Lesson Objectives
The student will:
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solve problems using the universal gas law, .
state Avogadro’s law of equal molecules in equal volumes.
calculate molar mass (M) from given mass,
temperature, pressure, and volume.
Vocabulary
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Avogadro's law
universal gas law
universal gas law constant (R)
Introduction
The individual gas laws (Boyle's, Charles's, and Gay-Lussac's) and the
combined gas law all require the quantity of gas remain constant. The
universal gas law (sometimes called the ideal gas law) allows us to make
calculations when there are different quantities of gas.
Avogadro’s Law
Avogadro’s law postulates that equal volumes of gas with the same
temperature and pressure contain the same number of molecules.
Mathematically, this would be written as: , where represents
the number of moles of molecules and represents a constant. This law
was known as Avogadro’s hypothesis for the first century of its
existence. Since Avogadro's hypothesis can now be demonstrated
mathematically, it was decided that it should be called a law instead of a
hypothesis. If we think about the definitions of the volumes, pressures,
and temperatures for gases, we can also develop Avogadro’s conclusion.
Suppose we have a group of toy robots that are all identical in strength.
They do not have the same size or look, but they all have exactly the
same strength at an assigned task. We arrange a tug-of-war between
groups of robots. We arrange the rope so that we can see one end of the
rope, but the other end disappears behind a wall. On the visible end, we
place eight robots to pull. On the other end, an unknown number of
robots will pull. When we say “go,” both sides pull with maximum
strength, but the rope does not move. How many robots are pulling on
the hidden end of the rope? Since each robot pulls with the exact same
strength, to balance the eight robots we know are on our end, there must
be eight robots on the other end. We can think of molecules as robots by
recognizing that molecules at the same temperature have exactly the
same striking power upon collision. As a result, equal volumes of gas
with equal pressures and temperatures must contain an equal number of
molecules.
In the early 1800s, the first attempts to assign relative atomic weights to
the atoms were accomplished by assigning hydrogen to have an atomic
mass of and by decomposing compounds to determine the mass
ratios in the compounds. Some of the atomic weights found this way
were accurate, but many were not. In the 1860s, Stanislao Cannizzaro
refined the process of determining relative atomic weights by using
Avogadro’s law. If gas and gas were heated to the same
temperature, placed in equal volume containers under the same pressure
(gas would be released from one container until the two containers had
the same pressure), Avogadro’s conditions would be present. As a result,
Cannizzaro could conclude that the two containers had exactly the same
number of molecules. The mass of each gas was then determined with a
balance (subtracting the masses of the containers), and the relationship
between the masses would be the same as the mass relationship of one
molecule of to one molecule of . That is, if the total mass of gas was and the total mass of gas was , then
Cannizzaro knew that one molecule of must have four times as much
mass as one molecule of . If an arbitrary value such as dalton was
assigned as the mass of gas , then the mass of gas on that same
scale would be daltons.
The Universal Gas Law Constant
We have considered three laws that examine how the volume of a gas
depends on pressure, temperature, and number of moles of gas (GayLussac's law requires a constant volume).
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Boyle’s law: at constant temperature and constant moles
of gas, where is a constant.
Charles’s law: at constant pressure and constant moles
of gas, where is a constant.
Avogadro’s law: at constant temperature and pressure,
where is the number of moles of gas and is a constant.
These three relationships can be combined to form the expression
, where is the combination of the three constants. The
equation is called the universal gas law (or the ideal gas law), and is
called the universal gas law constant. When the pressure is expressed in
atmospheres and the volume in liters, has the value
. You can convert the value of into
values for any set of units for pressure and volume. Moles, of course,
always have the unit moles, while the temperature must always be
Kelvin.
In our analysis of gas behavior, we have used a pair of assumptions that
are not always true. We have assumed that the volume of the actual
molecules in a gas is insignificant compared to the volume of the empty
space between molecules. We have treated the molecules as if they were
geometric points that took up no space. For most gases, this assumption
will be true most of the time, and the gas laws work well. If, however, a
gas is highly compressed (at very high pressure), the molecules will be
pushed together very closely, with much of the empty space between
molecules removed. Under such circumstances, the volume of the
molecules becomes significant, and some calculations with the gas laws
will be slightly off.
