Gas Laws
Mechanics & Properties of Matter 9:
Gas Laws
AIM
This section introduces a model to explain the behaviour of gases. The simple laws that
govern the behaviour of gases are introduced experimentally. Calculations are made to predict
the effects of changing the volume, pressure and temperature of a fixed mass of gas. The
meaning of the absolute zero of temperature is explained.
OBJECTIVES
On completing this unit, you should be able to:
• Describe how the kinetic model accounts for the pressure of a gas.
• State that the pressure of a fixed mass of gas at constant temperature is inversely
proportional to its volume.
• State that the pressure of a fixed mass of gas at constant volume is directly
proportional to its temperature measured in kelvin.
• State that the volume of a fixed mass of gas at constant pressure is directly
proportional to its temperature in kelvin.
• Carry out calculations to convert temperatures in °C to K and vice versa.
• Carry out calculations involving pressure, volume and temperature of a fixed mass
of gas using the general gas equation.
• Explain what is meant by absolute zero of temperature.
• Explain the pressure-volume, pressure-temperature and volume-temperature laws
qualitatively in terms of the kinetic model.
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Mechanics and Properties of Matter
Gas Laws
GAS LAWS
Kinetic Theory of Gases
The kinetic theory tries to explain the behaviour of gases using a model. The model considers
a gas to be composed of a large number of very small particles which are far apart and which
move randomly at high speeds, colliding elastically with everything they meet.
Volume
The volume of a gas is taken as the volume of the container. The volume
occupied by the gas particles themselves is considered to be so small as to be
negligible.
Temperature The temperature of a gas depends on the kinetic energy of the gas particles.
The faster the particles move, the greater their kinetic energy and the higher the
temperature.
Pressure
The pressure of a gas is caused by the particles colliding with the walls of the
container. The more frequent these collisions or the more violent these
collisions, the greater will be the pressure.
Note that the gas laws all hold true only if we are dealing with a fixed mass of gas - i.e. if the
number of molecules stays constant. It should be obvious that if some gas escapes, the
pressure will decrease, or the volume will decrease.
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Mechanics and Properties of Matter
Gas Laws
Relationship between pressure and volume of a fixed mass of gas at a fixed temperature.
Apparatus: Boyle’s Law apparatus, pump.
S ca le
T ra p p ed a ir co lu m n
B o u rd o n g a u g e
O il
P um p
Instructions
• Use the pump to increase the pressure on the column of trapped air.
• Seal the apparatus using the tap when the pressure is high.
• Record the length of the trapped air column and the corresponding pressure.
• Using the tap, slowly reduce the pressure on the oil and seal the apparatus at a new
value of length.
• Record the new value of length and corresponding pressure.
• Repeat for a range of values of length of trapped air column.
• Use an appropriate format to show the relationship between pressure and volume of
a gas at constant temperature.
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Mechanics and Properties of Matter
Gas Laws
Relationship between pressure and temperature of a fixed mass of gas at a fixed volume.
Apparatus: air-filled flask, thermometer and
glass tube bored through a rubber
bung, rubber tubing, bourdon
gauge, container of boiling water.
T h e r m o m e te r
Instructions
H o t w a te r
• Set up the apparatus as shown and
carefully heat the water to boiling.
• Record the temperature and
A ir -f ille d
f la sk
corresponding pressure.
• Remove the flask and allow to cool in
the air of the room.
• As the air in the flask cools, obtain a set
of corresponding values of temperature
and pressure for the air in the flask.
• Use an appropriate format to show the
relationship between pressure and
temperature of a gas at constant volume.
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G la s s tu b e
a n d ru b b er
tu b in g
Mechanics and Properties of Matter
Gas Laws
Relationship between Pressure and Volume of a Gas
For a fixed mass of gas at a constant temperature, the pressure of a gas is inversely
proportional to its volume.
P∝
Graph
1
V
P × V = constant
p
P1V1 = P2V2
p
1
V
V
0
0
Example
The pressure of a gas enclosed in a cylinder by a piston changes from 80 kPa to 200 kPa.
If there is no change in temperature and the initial volume was 25 litres, calculate the new
volume.
P1 = 80 kPa P1V1 = P2V2
V1 = 25 litres 80 × 25 = 200 × V2
P2 = 200 kPa V2 = 10 litres
V2 = ?
Relationship between Pressure and Temperature of a Gas
If a graph is drawn of pressure against temperature in degrees Celsius for a fixed mass of gas
at a constant volume, the graph is a straight line which does not pass through the origin.
When the graph is extended back until the pressure reaches zero, it crosses the temperature
axis at -273 oC. This is true for all gases.
p
T / oC
-2 7 3
Kelvin Temperature Scale
-273oC is called absolute zero and is the zero on the kelvin temperature scale. At a
temperature of absolute zero, 0 K, all particle motion stops and this is therefore the lowest
possible temperature.
One division on the kelvin temperature scale is the same size as one division on the celsius
temperature scale, i.e. temperature differences are the same in kelvin as in degrees celsius,
e.g. a temperature increase of 10°C is the same as a temperature increase of 10 K.
