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1
Temperature
1.1 Introduction
Many of the concepts of thermodynamics are familiar from everyday experience but are difficult to define in a scientifically rigorous way. Foremost of these is temperature. Everyone uses the
concepts of hotness and coldness and understands the meaning
of the air temperature given on the radio; however, few people can
give a definition of exactly what is meant by “the temperature of
the water was 68°F.” Introductory physics text usually explain
the concept of temperature of an object in terms of the kinetic
energy of the molecules which make up the object. Although this
approach is useful in gaining an intuitive understanding, it does
not meet our needs for two reasons. First, it is a microscopic
definition and so does not fit with the aims of thermodynamics.
Second, it does not provide any method to experimentally measure temperature. Thus it is the goal of this chapter to present a
rigorous, macroscopic definition of temperature, as well as to
demonstrate ways to measure it.
2
TEMPERATURE
1.2 Thermal Equilibrium and the Concept
of Temperature
The everyday concept of temperature is based on the sensory
perception of hotness and coldness. One can compare the temperature of two objects by touch and thus determine which has a higher
temperature. From this experience, we expect to be able to rank
objects by temperature. Therefore, temperature must be a scalar
quantity and can be specified by a single value.
A second observation based on experience is that if two objects
with different temperatures, as determined by touch, are brought
into contact, they will both change temperature. Eventually the
objects will reach a common temperature, which is between the
original temperatures. Similarly, if two objects with the same
temperature are brought into contact, neither will change temperature. These two observations will serve as the basis of our
scientific definition of temperature, but first we need several
definitions.
thermal contact
diathermal wall
thermal equilibrium
We say two objects are in thermal contact if they are separated
only by a wall which is a good conductor of heat, such as a thin
sheet of copper. Such a wall, called a diathermal wall in thermodynamics, prevents particles from passing through but allows
some energy to pass. When two objects are first brought into
thermal contact, they may exchange energy and so change temperature. Eventually the objects will stop changing. We can then
say they are in thermal equilibrium. Thus, we define thermal
equilibrium as follows:
Two objects are in thermal equilibrium if they do not
change when separated by a diathermal wall.
temperature
The concept of thermal equilibrium is used to define the concept
of temperature as follows:
The temperature is that property of an object that determines whether it is in thermal equilibrium with other
objects.
THERMAL E QUILIBRIUM AND
THE
C ONCEPT
OF
T EMPERATURE
For temperature to be a useful concept, we need the following
postulate:
Two objects in thermal equilibrium with a third object
will be in thermal equilibrium with each other.
The need for this postulate was recognized by R. H. Fowler in
1939 and we shall follow him in calling it the zeroth law of
thermodynamics. The first and second laws were already named
by this time.
3
This postulate seems selfevident and was taken for
granted by physicists
throughout the
development of
thermodynamics. We
cannot expect, however,
that relationships are
always transitive.
1.3 Thermometry
The above definition of temperature meets our need for a rigorous
physical definition but does not provide a way to measure and
quantify temperature. We need a way to measure temperature
which is based on an experimental definition which is reproducible. Such a definition is called an empirical temperature scale.
To define an empirical temperature scale, one must choose a
system to act as a thermometer and at least two reference
temperatures. Some measurable property of the thermometer,
called the thermometric property, must change with changing
temperature. All other properties of the system which affect this
thermometric property must be held constant. Several examples
of thermometers are considered below.
•Common Thermometers
The thermometers with which the student is most familiar are
probably liquid expansion thermometers. The basic design of this
type of thermometer, shown in figure 1.1, dates back to 1650,
when it was known as a Florentine thermometer. An arbitrary
scale was engraved on the sealed tube. Later, two reference
temperatures were chosen and the separation between the two
was divided into equal parts or degrees. As we will see later, many
different choices for reference temperatures and number of degrees were used.
empirical temperature
scale
thermometric property
4
TEMPERATURE
Figure 1.1: Florentine Thermometer.
sealed tube with
arbitrary scale
glass bulb filled with
liquid
The bimetallic strip thermometer is a second common type of
thermometer with which the student is probably familiar. This
thermometer, as shown in figure 1.2, consists of a coiled strip
consisting of two metals. The metals used have very different
coefficients of thermal expansion. As a result, a change in
temperature will cause the coil to expand or contract, moving a
needle on a calibrated dial.
Figure 1.2: Bimetallic Thermometer.
THERMOMETRY
5
•Precision Thermometers
Some of the systems and thermometric properties commonly used
to make precision measurements of temperature are summarized
in table 1.1. The most important of these, the gas thermometer,
is discussed in section 1.4. The physical theory needed for the
optical pyrometer and paramagnetic thermometer will be discussed in later chapters. Two of the remaining thermometers
merit special discussion.
