Lesson Guide: Mathematical Relationships between P, V, and T
In this lesson, you will be using the data and conclusions you drew from the previous
lessons to come up with a mathematical relationship between pressure, volume, and
temperature.
Guiding Question: How can we use a mathematical model for gases to
predict what will happen to them when conditions change?
: Marshmallows in the Bell Jar
Make a prediction: What will happen to the marshmallows if pressure inside the bell jar is
reduced?
Write down your observations:
Explain your observations and draw a picture:
Pressure and Volume: Boyle’s Law
In 1662, a British Scientist named Robert Boyle discovered the same thing you did yesterday. The
pressure and volume of a gas are inversely proportional to each other. If the pressure goes up,
the volume goes down. This relationship is known as Boyle’s law.
Imagine a sample of gas in a syringe:
Before squeezing the syringe, we’ll call the pressure P1 and the volume V1. After squeezing the
syringe, the same amount of gas takes up less space. The new pressure is P2 and the new volume
is V2.
Here’s an equation that relates the syringe before and after squeezing:
P1V1 = P2 V2
We can use this equation to predict what will happen to a gas if the pressure
or volume is changed.
What variables must remain the same in order for this equation to work?
Let’s see how we might use this equation to predict what will happen to a gas:
A 2.0 L container of nitrogen gas has a pressure of 1.5 atm. What is the new pressure of the
system if the volume is increased to 4.5 L?
First identify your variables (include the units!):
P1 =
P2 =
V1 =
V2 =
Next: Solve for the unknown P2:
P1 V1 = P2 V2
(
atm) (
L) = (P2) (
L)
Then, get P2 by itself to solve the equation.
P2 = (
atm) (
L)
=
atm
L
Your turn:
1) If I have 5.6 L of gas in a piston at a pressure of 1.0 atm and compress the gas until its
volume is 2.8 L, what will the new pressure inside the piston be?
P1 =
P2 =
V1 =
V2 =
2) A balloon has a volume of 400 L when the pressure is 0.02 atm. What will the volume be at
a pressure of 1.0 atm?
P1 =
P2 =
V1 =
V2 =
: The Egg on a Diet
Write down your observations: (What happened to the egg when your teacher set it on top of
the flask shortly after lighting a fire inside?)
Explain your observation and draw a picture:
Temperature and Volume: Charles’s Law
Unlike the pressure and volume relationship, you know that there is a direct relationship between
temperature and volume. If the temperature of a gas goes up, the volume probably goes up as
well. This assumes that the container the gas is in is flexible (like a syringe or balloon). In 1787
Jacques Charles described this relationship mathematically. Notice it looks a little different from
Boyle’s law:
Charles’s Law: If pressure and the number of particles of gas stay the
same, the volume is proportional to the Kelvin temperature.
Caution: For Charles’s law to work, temperature has to be measured in Kelvin, and the pressure
has to stay the same.
Example: Let’s use this to predict what happens to a balloon when it is cooled down. Imagine a
2.0 Liter balloon at 293 K that is taken outside where the temperature is 263 K. What is the
new volume of the balloon?
Step 1: Identify variables (include the units!):
V1 =
V2 =
T1 =
T2 =
Step 2: Solve for unknown:
_
L
K
=
___V2___
K
Then, get V2 by itself to solve the equation.
(
L) (
V2
=
K
K)
=
L
V2
Your turn:
3)
A 500.0 mL balloon is cooled with liquid nitrogen from a temperature of 298 K to a new
temperature of 78 K. What is the new volume of the balloon?
V1 =
V2 =
T1 =
T2 =
4)
A sample of gas has a volume of 25.0 L outside on a cold day (-20°C). What will the new
volume of the gas be if brought inside to a warm room (25°C)? (*Note: Temperature is
given in °C)
V1 =
V2 =
T1 =
T2 =
Pressure and Temperature: Gay-Lussac’s Law
Gay-Lussac’s Law: If volume and the number of gas particles remain
the same, the pressure is proportional to the Kelvin temperature.
P1
P
= 2
T1
T2
Caution: For Gay-Lussac’s law to work, temperature has to be measured in Kelvin and the volume
has to stay the same.
