Principles of Technology
Ch 8 Internal Energy & Properties of Matter 2
Name_______
Key Objectives:
At the conclusion of this chapter you’ll be able to:
• State the equations for Boyle’ and Charles’s laws, and solve problems using them.
• Interpret graphs that illustrate Boyle’s and Charles’s laws.
• State the equation for the combined gas law, and solve problems using it.
• State the relationship for the ideal gas law.
• Define the term ideal gas, and list the properties of an ideal gas according to the
kinetic-molecular theory (KMT) of gas behavior.
• Define pressure, volume, and temperature as they are explained by the KMT.
• State the conditions under which real gases exhibit ideal behavior.
8.4 THE GAS PHASE
Matter commonly exists in the solid, liquid, or gas phase. The phase of a substance is
usually recognized by the characteristics of the substance’s shape and volume. What
phase a particular sample of matter is in depends on the nature of the sample, its
temperature, and the pressure exerted on it.
Gases have neither definite shape nor definite volume. In our study of gases, we will
assume that gases behave ideally. (On page 160 we will indicate exactly what we mean by
the term ideal behavior.) Although no gas is truly ideal, many samples of gases exhibit
ideal behavior under appropriate conditions. We can make this assumption because the
“laws” that govern ideal gas behavior are very simple.
To describe the behavior of a sample of an ideal gas, we need to know four
characteristics of the sample: the pressure it exerts, its volume, its temperature, and
the number of particles (i.e., the mass) it contains.
We have already learned how volume and temperature are measured. The number of
particles contained in substances of any type (including a gas) is measured in terms of a
unit known as the mole (mol). One mole stands for a specific number (Avogadro’s number
= 6.02 x 1023), just as the term dozen stands for the specific number 12. For the
present, however, we will use mass to describe gas behavior since it is directly related to
the number of particles present.
Pressure is defined as the force per unit area of surface:
P= F/A
Pressure gives us a means of describing how a force is distributed over an entire
surface. In the SI metric system, the unit of pressure is the newton per square meter
(N/m also known as the pascal (Pa). The pascal is a very small unit, and under normal
atmospheric conditions air exerts a pressure of approximately 1.01 x l05 pascals. This
value is known as standard pressure.
ASSESSMENT QUESTION 1
All of the following are true EXCEPT:
A. What phase a particular sample of matter is in depends on the nature of the
sample, its temperature, and the pressure exerted on it.
B. To describe the behavior of a sample of an ideal gas, we need to know four
characteristics of the sample: the pressure it exerts, its volume, its temperature,
and the number of particles (i.e., the mass) it contains.
C. Pressure gives us a means of describing how a force is distributed over an entire
surface.
D. The number of particles contained in substances of any type (including a gas) is
measured in terms of a unit known as the mole (mol). One mole stands for a dozen
particles.
BOYLE’S LAW
Boyle’ law states the relationship between
the volume of a gas and its pressure (at
constant temperature and mass). The table
below gives the measurements
obtained in one volume-pressure experiment
carried out on an ideal gas at constant
temperature:
According to the first four pairs of data
given in the table, the volume varies
inversely with the pressure at constant
temperature, assuming that the mass is also
constant. In other words, an increase in pressure is accompanied by a decrease in volume.
Moreover, the product of pressure and volume is constant under the condition known as
Boyle’s Law.
Pressure∙Volume = constant (at contant temperature and mass)
In symbolic form: PV= k
If we graph the data given in the table, we produce a curve (shown below known as a
rectangular hyperbola. This curve is characteristic of inverse relationships. Each point on
the graph obeys the Boyle’s law equation:
PV = constant.
The alternative form for expressing Boyle law:
P1V1 = P2V2
It is very useful for solving mathematical problems involving pressure and volume at
constant temperature and mass.
PROBLEM
Calculate the volume of the gas in the table on the graph when the pressure is 20,000
pascals.
SOLUTION
Boyle law indicates that we can use any other set of values in the table to solve this
problem. We will use the third set. Then:
P1V1 = P2V2
(4000 Pa)(0.02 m3)= (20,000 Pa) V2
ASSESSMENT QUESTION 2
Calculate the volume (V2) of the gas with an initial pressure of 500 Pa (P1) and 0.05 m3
(V1) when the pressure is 3,000 Pa (P2).
P1V1 = P2V2
(500 Pa)(0.05 m3) = (3,000 Pa) V2
V2 = (500 Pa)(0.05 m3) / (3,000 Pa)
A. 0.0083 m3
B. 0.27 m3
C. 99 m3
D. 750 m3
Charles’s Law
To examine the relationship between the (Kelvin) temperature
and the volume of an ideally behaving gas, we must hold the
pressure and the mass constant. The following table provides
data for such an investigation.
From data in the table, we can draw the following conclusion:
The volume of the gas at constant pressure and mass is directly
proportional to the Kelvin temperature. The graph illustrates this relationship.