Note the unit of the kelvin scale is the kelvin, K. It is not degrees kelvin, °K!
Converting Temperatures Between °C and K
Converting °C to K
Converting K to °C
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add 273
subtract 273
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Mechanics and Properties of Matter
Gas Laws
If the graph of pressure against temperature is drawn using the kelvin temperature scale, zero
on the graph is the zero on the kelvin temperature scale and the graph now goes through the
origin.
p
T /K
0
For a fixed mass of gas at a constant volume, the pressure of a gas is directly proportional to
its temperature measured in kelvin (K).
p ∝ T
p
= constant
T
p1
p
= 2
T1
T2
Example
Hydrogen in a sealed container at 27 °C has a pressure of 1.8 × 105 Pa. If it is heated to a
temperature of 77 °C, what will be its new pressure?
p1 = 1.8 × 105 Pa
T1 = 27 °C = 300 K
p2 = ?
T2 = 77 °C = 350 K
p2 = 2.1 × 105 Pa
Relationship between Volume and Temperature of a Gas
If a graph is drawn of volume against temperature, in degrees celsius, for a fixed mass of gas
at a constant pressure, the graph is a straight line which does not pass through the origin.
When the graph is extended until the volume reaches zero, again it crosses the temperature
axis at -273 °C. This is true for all gases.
V
T / oC
-2 7 3
If the graph of volume against temperature is drawn using the kelvin temperature scale, the
graph now goes through the origin.
V
T /K
0
For a fixed mass of gas at a constant pressure, the volume of a gas is directly proportional to
its temperature measured in kelvin (K).
V∝T
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V
= constant
T
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V1
V2
=
T1
T2
Mechanics and Properties of Matter
Gas Laws
Example
400 cm3 of air is at a temperature of 20 °C. At what temperature will the volume be 500 cm3
if the air pressure does not change?
V1 = 400 cm3
T1 = 20 °C = 293 K
V2 = 500 cm3
T2 = ?
V1
V
= 2
T1
T2
400
500
=
293
T2
T2 = 366 K = 93 °C (convert back to temperature
scale in the question)
Combined Gas Equation
By combining the above three relationships, the following relationship for the pressure,
volume and temperature of a fixed mass of gas is true for all gases.
P× V
= constant
T
p 1 V1
p 2 V2
=
T1
T2
Example
A balloon contains 1.5 m3 of helium at a pressure of 100 kPa and at a temperature of 27 °C.
If the pressure is increased to 250 kPa at a temperature of 127 °C, calculate the new volume
of the balloon.
p1 = 100 kPa
V1 = 1.5 m3
T1 = 27 °C = 300 K
p2 = 250 kPa
V2 = ?
T2 = 127 °C = 400 K
Strathaven Academy
100 × 1.5
300
=
200 × V2
400
V2 = 0.8 m3
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Mechanics and Properties of Matter
Gas Laws
Explaining the gas laws in terms of the kinetic model
These abbreviated ‘6-point’ versions may help you to memorise the main points in the
explanations:
1. The pressure-volume law (temperature constant)
• If volume is decreased,
• particles will hit container walls more often,
• and at the same time, the area of walls is decreased.
• so average force on walls will increase.
• Since P = F/A, pressure will increase.
• So when volume decreases, pressure increases
2. The pressure-temperature law (volume constant)
• If temperature is increased,
• kinetic energy and therefore speed of particles will increase.
• Particles collide with walls both harder and more often,
• so average force on container walls will increase.
• Since P = F/A, pressure will increase.
• So when temperature increases, pressure increases.
3. The volume-temperature law (pressure constant)
• If temperature is increased,
• kinetic energy and speed of particles will increase.
• Particles collide with walls harder.
• If pressure, P = F/A, is to remain constant, the increased force must be
spread over a greater area.
• The volume of the container must therefore increase.
• So when temperature increases, volume increases.
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Mechanics and Properties of Matter
Gas Laws
Gas laws
Pressure and volume (constant temperature)
21. 100 cm of air is contained in a syringe at atmospheric pressure (105 Pa).
If the volume is reduced to a) 50 cm3 or b) 20 cm3 without a change in
temperature, what will be the new pressures?
3
22. If the piston in a cylinder containing 300 cm3 of gas at a pressure of 105 Pa is moved
outwards so that the pressure of the gas falls to 8 × 104 Pa, find the new volume of the
gas.
23. A weather balloon contains 80 m3 of helium at normal atmospheric pressure of 105 Pa.
What will be the volume of the balloon at an altitude where air pressure is 8 × 104 Pa?
24. The cork in a pop-gun is fired when the pressure reaches 3 atmospheres. If the plunger is
60 cm from the cork when the air in the barrel is at atmospheric pressure, how far will
the plunger have to move before the cork pops out?
60 cm
25. A swimmer underwater uses a cylinder of compressed air which holds 15 litres of air at a
pressure of 12000 kPa.
(a) Calculate the volume this air would occupy at a depth where the pressure is
200 kPa.
(b) If the swimmer breathes 25 litres of air each minute at this pressure, calculate how
long the swimmer could remain at this depth (assume all the air from the cylinder
is available).