Example: Let’s use this to predict what will happen to a gas in a closed container:
A 3.0 L sealed container of nitrogen gas has a pressure of 2.0 atm at 298 K. It is placed in a dark
storage room, where the temperature is 273 K. What is the new pressure of the system?
Step 1: Identify variables (include the units!):
P1 =
P2 =
T1 =
T2 =
Was there any unnecessary/extraneous information in this question? _______________________
Step 2: Solve for the unknown:
P1
P2
=
T1
T2
atm =
K
P2_____
K
Then, get P2 by itself to solve the equation.
(
atm) (
K)
= P2
K
P2 =
atm
Your turn:
5) A sample of gas in a closed container has an initial pressure of 1.4 atm at 264 K. The gas is
heated to a temperature of 298 K. What is the new pressure of the system?
P1 =
P2 =
T1 =
T2 =
6)
A sealed container of gas has an initial pressure of 1.8 atm at 273 K. The gas is heated
until the internal pressure reaches 2.1 atm. What is the new temperature of the gas?
P1 =
P2 =
T1 =
T2 =
Conflicting outcomes –
Combined Gas Law:
In some circumstances, like your warm up problem, more than two gas variables change at once.
For example, if you released a balloon at sea level and it rose into the atmosphere; both the
changes in pressure and temperature would affect the volume of the balloon. Charles’s law would
indicate that the volume of the balloon would shrink with the decreasing temperature at a higher
altitude. But Boyle’s law would indicate that as the pressure outside the balloon decreases, the
balloon would also expand in volume. There are two conflicting outcomes. Will the balloon expand
or shrink?
Combined Gas Law: If you know the relationship among temperature (T), pressure (P), and
volume (V) of a gas, you can determine the volume of the gas for other pressures and
temperatures.
Example: A child has a toy balloon with a volume of 1.80 L. The temperature of the balloon
when it was filled was 20°C and the pressure was 1.00 atm. If the child were to let go of the
balloon and it rose 3 kilometers into the sky where the pressure is 0.667 atm and the temperature
is -10°C, what would the new volume of the balloon be?
Step 1: Identify variables (include the units!)
P1 =
T1 =
V1 =
P2 =
°C or
K
T2 =
V2 =
°C or
K
Step 2: Solve for V2
(
atm) (
L)
=
(
K
V2 =
atm) ( V2 )
K
Liters
7. A 1.5 L sample of gas has a pressure of 1.5 atm at 15 C. If the volume is increased to 3.5 L
and the temperature is increased to 30 C, what is the new pressure?
P1 =
T1 =
V1 =
P2 =
K
T2 =
V2 =
K
Name:__________________________ Period:______ Date:___________
Gas Law Practice Problems:
1.
Examine the following set of data:
X
Y
2
4
4
8
6
12
Which of the equations below best describes the relationship shown?
X1 X 2

Y1 Y 2
2.
X1Y1 = X2Y2
other:
Examine the following set of data:
X
Y
8
8
16
4
2
32
Which of the equations below best describes the relationship shown?
X1 X 2

Y1 Y 2
X1Y1 = X2Y2
other:
3. If you have 5.6 L of gas in a piston at a pressure of 1.5 atm and compress the gas
until its volume is 4.8 L, what will the new pressure inside the piston be?
4. The temperature of air in a balloon at sea level (1.0 atm) is 25°C. If you take the
balloon with you to Denver, where the air pressure is 0.85 atm, what will the new
temperature inside the balloon be if the volume remains constant?
5. A 2.0 L container of nitrogen has a pressure of 3.2 atm and a temperature of 32°C.
What would the new temperature of the nitrogen be if you decreased the pressure
to 1.0 atm and enlarged the container to 3.0 L?
6. Deadly chlorine gas occupies a volume of 25 mL at 300 K. What volume will it occupy
at 600 K?
7. A bag of potato chips is packaged at sea level (1.00 atm) and has a volume of 315 mL.
If this bag of chips is transported to Denver (0.775 atm), what will the new volume of the
bag be?