The mathematical form of this relationship is as
follows:
Volume / Temperature = constant
V/Tk = k
We can write this relationship, known as Charles’s
law, in its equivalent form:
V1 / T1 = V2 / T2
PROBLEM
The volume of an ideally behaving gas is 0.030 meter3
(V1) at 500. K (T1).
What volume (V2) will the gas occupy at 200 K (T2), pressure and mass remaining
constant?
SOLUTION
V1 / T1 = V2 / T2
0.030 meter3 / 500. K = V2 / 200 K
V2 = 0.012 meter3
ASSESSMENT QUESTION 3
The volume of an ideally behaving gas is 0.70 meter3 (V1) at 600. K (T1).
What volume (V2) will the gas occupy at 400 K (T2), pressure and mass remaining
constant?
V1 / T1 = V2 / T2
0.70 meter3 / 600. K = V2 / 400 K
V2 = (0.70 meter3 / 600 K)∙400 K
A. 0.47 m3
B. 1.05 m3
C. 190 m3
D. 1680 m3
Gay Lussac's Law
Gay-Lussac's Law states that the pressure of a sample of gas at constant volume, is
directly proportional to its temperature in Kelvin.
According to Gay-Lussac's Law:
P1 / T1 = constant
After the change in pressure and temperature,
P2 / T2 = constant
Combine the two equations:
P1 / T1 = P2 / T2
PROBLEM
The pressure of an ideally behaving gas is 2500 Pa (P1) at 500. K (T1).
What pressure (V2) will the gas exert at 200 K (T2), temperature and mass remaining
constant?
SOLUTION
P1 / T1 = P2 / T2
2500 / 500. K = P2 / 200 K
P2 = 1000 Pa
ASSESSMENT QUESTION 4
The pressure of an ideally behaving gas is 500 Pa (P1) at 450. K (T1).
What pressure (V2) will the gas exert at 210 K (T2), temperature and mass remaining
constant?
P1 / T1 = P2 / T2
500 Pa / 450 K = P2 / 210 K
P2 = (500 Pa / 450 K)∙210 K =
A. 87 Pa
B. 160 Pa
C. 230 Pa
D. 350 Pa
ASSESSMENT QUESTION 5
All of the following are true EXCEPT:
A. According to Boyle’ law (at constant temperature and mass) an increase in
pressure is accompanied by a decrease in volume of a gas.
B. According to Charles’ law the volume of the gas at constant pressure and mass is
directly proportional to the Kelvin temperature.
C. Gay-Lussac's law states that the pressure of a sample of gas at constant volume,
is directly proportional to its temperature in Kelvin.
D. According to Charles’ law and Gay-Lussac's law if we increase the temperature of
a gas we will decrease the pressure and the volume of a gas if mass remains
constant.
Combined Gas Law
It is possible to combine Boyle’s, Charles’s and Gay-Lussac's laws into a single
relationship that holds as long as the mass is constant:
(Pressure∙Volume) / Temperature (K) = constant
In symbolic form: (PV) / T = k
We can write this relationship, known as the combined gas law, in its equivalent
form:
(P1V1) / T1 = (P2V2) / T2
PROBLEM
A gas occupies 0.6 cubic meter (V1) at 5 x 104 pascals (P1) and 400 K (T1).
What volume (V2) does it occupy at 4 x 104 pascals (P2) and 200 K (T2), mass remaining
constant?
SOLUTION
(P1V1) / T1 = (P2V2) / T2
(5 x 104 pascals ∙ 0.6 cubic meter) / 400 K = (4 x 104 pascals∙V2) / 200 K
V2 = 0.375 m3 (0.4 m3 to one significant digit)
ASSESSMENT QUESTION 6
A gas occupies 0.9 cubic meter (V1) at 2 x 104 pascals (P1) and 600 K (T1).
What volume (V2) does it occupy at 8 x 104 pascals (P2) and 300 K (T2), mass remaining
constant?
(P1V1) / T1 = (P2V2) / T2
(2 x 104 pascals∙0.9 cubic meter) / 600 K = (8 x 104 pascals∙V2) / 300 K
V2 = (2 x 104 pascals∙0.9 cubic meter∙300 K) / (600 K∙8 x 104 pascals) =
A. 0.08 m3
B. 0.1 m3
C. 5 m3
D. 20 m3
Ideal Gas Law
We now turn our attention to how the combined gas law relates to the number of gas
particles. Since the mass of a substance is directly related to the number of molecules
present, we can reformulate the combined gas law to include the number of particles. We
need only realize that adding more gas molecules at constant temperature and pressure
will increase the volume proportionally a fact known as Avogadro law. The combined law
now takes this form:
(Pressure∙Volume) / (Number of Particles∙Temperature (K)) = constant
In symbolic form: (PV) / nT = k
The number of particles (n) is measured in mole., The constant (R) is known as the
universal gas constant. In the SI metric system, its value is 8.315 joules per mole per
kelvin (J/mol K). (You should convince yourself, as an exercise, that the units of pressure
times volume are equivalent to a joule.)
The ideal gas law is usually written in its equivalent form:
PV= nRT
ASSESSMENT QUESTION 7
All of the following are true EXCEPT:
A. It is impossible to combine Boyle’s, Charles’s and Gay-Lussac's laws into a single
relationship that includes the number of particles.
