Lesson Objectives
The student will:
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state Boyle's law, Charles's law, and Gay-Lussac's law.
solve problems using Boyle's law, Charles's law, and Gay-Lussac's
law.
state the combined gas law.
solve problems using the combined gas law.
Vocabulary
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Boyle's law
Charles's law
combined gas law
Gay-Lussac's law
Introduction
The gas laws are mathematical expressions that relate the volume,
pressure, temperature, and quantity of gas present. They were
determined from the results of over 100 years of experimentation. They
can also be derived logically by examining the present day definitions of
pressure, volume, and temperature.
Boyle’s Law
Gases are often characterized by their volume, temperature, and
pressure. These characteristics, however, are not independent of each
other. Gas pressure is dependent on the force exerted by the molecular
collisions and the area over which the force is exerted. In turn, the force
exerted by these molecular collisions is dependent on the absolute
temperature. The relationships between these characteristics can be
determined both experimentally and logically from their mathematical
definitions.
The relationship between the pressure and volume of a gas was first
determined experimentally by an Irish chemist named Robert Boyle
(1627-1691). The relationship between the pressure and volume of a gas
is commonly referred to as Boyle’s law.
When we wish to observe the relationship between two variables, it is
absolutely necessary to keep all other variables constant so that the
change in one variable can be directly related to the change in the other.
Therefore, when the relationship between the volume and pressure of a
gas is investigated, the quantity and temperature of the gas must be held
constant so that these factors do not contribute to any observed changes.
You may have noticed that when you try to squeeze a balloon, the
resistance to squeezing becomes greater as the balloon becomes smaller.
That is, the pressure inside the balloon becomes greater when the
volume is reduced. This phenomenon can be studied more carefully with
an apparatus like the one shown below. This device is a cylinder with a
tightly fitted piston that can be raised or lowered. There is also a
pressure gauge fitted to the cylinder so that the gas pressure inside the
cylinder can be measured. The amount of gas of gas inside the cylinder
cannot change, and the temperature of the gas is not allowed to change.
In the picture on the left, a 4.0-liter volume of gas is exerts a pressure of
. If the piston is pushed down to decrease the volume of the gas
to , the pressure of the gas is found to be . The piston
can be moved up and down to positions for several different volumes,
and the pressure of the gas can be for each of the volumes.
We might note from casual observation of the data that doubling volume
is associated with the pressure being reduced by half. Likewise, if we
move the piston to cause the pressure to double, the volume is halved.
We can analyze this data mathematically by adding a fourth column to
our table – namely, a column showing the product of multiplying
pressure times volume for each trial.
The data in the last column shows that with constant temperature and
quantity of gas, the pressure times the volume for this sample of gas
yields a constant. A mathematical constant (often represented by ) is a
number that does not change even when other quantities in the formula
do change. The value of will change if a different quantity of gas is
used or if the trials are carried out at a different temperature, but for a
particular mass of a particular gas at a particular temperature, the value
of will always be the same. A subscript 1 is used to distinguish this
constant from the constants of other gas laws. This relationship can be
shown in a mathematical equation.
This equation is a mathematical statement of Boyle’s law. This particular
equation demonstrates what is called an inverse proportionality. When
one of the variables is increased, the other variable will decrease by
exactly the same factor. This relationship can be easily seen in a graph
like the one shown below.
This result matches our logic intuition. If the pressure a gas exerts is
equal to the force divided by the area over which it is exerted, we would
expect the pressure to increase when we decrease the area but keep the
force constant. Similarly, if we maintain the same number of molecules
of gas and we keep the same temperature, we expect the total force
exerted by the molecules to be the same. As a result, if we expand the
volume of the gas, which increases the area over which the force is
exerted, we would expect the pressure to decrease.
This video is a laboratory demonstration of Boyle's Law (4c): http://
www.youtube.com/watch?v=J_I8Y-i4Axc (1:38).
Charles’s Law
The relationship between the volume and temperature of a gas was
investigated by a French physicist, Jacques Charles (1746-1823). The
relationship between the volume and temperature of a gas is often
referred to as Charles’s law.
An apparatus that can be used to study the relationship between the
temperature and volume of a gas is shown below. Once again, the
sample of gas trapped inside a cylinder so that no gas can get in or out.
Thus, we would have a constant mass of gas inside the cylinder. In this
setup, we would also place a mass on top of a movable piston to keep a
constant force pushing against the gas. This guarantees that the gas
pressure in the cylinder will be constant. If the pressure inside increases,
the piston will be pushed up until the inside pressure becomes equal to
the outside pressure. Similarly, if the inside pressure decreases, the
outside pressure will push the cylinder down, decreasing the volume
until the two pressures again become the same. This system guarantees
constant gas pressure inside the cylinder.
With this set up, we can adjust the temperature and measure the volume
at each temperature to produce a data table similar to the one we created
for comparing pressure and volume. The picture on the left in the
diagram above shows the volume of a sample of gas at , while the
picture on the right shows the volume when the temperature has been
raised to .
In order to find a constant from this data, it was necessary to divide each
volume with the corresponding Kelvin temperature. The mathematical
expression for Charles’s Law is:
This relationship is to be expected if we recognize that we are increasing
molecular collisions with the walls by raising the temperature. The only
way to keep the pressure from increasing is to increase the area over
which that force is exerted. This mathematical relationship is known as a
direct proportionality. When one variable is increased, the other variable
also increases by exactly the same factor.