Pressure and temperature (constant volume)
26. Convert the following celsius temperatures to kelvin.
a) -273 °C b) -150 °C c) 0 °C
d) 27 °C
e) 150 °C
27. Convert the following kelvin temperatures to celsius.
a) 10 K
b) 23 K
c) 100 K
d) 350 K
e) 373 K
28. A cylinder of oxygen at 27 °C has a pressure of 3 × 106 Pa. What will be the new pressure
if the gas is cooled to 0 °C?
29. An electric light bulb is designed so that the pressure of the inert gas inside it is
100 kPa (normal air pressure) when the temperature of the bulb is 350 °C. At what
pressure must the bulb be filled if this is done at 15 °C?
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Mechanics and Properties of Matter
Gas Laws
30. The pressure in a car tyre is 2.5 × 105 Pa at 27 °C. After a long journey the pressure has
risen to 3.0 × 105 Pa. Assuming the volume has not changed, what is the new
temperature of the tyre?
31. A compressed air tank which at room temperature of 27 °C normally contains air at 4
atmospheres is fitted with a safety valve which operates at 10 atmospheres.
During a fire the safety valve was released. Estimate the average temperature of the
air in the tank when this happened.
32.
(a) Describe an experiment to find the relationship between the pressure and
temperature of a fixed mass of gas at constant volume. Your answer should
include:
(i) a labelled diagram of the apparatus
(ii) a description of how you would use the apparatus
(iii) the measurements you would take.
(b) Use the following results to plot a graph of pressure against temperature in °C
using axes as shown below.
Pressure / kPa
91
98
104
110
117
Temperature / ° C
0
20
40
60
80
-300
p / kPa
0
1
0
0 T/°C
(i) Explain why the graph you have drawn shows that pressure does not vary
directly as celsius temperature.
(ii) Explain how the graph can be used to show direct variation between
temperature and pressure if a new temperature scale is introduced.
(iii) Use the graph to estimate the value in °C of the zero on this new temperature
scale.
(c) Use the particle model of a gas to explain the following:
(i) why the pressure of a fixed volume of gas decreases as its temperature
decreases
(ii) why the pressure of a gas at a fixed temperature decreases if its volume
increases
(iii) what happens to the molecules of a gas when Absolute Zero is reached.
Volume and temperature (constant pressure)
33.
Describe an experiment to find the relationship between the volume and temperature
of a fixed mass of gas at constant pressure. Your description should include:
(a) a diagram of the apparatus used
(b) a note of the results taken
(c) an appropriate method to find the relationship using the results.
34.
100 cm3 of a fixed mass of air is at a temperature of 0 °C. At what temperature will
the volume be 110 cm3 if its pressure remains constant?
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Mechanics and Properties of Matter
Gas Laws
35.
Air is trapped in a glass capillary tube by a bead of mercury. The volume of air is
found to be 0.10 cm3 at a temperature of 27 °C. Calculate the volume of air at a
temperature of 87 °C.
36.
The volume of a fixed mass of gas at constant temperature is found to be 50 cm3.
The pressure remains constant and the temperature doubles from 20 °C to 40 °C.
Explain why the new volume of gas is not 100 cm3.
General gas equation
37.
Given, for a fixed mass of gas, P ∝ T and P ∝ 1/V, derive the general gas equation.
38.
Find the unknown quantity from the readings shown below for a fixed mass of gas.
(a) P1 = 2 × 105 Pa
V1 = 50 cm3
T1 = 20 °C
5
P2 = 3 × 10 Pa
V2 = ?
T2 = 80 °C
5
3
(b) P1 = 1 × 10 Pa
V1 = 75 cm
T1 = 20 °C
P2 = 2.5 × 105 Pa
V2 = 100 cm3
T2 = ?
5
3
(c) P1 = 2 × 10 Pa
V1 = 60 cm
T1 = 20 °C
P2 = ?
V2 = 80 cm3
T2 = 150 °C
5
3
d)
P1 = 1 × 10 Pa
V1 = 75 cm
T1 = ?
P2 = 2.5 × 105 Pa
V2 = 50 cm3
T2 = 40 °C
39.
A sealed syringe contains 100 cm3 of air at atmospheric pressure (105 Pa) and a
temperature of 27 °C. When the piston is depressed the volume of air is reduced to 20
cm3 and this produces a temperature rise of 4 °C. Calculate the new pressure of the
gas.
40.
Calculate the effect the following changes have on the pressure of a fixed mass of gas.
(a) Its temperature (in K) doubles and volume halves.
(b) Its temperature (in K) halves and volume halves.
(c) Its temperature (in K) trebles and volume quarters.
41.
Calculate the effect the following changes have on the volume of a fixed mass of gas.
(a) Its temperature (in K) doubles and pressure halves.
(b) Its temperature (in K) halves and pressure halves.
(c) Its temperature (in K) trebles and pressure quarters.
42.
Explain the pressure-volume, pressure-temperature and volume-temperature laws
qualitatively in terms of the kinetic model.
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Mechanics and Properties of Matter