B. We need only realize that adding more gas molecules at constant temperature and
pressure will increase the volume proportionally a fact known as Avogadro law.
C. (R) is known as the universal gas constant. In the SI metric system, its value is
8.315 joules per mole per kelvin (J/mol K).
D. Units of pressure times volume are equivalent to a joule.
PROBLEM
A gas occupies 0.6 cubic meter (V) at 5 x 104 pascals (P) and 400 K (T).
What is the quantity of gas, in moles (n)?
R = 8.315 J/mol K
SOLUTION
PV= nRT
n = PV / RT
n = (0.6 cubic meter∙5 x 104 pascals) / (8.315 J/mol K∙400 K)
n = 9.2 moles
ASSESSMENT QUESTION 8
A gas occupies 0.2 cubic meter (V) at 3 x 104 pascals (P) and 800 K (T).
What is the quantity of gas, in moles (n)?
R = 8.315 J/mol K
PV= nRT
n = PV / RT
n = (0.2 cubic meter∙ 3 x 104 pascals) / (8.315 J/mol K∙800 K) =
A. 0.06 moles
B. 0.9 moles
C. 2 moles
D. 10 moles
The Kinetic Theory (KMT) of Gas Behavior
So far, we have described gas behavior in terms of experimental laws, that is, laws that
have been investigated in the laboratory We have assumed that all of the gases behaved
ideally, but we never actually defined the term ideal behavior.
Now we take a different approach: We propose a model of an ideal gas, and we examine
the model according to the basic laws of physics. An ideal gas is a collection of particles
that:
• have mass but negligible volume;
• move randomly in straight lines;
• are not subject to any attractive or repulsive forces (except during collisions with each
other or with the walls of their container);
• collide in a perfectly elastic fashion, that is, both linear momentum and kinetic energy
are conserved during collisions.
From the KMT model of gas behavior, the following facts are obtained:
• The volume occupied by an ideal gas is essentially the volume of its container.
• The pressure exerted by an ideal gas is related to the number of collisions that the
particles make with the walls of the container in a given amount of time.
• The absolute (Kelvin) temperature of a gas is directly proportional to the average
kinetic energy of the gas particles. (In other words, if the absolute temperature is
doubled, the average kinetic energy of the particles will also be doubled.)
ASSESSMENT QUESTION 9
All of the following are true EXCEPT:
A. An ideal gas is a collection of particles that move randomly in straight lines and
are not subject to any attractive or repulsive forces (except during collisions with
each other or with the walls of their container).
B. An ideal gas is a collection of particles that have mass but negligible volume and
collide in a perfectly elastic fashion, that is, both linear momentum and kinetic
energy are conserved during collisions.
C. The volume occupied by an ideal gas is essentially the volume of its container and
the pressure exerted by an ideal gas is related to the number of collisions that
the particles make with the walls of the container in a given amount of time.
D. If the absolute temperature is tripled, the average kinetic energy of the
particles will be 1/3.
All of the experimental gas laws developed in this chapter can be derived mathematically
as a direct result of the KMT model. Therefore, we can conclude that the KMT is a wellconstructed model for explaining the behavior of ideal gases.
Real gases, such as oxygen, hydrogen, helium, and carbon dioxide, may or may not exhibit
ideal behavior, depending on the conditions of the environment. For a real gas to behave
ideally, its particles must be relatively far apart and must be moving at relatively high
speeds. Under these conditions, the particles will exert only negligible forces on one
another and the volume of the particles will be negligible in relation to the volume of the
container.
The behavior of real gases is most nearly ideal under conditions of high temperatures
and low pressures.
ASSESSMENT QUESTION 10
All of the following are true EXCEPT:
A. All of the experimental gas laws developed in this chapter can be derived
mathematically as a direct result of the KMT model. Therefore, we can conclude
that the KMT is a well-constructed model for explaining the behavior of ideal
gases.
B. Real gases, such as oxygen, hydrogen, helium, and carbon dioxide, never exhibit
ideal behavior, regardless of the conditions of the environment.
C. For a real gas to behave ideally, its particles must be relatively far apart and
must be moving at relatively high speeds. Under these conditions, the particles
will exert only negligible forces on one another and the volume of the particles
will be negligible in relation to the volume of the container.
D. The behavior of real gases is most nearly ideal under conditions of high
temperatures and low pressures.
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Conclusion
The gas phase is, in many respects, the simplest phase of matter.
Scientists such as Boyle and Charles developed experimental laws that demonstrate
gas behavior under different conditions.
These laws relate pressure, volume, and temperature.
The universal gas law com bines Boyle’s and Charles’s laws and relates these
quantities to the number of gas particles present.
Gas behavior can be explained by constructing a model of an ideal gas and applying
the laws of physics to the model.
This construction is known as the kinetic-molecular theory.
With this theory, the pres sure, temperature, and volume of a gas are explained in
molecular terms and the ideal gas law is derived from physical principles.