CLASS 10. TEMPERATURE AND ATOMS
10.1. INTRODUCTION
Boyle’s understanding of the pressure-volume relationship for gases occurred in the late 1600’s.
The relationships between volume and temperature, and between pressure and temperature were not
determined until the 1800’s, primarily because scientists needed to learn how to separate different
gases.
10.2. GOALS
• Be able to convert temperatures between Celsius, Kelvin and Fahrenheit scales;
• Understand the physical meaning of the Kelvin temperature scale;
• Be able to explain and use Charles and Gay-Lussac’s Laws; and
• Understand how temperature relates to the motion and kinetic energy of atoms.
10.3. A MACROSCOPIC VIEW OF TEMPERATURE
10.3.1. Definition and Units. The symbol T represents temperature. There are three units for
temperature. The British unit is included because it is used so commonly in this country.
• degrees Fahrenheit (°F)
• degrees Celsius (°C)
• degrees Kelvin (K)
A degree sign is used for Celsius and Fahrenheit scales, but not for the Kelvin scale. Most of the
world uses the Celsius scale, with the exception of a few third-world countries and the US, which
use the Fahrenheit scale. The Kelvin scale is used primarily by scientists. Figure 10.1 compares the
temperature scales. Standard temperature is defined as zero °C or 273 K. Converting between the
different systems requires the following equations:
To convert from Celsius to Fahrenheit:
° F = ( 95 × °C ) + 32
(10.3.1)
To convert from Fahrenheit to Celsius:
°C =
( °F − 32 )
9
5
(10.3.2)
To convert from Celsius to Kelvin:
K = °C + 273
(10.3.3)
°C = K − 273
(10.3.4)
To convert from Kelvin to Celsius:
Celcius
Fahrenheit
Kelvin
100 °C
212 °F
373 K
20 °C
70 °F
293 K
0 °C
32 °F
273 K
-273 °C
-459 °F
0K
Figure 10.1: Comparison of the Fahrenheit, Celsius and Kelvin Scales
10.4. CHARLES’ LAW (OR GAY-LUSSAC’S LAW?)
There was significant disagreement about how the volume of a gas changed when the temperature
changed because different experiments often produced different results. This likely is because
different types of gases had different amounts of water vapor in them and the water vapor would
have a significant impact on the temperature-dependence of the gas.
10.4.1. Gay-Lussac’s Law. Joseph Gay-Lussac (1778 – 1850) was a careful experimenter and was
able to exclude most of the water vapor from his apparatus. His results thus were more accurate than
previous experiments. In 1802, he showed that
V
= constant
T
(10.4.1)a
Like Boyle’s Law, this relationship requires some qualification: the volume-temperature law holds
only when the pressure is constant and it works only for a subset of gases called ideal gases. We
will study ideal gases at the end of this chapter; however, the important thing to know right now is
that real gases behave like ideal gases at low pressures. We can write the volume-temperature law as
a comparison of the volume and the temperature at two different times:
V1 V2
=
T1 T2
(10.4.1)b
Important: These equations only work if you use temperatures in Kelvin units. If you use
Celsius units, you WILL get the wrong answer.
10.4.2. Charles’ Contribution. Gay-Lussac was not the first to discover this law. Jacques
Alexandre César Charles (1746-1823) discovered this volume-temperature relationship in 1787
(fifteen years earlier), but did not published it. Charles, who became interested in science when he
met the American ambassador to France, Benjamin Franklin, found that oxygen, nitrogen, hydrogen,
carbon dioxide, and air expand the same amount over the same 80-degree interval. Gay-Lussac did
Volume
improve on Charles’ work and was the first to
publish.
Charles did not measure the
Gas 2
coefficient of expansion, and the presence of
water in the apparatus and the gases
themselves caused Charles to obtain results
Gas 1
indicating unequal expansion for watersoluble gases. Gay-Lussac did cite Charles’
earlier results, and this has resulted in this
law being called either Gay-Lussac’s Law or
Charles’ Law.