Historical note: In addition to exploring the relationship between
volume and temperature for gases, Jacques Charles was also the first
person to fill a large balloon with hydrogen gas and take a solo balloon
flight.
This video is a laboratory demonstration of Charle's Law (4c): http://
www.youtube.com/watch?v=IkRIKGN3i0k (4:02).
Gay-Lussac’s Law
The relationship between temperature and pressure was investigated by
the French chemist, Joseph Gay-Lussac (1778-1850). An apparatus that
could be used for this investigation is shown below. In this case, the
cylinder does not have a movable piston because it is necessary to hold
the volume, as well as the quantify of gas, constant. This apparatus
allows us to alter the temperature of a gas and measure the pressure
exerted by the gas at each temperature.
After a series of temperatures and pressures have been measured, Table
below can be produced. Like Charles’s Law, in order to produce a
mathematical constant when operating on the data in the table, we must
divide pressure by temperature. The relationship, again like Charles’s
Law, is a direct proportionality.
The mathematical form of Gay-Lussac’s Law is:
This relationship demonstrates that when the temperature is increased,
the pressure must also increase to maintain the value of the constant, .
If the area the molecules are occupying is kept the same, the collisions
of the molecules with the surroundings will increases as the temperature
increases. This results in a higher pressure.
Standard Conditions for Temperature and Pressure (STP)
It should be apparent by now that expressing a quantity of gas simply by
stating its volume is inadequate. Ten liters of oxygen gas could contain
any mass of oxygen from to depending on
the temperature and pressure of the gas. Chemists have found it useful to
choose a standard temperature and pressure with which to express gas
volume. The standard conditions for temperature and pressure (STP)
were chosen to be and (
).
You will commonly see gas volumes expressed as of gas
under standard conditions or of gas at STP. Once you know
the temperature and pressure conditions of a volume of gas, you can
calculate the volume under other conditions.
This video is a black board presentation of some ideal gas law
calculations and it includes the definition of standard temperature and
pressure (4d): http://www.youtube.com/watch?v=GwoX_BemwHs
(13:01).
The Combined Gas Law
Boyle’s law shows how the volume of a gas changes when its pressure is
changed with the temperature held constant, while Charles’s law shows
how the volume of a gas changes when the temperature is changed with
the pressure held constant. Is there a formula we can use to calculate the
change in volume of a gas if both pressure and temperature change? The
answer is yes, as we can use a formula that combines both Boyle’s law
and Charles’s law.
Boyle's law states that for a sample of gas at constant temperature, every
volume times pressure trial will yield the same constant. We use the
subscript 1 to represent one set of conditions, and the subscript 2 to
represent a second set of conditions,
We can find a similar expression for Charles's law:
Combining the two equations yields:
The terms in this equation are rearranged and are commonly written in
the form shown below. This equation is also known as the combined gas
law.
When solving problems with the combined gas law, temperatures must
always be in Kelvin. The units for pressure and volume may be any
appropriate units, but the units of pressure must be the same for and
, and the units of volume for and must also be the same.
Example:
A sample of gas has a volume of and its pressure is occupy at STP?
. liters when its temperature is
. What volume will the gas
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 of gas occupies under standard conditions. What
temperature would be required for this sample of gas to occupy
and exert a pressure of ?
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 of oxygen gas under standard conditions has a
density of . What is the density of oxygen gas at and
?
Solution:
You can find the mass of oxygen in the sample by
multiplying volume times density, which yields a mass of Changing the temperature and/or pressure of a sample of gas changes its
volume and therefore its density, but it does not change the mass.
Therefore, when the new volume of the gas is found, the mass of oxygen
gas in it will still be The density under the new conditions
can be found by dividing the mass by the volume the gas now occupies.
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.
Lesson Summary
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Boyle's law states that for a gas at constant temperature, volume is
inversely proportional to pressure.
Charles's law states that for a gas at constant pressure, volume in
directly proportional to temperature.
Gay-Lussac's law states that for a gas at constant volume, pressure
is directly proportional to temperature.
The volume of a mass of gas is dependent on the temperature and
pressure. Therefore, these conditions must be given along with the
volume of a gas.
Standard conditions for temperature and pressure are and
.
The combined gas law relates the temperature, pressure, and
volume of a gas.
Review Questions
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When a sample of gas is placed in a larger container at the same
temperature, what happens to the total force of the molecules
hitting the walls?
When a sample of gas is placed in a larger container at the same
temperature, what happens to the pressure exerted by the gas?
If and are quantities that are related to each other by inverse
proportion, what will the value of become when the value of is increased by a factor of five?
Under what conditions will the value for the constant, , change
in the equation for Boyle’s Law, .
A sample of gas has a volume of under a pressure of
. What will be the new volume of the gas if the
pressure is reduced to at constant temperature?
A graph is made illustrating Charles’s Law. Which line would be
appropriate assuming temperature is measured in Kelvin?
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At constant pressure, the temperature of a sample of gas is
decreased. Will the volume of the sample
a increase
b decrease
c remain the same?
8 A sample of gas has its temperature increased from to
at constant pressure. If its volume at was ,
what is its volume at ?
9 A gas is confined in a rigid container and exerts a pressure of
at a temperature of . To what temperature
must this gas be cooled in order for its pressure to become
? Express this temperature in .
10 What is the abbreviation used to indicate standard conditions for
temperature and pressure?
11 A sample of gas has a volume of at standard conditions.
Find its volume at and .
12 A sample of gas has a volume of at and .
To what temperature must the gas be cooled in order for its volume
to become at a pressure of ?