Another assumption that we have used is that the molecules are not
attracted to each other so that every collision is a perfectly elastic
collision. In other words, we have assumed that no energy is lost during
the collision. This assumption also works well most of the time. Even
though the molecules do have some attraction for each other, usually the
temperature is high enough that the molecular motion readily overcomes
any attraction and the molecules move around as if there were no
attraction. If, however, we operate with gases at low temperatures
(temperatures near the phase change temperature of the gas), the
molecular attractions have enough effect to cause our calculations to be
slightly off. If a gas follows the ideal gas laws, we say that the gas
behaves ideally. Gases behave ideally at low pressure and high
temperature. At low temperatures or high pressures, gas behavior may
become non-ideal. As a point of interest, if you continue your study of
chemistry, you will discover that there is yet another gas law equation
for these non-ideal situations.
Example:
A sample of nitrogen gas has a volume of at and
a pressure of . How many moles of nitrogen are present in this
sample?
Solution:
Step 1: Assign known values to the appropriate variable.
Step 2: Solve the combined gas law for the unknown variable.
Step 3: Substitute the known values into the formula and solve for the
unknown.
Example:
of methane gas are placed in a rigid container and heated to . What pressure will be exerted by the
methane?
Solution:
Step 1: Assign known values to the appropriate variable.
Step 2: Solve the combined gas law for the unknown variable.
Step 3: Substitute the known values into the formula and solve for the
unknown.
Example:
A sample gas containing of helium at a pressure of
is cooled to . What volume will the gas occupy under
these conditions?
Note: If we are to use for the value of
, then the pressure must be converted from torr to atm.
Solution:
Step 1: Assign known values to the appropriate variable.
Step 2: Solve the combined gas law for the unknown variable.
Step 3: Substitute the known values into the formula and solve for the
unknown.
For a video example of solving an ideal gas law problem (4h), see http://
www.youtube.com/watch?v=JEsfU7ogbVQ (4:44).
Molar Mass and the Universal Gas Law
The universal gas law can also be used to determine the molar mass of
an unknown gas provided the pressure, volume, temperature, and mass
are known. The number of moles of a gas, , can be expressed as grams/
molar mass. If we substitute for in the universal gas law, we get
which can be re-arranged to .
Example:
grams of an unknown gas occupy under standard
conditions. What is the molar mass of the gas?
Solution:
Density and the Universal Gas Law
The density of a gas under a particular set of conditions is the mass of
the sample of gas divided by the volume occupied at those conditions,
. In the universal gas law, both the mass of the sample of gas
and the volume it occupies are represented. We can substitute density for
where it appears in the equation and produce an equation that
contains density instead of mass and volume. For example, in the
equation , mass appears in the numerator and volume appears
in the denominator. We can substitute for those two variables and
obtain the equation .
Example:
The density of a gas was determined to be and . What is the molar mass of the gas?
at Solution
Lesson Summary
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Avogadro’s law states that equal volumes of gases under the same
temperature and pressure contain equal numbers of molecules.
The universal gas law, also known as the ideal gas law, relates the
pressure, volume, temperature, and number of moles of gas:
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The universal gas law assumes that the gas molecules are
geometric points that take up no space and that they undergo
perfectly elastic collision.
The universal gas law can also be used to determine the molar
mass and the density of an unknown gas.
Further Reading / Supplemental Links
This video reviews how to perform ideal gas law (universal gas law)
calculations.
• http://video.google.com/videoplay?
docid=-2010945058854921425#
Review Questions
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What conditions of temperature and pressure cause gases to
deviate from ideal gas behavior?
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What volume will of hydrogen gas occupy at
of pressure and ?
How many moles of gas are required to fill a volume of
at and ?
What is the molar mass of a gas if its density is at STP?