressure
plate, they would exert exactly the same force. If the balls were
invisible, we would not be able to determine which one had struck the
plate.
Now suppose we have three one-liter containers labeled A, B, and C.
Container A holds of gas at , container holds
of gas at , and container holds of
gas at . The pressure in each of the separate containers can be
calculated with .
The sum of these three pressures is If all three gases are placed in one of the containers at , the
pressure in the single container can also be calculated with
. (Remember that with all three gases in the same
container, the number of moles is .
You can quickly verify that the total pressure in the single container is
the sum of the individual pressures the gases would exert if they were
alone in their own container. The pressure exerted by each of the gases
in a mixture is called the partial pressure of that gas. Hence, Dalton’s
law is known as the law of partial pressures.
This video presents a discussion of various relationships involved in
Dalton's Law of partial pressures (4i): http://www.youtube.com/watch?
v=vPWFjmX-1aI (6:22).
Graham’s Law of Diffusion
It was mentioned earlier that gases will spread out and occupy any and
all space available. The spreading out and mixing of a substance is
called diffusion. The small spaces available in liquid structure allow
diffusion to occur only slowly. A drop of food coloring in a glass of
water (without stirring) might take half an hour or more to spread
through the water evenly. With a solid structure we would expect no
diffusion or mixing at all, although there are stories of lead and gold
bricks stacked together for many years that showed a few molecules
have exchanged on the surfaces. Even if the story is true, diffusion in
solids is negligible. In comparison, diffusion is very rapid in gases. Not
even the presence of other gases offer much obstacle to gases. If
someone opens a bottle of perfume or ammonia from across the room, it
is only a matter of minutes before you smell it. The molecules evaporate
from the liquid in the bottle and spread throughout the room quickly.
It turns out that diffusion is not the same for all gases. If bottles of strong
smelling substances are opened at the same time from across the room,
the odors will not reach you at the same time. If we pick the strong
smelling substances ammonia and acetone and
opened bottles of these substances, you will always smell the ammonia
before the acetone. If we think about this situation a little more, we can
probably predict this behavior. Since the two liquids are released in the
same room, they are likely to be at the same temperature, so the
molecules of both liquids will have the same kinetic energy. Remember
that when molecules have the same kinetic energy, the larger molecules
move slower and the smaller ones move faster. If we consider the molar
mass of these two substances, the molar mass of ammonia is ,
and the molar mass of acetone is . We would have realized,
then, that the ammonia molecules are traveling quite a bit faster than the
acetone molecules in order for them to have the same kinetic energy. As
a result, we would have expected the ammonia to diffuse through the
room faster.
A demonstration commonly used to show the different rates of diffusion
for gases is to dip one cotton ball in an ammonia solution and another
cotton ball in a dilute solution and stuff both cotton balls at
opposite ends of a glass tube. This setup is shown in the diagram below.
When and react, they form , a white powdery
substance. Molecules of and will escape the cotton balls at
opposite ends of the tube and diffuse through the tube toward each other.
Since the molar mass of is slightly more than double that of ,
the will travel further down the tube than the by the time they
meet.
As a result, the white cloud of always forms closer to the end. If this experiment is done carefully and the distances are measured
accurately, a reasonable ratio for the molar masses of these compounds
can be determined.
Thomas Graham (1805-1869), an English chemist, studied the rates of
diffusion of different gases and was able to describe the relationships
quantitatively in what is called Graham’s law:
Under the same conditions of temperature and pressure, gases diffuse at
a rate inversely proportional to the square root of the molecular masses.
Mathematically, this expression would be written as:
The video demonstrates diffusion in gases and liquids. It clearly shows
the differences in rate of diffusion due to molar mass as well as higher
temperatures (4b): http://www.youtube.com/watch?v=H7QsDs8ZRMI
(4:57).
Lesson Summary
•
At STP, one mole of any gas occupies •
•
When gases are mixed and do not react chemically, the total
pressure of the mixture of gases will be equal to the sum of the
partial pressures of the individual gases (Dalton's law of partial
pressures).
Gases diffuse at rates that are inversely proportional to the square
roots of their molecular masses (Graham's law).
Further Reading / Supplemental Links
This video shows an experiment that demonstrates the molar volume of
a gas at STP is 22.4 liters.
• http://www.youtube.com/watch?v=4b852VIEkHQ
Review Questions
1
2
3
4
5
A container of helium gas at pressure and a
container of hydrogen gas at are both
transferred into a container containing nitrogen gas at
. What is the final pressure in the final container holding
all three gases (assuming no temperature change)?
For the situation described in problem #1, what will be the partial
pressure of the helium in the final container?
At STP, how many molecules are in of gas?
If of gas A at STP and of gas at STP are
both placed into a evacuated container at STP, what
will the pressure be in the container?
Consider the gases and . Which of the following will be
nearly identical for the two gases at and ? i. average
molecular speed ii. rate of effusion through a pinhole iii. density
a i only
b iii only
c i and ii only
d ii and iii only
6
e i, ii, and iii
The density of an unknown gas at and is
determined to be . Which of the following gases is the
unknown most likely to be?
a b c d e