10.4.3. The Meaning of the Kelvin Scale.
When Boyle found the relationship between
-273 °C
pressure and volume, he had only air with
Temperature
which to experiment. Scientists in the early
1800s had the benefit of being able to study Figure 10.2: The relationship between volume and
different gases. They thus could compare temperature. The dashed line is an extrapolation of
and contrast the behavior of different gases. the solid line and starts at the temperature where
A plot of the volume as a function of the gas changes into a liquid.
temperature (Figure 10.2) shows that (until
the gas gets so cold that it becomes a liquid) the plot is a straight line. If you extrapolate the line
back to zero volume, the lines for many of the gases extrapolate back to the same intercept with the
temperature axis.
This special temperature at which the volumes go to zero is -273°C, which is 0 K. This is the
reason the Kelvin temperature scale is special – it is the only one of the three temperature scales not
based on arbitrary standards such as water boiling or freezing. 0 K corresponds to the lowest
possible temperature possible.
10.5. SOLVING PROBLEMS INVOLVING TEMPERATURE AND VOLUME
Consider a balloon filled with air. At room temperature, I have one volume. If I put the balloon in
the freezer for a little while, the balloon will shrink. If I heat the balloon, it will grow larger. We
can use the volume-temperature relationship to calculate how much the volume will change when we
cool a gas.
EXAMPLE 10.1: If the temperature of a balloon filled with air doubles, what happens to the volume?
Draw a picture
V2
V1
T1 < T2
known:
T1 = temperature before
V1 = volume before
T2 = temperature after
T2 = 2T1
need to find:
T2
V2= volume after
V1 V2
=
T1 T2
Equation to use:
T2
V1
T1
Solve for the unknown
V2 =
Plug in numbers.
V2 = 2V1
Answer:
The volume will also double.
EXAMPLE 10.2: A volume of 26.0 L of gas is at a temperature of 30.0°C. If the temperature decreases to
10.0°C, what is the volume of the gas?
Draw a picture
V1 = 26.0 L
V2
T1 = 30 °C
known:
T1 = temperature before = 30.0°C
V1 = volume before = 26.0 L
T2 = temperature after = 10.0°C
need to find:
T1 = 30.0 + 273 = 303.0 K
T2 = 10.0 + 273 = 283.0 K
V2 =
Solve for the unknown
Plug in numbers.
V2= volume after
V1 V2
=
T1 T2
Equation to use:
Important: you have to convert all of
the temperatures into Kelvin!
T2 = 10 °C
T2
V1
T1
303.0 K
283.0 K
= 27.837455830 L
V2 = ( 26.0 L )
V2 = 27.8 L
Answer: (3 s.f.)
Check that this makes sense. If the temperature increases, we know that the volume should increase. If I
had gotten a smaller volume, I would know I had done something wrong.
10.6. GAY-LUSSAC’S LAW
The origin of this law is not known exactly, but Gay-Lussac found the following to hold true:
P
= constant
T
(10.6.1)a
This can be written the same way the pressure-volume relationship was written:
P1 P2
=
T1 T2
(10.6.1)b
Again, all temperature s must be in Kelvin, and this is valid only when the volume is held constant
and only for ideal gases.
EXAMPLE 10.3: A gas at a pressure of 1.30 atm is at a temperature of 30.0°C. If the pressure is
increased to 2.50 atm, what is the temperature of the gas? Assume that the volume remains
constant
P1, = 1.30 atm
T1 = 30.0 °C
P2, = 2.50 atm
T2 = ?
Draw a picture
T1 = temperature before = 30.0°C
known:
P1 = pressure before = 1.30 atm
P2 = pressure after = 2.50 atm
T2 = temperature after
need to find:
P1 P2
=
T1 T2
Equation to use:
Remember:
you have to convert all of the
temperatures into Kelvin!
Solve for the unknown
T1 = 30.0 + 273 = 303 K
T2 =
P2
T1
P1
Plug in numbers.
⎛ 2.50 atm ⎞
T2 = ⎜
⎟ 303K
⎝ 1.30 atm ⎠
= 582.692307692 K
Answer: (3 s.f.)
T2 = 583 K
Check that this makes sense. If the pressure increases, I expect the temperature to increase, so this answer
makes sense.
10.7. A MICROSCOPIC THEORY OF TEMPERATURE
Remember that Bernoulli’s kinetic theory made specific predictions for the temperature, pressure
and volume dependence of gases. The third assumption of Bernoulli’s theory, which we delayed
discussing until now, was that the temperature of a gas was proportional to the speed at which the
gas particle moved.