First is the thermocouple, which consists of two wires made of
dissimilar metals in contact with each other. A small thermal
electric potential is produced at the junction. This emf is temperature-dependent and, therefore, can be used as a thermometric
property. For a more accurate temperature measurement, the
difference in emf from a thermocouple at the test temperature and
a second at a reference temperature, often 0°C, can be measured
by a microvoltmeter or a potentiometer. This difference in emf, φ,
is generally between 5 and 50µV/K and does not vary linearly with
the temperature. The thermocouple must be calibrated at a
number of known temperatures. The variation of φ as a function
of the Celsius temperature can usually be fit to a cubic equation,
φ = a + bt + ct 2 + dt 3 ,
(1.1)
to within the desired accuracy over a large range of temperatures.
As an example, a commonly used thermocouple is constructed
from platinum and 90% platinum, 10% rhodium alloy. It can be
fit to a cubic function to an accuracy of ±0.1K over a range of 400K.
Thermometer
Thermometric Property
Constant Volume Gas
pressure
Acoustic
speed of sound
Range of Use
2K < T < 30K
Thermal Noise
thermal noise voltage
0.01K < T < 1K
Paramagnetic
magnetic susceptibility
0.001K < T < 1K
Thermocouple
thermal emf
150K < T < 2000K
Electrical Resistance
electrical resistance
10K < T < 1000K
Optical Pyrometer
radiant emittance
T > 1000K
Table 1.1 : Some thermometers, thermometric properties, and typical
ranges of application.
This phenomenon was
discovered by the
German physicist
Thomas Johann
Seebeck (1770-1831) in
1821 and is called the
Seebeck effect.
6
TEMPERATURE
Thermocouple thermometers have a number of advantages which
makes their use widespread. The junctions and wires can be made
very small so that the heat capacity of the thermometer does not
significantly affect the experiment and the thermocouple can
reach the temperature of the sample very quickly. With modern
semiconductor technology, it is inexpensive to produce a combination microvoltmeter and dedicated microprocessor which can
measure φ, convert it to a temperature using a cubic or quadratic
equation of fit, and display the result. This type of thermometer
is now common not only in physics labs but also in homes.
The second type of precision thermometer we need to consider is
the electric resistance thermometer. The most important of these
uses the resistance of platinum as the thermometric property.
This resistance increases with increasing temperature and can be
very accurately measured. The most common experimental
approach is shown in figure 1.3. The current through the circuit
is held constant by varying a rheostat (variable resistor) while the
voltage drop across the resistor is measured using a potentiometer. Since the variation of resistance with temperature is not
linear, the thermometer is calibrated at several known temperatures. Thermometers using platinum or other metals can be very
accurate but have several inherent problems. First, since a
current is running through the resistor, there is heat being
dissipated into the sample. This can be significant problem if the
sample is very small. Second, the resistor requires very careful
handling since small physical changes in the resistor can affect
the reproducibility of the measurement.
resistor
in sample
battery
to
potentiometer
µA
rheostat
Figure 1.3: Basic circuit for electric resistance thermometry.
THERMOMETRY
7
Semiconductors may also be used in resistance thermometry. For
these materials, the resistance decreases approximately exponentially with increasing temperatures. Semiconductor thermometers are commonly used for precision measurements at very
low temperatures. They are also used at higher temperatures in
inexpensive digital thermometers and as thermostats.
•Temperature Scales
In addition to specifying the system and thermometric property
used as a thermometer, it is necessary to specify either two
reference temperatures or one reference temperature and the size
of the degree to completely define an empirical temperature scale.
Many scales have been used historically, but the most familiar are
the Fahrenheit and Celsius scales.
The Fahrenheit scale has a long history beginning with Isaac
Newton, who suggested using the melting point of ice as the lower
reference temperature, 0, and the temperature of a healthy male
as the higher reference temperature. He suggested dividing this
range into 12 parts. Fahrenheit made two improvements. First,
he was able to produce a lower temperature for his zero point by
using a mixture of salt and ice. Second, because of his skill in
making mercury-in-glass thermometers, he was able to measure
much smaller differences in temperatures. Following the usual
practice of the times, he divided his degrees into eighths. As part
of his investigations, he measured the melting point of ice as 32°F
and the boiling point of water as 212°F. Soon after his death, these
two temperatures were adopted as the fixed reference temperatures. The widespread use of this scale resulted primarily from
the high quality thermometers produced by Fahrenheit and his
successors.