10.7.1. Dependence of Pressure on Temperature. This model can account for why pressure
increases when temperature increases: As temperature increases, the speeds of the gas particles
increase. The gas particles therefore collide more frequently with the wall and exert greater force on
the vessel wall.
10.7.2. Dependence of Volume on Temperature. This model can account for why volume
increases when temperature increases: As temperature increases, the speeds of the gas particles
increase. If the walls of the vessel holding the gas can move, the more frequent collisions will cause
the walls of the vessel to move further outward.
10.7.3.Websites. A couple nice websites that show animations of the molecular properties of ideal
gases are:
• http://www.phy.ntnu.edu.tw/java/idealGas/idealGas.html
• http://www.falstad.com/gas/
• http://streaming.lbcc.cc.ca.us/chemistry/chm3d03s.mov
10.7.4. Relationship between Average Kinetic Energy and Average Speed. In Bernoulli’s theory,
the average kinetic energy KE of a gas is given by
KE = 12 Nm ( vav )
2
(10.7.1)
where N is the number of gas particles, each gas particle has a mass m, and vavg is the average speed
of the gas particles. Kinetic energy is energy of motion. The kinetic energy is larger when the gas
particles are moving faster. The units of kinetic energy are joules (J), where
kg m 2
1 J = 1 2 = 1 Nm
s
Bernoulli’s theory predicts that the average kinetic energy of a gas also should be proportional to
the temperature.
KE = 32 NkT
(10.7.2)
where N again is the number of gas particles, and k is Boltzmann’s constant, which has a value of
1.38x10-23 KJ The temperature must be in Kelvin for this equation to work.
EXAMPLE 10.4: A gas at a temperature of 20.0°C contains 6.02×1023 gas particles. What is its average
kinetic energy?
Draw a picture
known:
No picture is necessary
T = temperature = 20.0°C
N = number of atoms =6.02×1023
k = 1.38×10−23
J
K
Equation to use:
Remember: you have to convert the
temperature into Kelvin!
need to find:
KE
energy.
KE = 32 NkT
T = 20.0 + 273
= 293 K
= average kinetic
KE =
Plug in numbers.
3
2
( 6.022 ×10 )(1.38 ×10
23
−23 J
K
) ( 293 K )
= 3652.40322 J
KE = 3.65 × 103 J
Answer: (3 s.f.)
10.7.5. Relationship between average speed and temperature. We can combine Equations (10.7.2)
and (10.7.1):
1
2
m ( vavg ) = 32 kT
2
(v )
avg
vavg =
2
=
3kT
m
3kT
m
(10.7.3)
Equation (10.7.3) emphasizes the correlation between the speed of the gas particles and the
temperature. As the temperature decreases, the average speed of the gas particles also decreases. At
the time, scientists didn’t know the mass of the individual gas particles; however, that information
was about to be determined.
EXAMPLE 10.5: What is the average speed of a hydrogen molecule at 293 K? (The mass of a hydrogen
molecule is 3.34 × 10-27 kg).
Draw a picture
known:
T1 = temperature = 293 K
m = 3.34×10−27 kg
Equation to use:
Plug in numbers.
No picture necessary
vavg = average speed
need to find:
vavg =
3kT
m
vavg =
3kT
m
=
3 (1.38 × 10−23
J
K
) 293K
3.34 × 10−27 kg
= 1905.7272646
J
kg
J
kg m 2
=
kg
kg s 2
The units aren’t obvious, so we need to show that
they work out to a unit appropriate for velocity.
m2
s2
m
=
s
=
Answer: (3 s.f.)
vavg = 1.90 × 103
m
s
10.8. SUMMARIZE
10.8.1. Definitions: Define the following in your own words. Write the symbol used to represent
the quantity where appropriate.
1. Kinetic Energy
2.
Standard temperature
10.8.2. Equations: For each question: a) Write the equation that relates to the quantity b) Define
each variable by stating what the variable stands for and the units in which it should be expressed,
and c) State whether there are any limitations on using the equation.
1. The equations that allow you to convert between Fahrenheit and Celsius temperatures
2.
The equations that allow you to convert between Kelvin and Celsius temperatures
3.
The relationship between the volume and the temperature of a gas.
4.
The relationship between the pressure and the temperature of a gas.
5.
The relationship between the average kinetic energy and the average speed of gas particles
according to kinetic theory.
6.
The relationship between average kinetic energy and temperature.
7.
The relationship between the average speed of gas particles and their temperature.
10.8.3. Concepts: Answer the following briefly in your own words.
1. Compare the macroscopic and microscopic views of temperature. How does temperature relate
to molecular motion?
2.
What is the physical significance of the Kelvin temperature scale?
3.
Explain how the Kelvin temperature scale relates to the volume vs. temperature relationship.
4.
When using either the pressure-temperature or the volume-temperature relationships, you
always have to remember one thing about making calculations. What is it?
5.
Can you ever have a negative temperature on the Kelvin scale? Why not?
10.8.4. Your Understanding
1. What are the three most important points in this chapter?
2.
Write three questions you have about the material in this chapter.
10.8.5. Questions to Think About
1. When we studied Boyle’s Law, we had to plot P vs. 1/V to get a linear relationship. Why can
we plot V vs. T and have a linear relationship?
2. Why did we have to extrapolate the volume vs. temperature graphs back to zero volume?
3. Why does water boil at lower temperatures at high altitudes?
4. Is it meaningful to say that an object at a temperature of 200°C is twice as hot as one at 100°C?
5. In the table of densities in class 4, the densities for the gases are specified at a particular
temperature. Why is this done? Would the density of a gas change with temperature?
6. You have a Fahrenheit thermometer and a Celsius thermometer in the same room. They show
the same temperature. What is the temperature of the room?
10.8.6. Problems
1. Convert: 261 K to °C; -50 °C to K; 70 °F to °C; 0 °F to °C.
2. The record low temperature in Lincoln is -33 °F. What is this temperature in °C and in K?
3. The record high temperature in Lincoln is 115 °F. What is this temperature in °C and in K?
4. What is normal body temperature in °C?
5. A gas is held at a pressure of 4.58 atm at a temperature of 25.6°C. If the temperature is
changed to -46.0°C, what is the new pressure?
6. A gas of volume 545 cm3 has a temperature of 34.0 °C. If the volume is reduced to 305 cm3,
what is the new temperature?
7. What is the average speed of gas particles at room temperature, assuming that the gas particles
have a mass of 3.321×10-27 kg?
8. A gas has a temperature of 40°C. What is the average kinetic energy of one molecule of that
gas?
9. What is the kinetic energy of a H2 molecule at a temperature of 67°C?
10. At constant pressure, the volume of a gas sample is direction proportional to: a) the size of its
gas particles, b) its Fahrenheit temperature, c) its Celsius temperature, or d) its temperature in
Kelvin.
11. The volume of a gas sample is increased while its temperature is held constant. The gas exerts
a lower pressure on the walls of its container because its molecules strike the walls a) less
often, b) with lower speed, c) with less energy and d) with less force.
12. 1250 cm3 of hydrogen is at 0°C and ordinary atmospheric pressure.
a) If the pressure is tripled while the temperature is held constant, what is the volume of the
gas?
b) If the temperature then increases to 273°C while the pressure is held constant, what will the
volume be?
13. An air tank used for scuba diving has a pressure valve that is set to open if the pressure inside
the tank reaches 28 MPa. The normal pressure in the tank when full at 20°C is 20 MPa. If the
tank is heated after being filled to 20 MPa, at what temperature will the safety valve open?
14. To what temperature in Celsius must a gas sample initially at 20°C be heated if its volume is to
double while its pressure remains the same?
15. To what temperature must a gas sample initially at 27°C be raised to double the average kinetic
energy of its molecules?
16. The average speed of a hydrogen molecule at 20°C is about 1.6 kms . A carbon dioxide molecule
has about 22 times the mass of a hydrogen molecule. What is the average speed of a carbon
dioxide molecule at 20°C?
PHYS 261 Spring 2007
HW 11
HW Covers Class 10 and is due February 2nd, 2007
1.
2.
(2 pts) 1.250×103 cm3 of hydrogen gas is at 0°C and 1.00 atm pressure.
a) If the pressure is tripled while the temperature is held constant, what is the volume of the
gas?
b) After the change in part a), the temperature increases to 273°C while the pressure is held
constant. What is the new volume?
A particular gas at a temperature T has an average speed given by Equation (10.7.3). By how
much does the temperature have to change if the average speed is to double?