Dividing the temperature interval between the freezing point and
boiling point of water into 100 degrees was proposed by several
scientists prior to 1740. The resulting scale, which also depends
on the choice of thermometric property, is called a centigrade
scale. The most commonly used temperature scale today is the
Celsius temperature scale, in which the freezing point of water is
0°C and the boiling point of water is (approximately) 100°C.
Gabriel Daniel
Fahrenheit (1686-1739),
German craftsman and
physicist.
The system was named in
honor of Anders Celsius
(1701-1744), a Swedish
astronomer who
suggested the use of a
centigrade scale. In his
scale, the boiling point
was 0 and the freezing
point was 100.
8
This scale was suggested
by René Antoine
Ferchault de Réaumur
(1683-1757), a French
physicist, and found
common acceptance in
pre-revolutionary France.
TEMPERATURE
A third system still in use in parts of Europe is the Réaumur scale.
This scale uses the same fixed reference temperatures but divides
the interval into 80 degrees, with the freezing point being 0°Ré
and boiling point being 80°Ré.
1.4 Gas Thermometry and Absolute
Temperature
In an attempt to extend the range of temperature and to avoid
some of the arbitrariness of the choice of mercury as a standard,
physicists devised several thermometers which use a gas as the
thermometric substance. The most important of these is the
constant volume gas thermometer.
•The Constant Volume Gas Thermometer
Figure 1.4 is a diagram of a simple constant volume gas thermometer. The thermometric gas is contained in a bulb made of metal,
which is connected to a mercury manometer. The mercury
reservoir is raised or lowered to keep the volume of the gas
constant. The pressure of the gas, which is equal to atmospheric
pressure plus ρgh, is used as the thermometric property.
The thermometer shown in the figure, while sufficient for lowprecision measurements at moderate temperatures, suffers from
a number of technical problems which must be overcome in
designing a thermometer for precision work.
These include the following:
1. The gas in the tube and above the mercury is at a
temperature different from the gas in the bulb and the
sample. The gas does not have uniform temperature.
This problem may be overcome by using a second gas,
separated from the thermometric gas by a diaphragm,
as the manometric gas.
2. The bulb and tube change volume (slightly) with changes
in temperature and pressure. The pressure effects can
be largely eliminated by surrounding the bulb and
THERMOMETRY
9
tubing with a gas, often the manometric gas, whose
pressure is varied to equal that of the thermometric
gas.
3. The density of the mercury varies with the pressure
and temperature. The temperature dependence can be
eliminated by maintaining the mercury at a standard
temperature. This heats the manometric gas near the
mercury, but the conductivity of gases is low enough
that little heat flows into the bulb. The pressure
variation must be measured and calibrated, and a
correction applied to the final measurement.
4. At very low temperatures, the gas may adhere to or be
absorbed by the walls of the bulb or tube. This effect
must be measured and a correction applied.
h
gas bulb
to be placed in
sample
mercury
manometer
flexible rubber
tubing
Figure 1.4: Simple constant volume gas thermometer in which the
mercury reservoir is raised or lowered so that the left meniscus
remains at the same height. The difference in column heights, h, gives
the gauge pressure.
10
TEMPERATURE
A temperature scale using the pressure of a constant volume gas
thermometer as the thermometric property still suffers from one
difficulty— different gases give slightly different temperatures.
To eliminate this problem, we use the fact that all gases act like
an ideal gas at low enough pressure.
•The Ideal Gas Temperature Scale
The concept of an ideal
gas is discussed in
chapter 2.
absolute zero
absolute temperature
The ideal gas temperature scale is defined using a constant
volume gas thermometer. The thermometer is used to measure
the pressure, Pi, at the freezing point of water and the pressure,
P, at the temperature of the sample. The measurements are
repeated with a lower density of gas in the bulb, which will reduce
both Pi and P. When extrapolated to Pi=0, the ratio of pressures
is found to be independent of the gas used. This limit is used as
the thermometric property. Thus
T = lim P .
T i Pi→ 0 Pi
This equation also specifies one of the reference temperatures as
T=0 when P=0. This temperature is called absolute zero and a
temperature scale with zero at absolute zero is called an absolute
temperature scale. No gas can be used to actually measure
temperatures at very low temperatures, so absolute zero on the
ideal gas scale exists only as an extrapolation.
To complete the specification of the temperature scale, we must
specify a second fixed temperature or the size of the degree.
Originally the size of the degree was chosen to be equal to a Celsius
degree. This requires
Ts - Ti = 100 .
(1.2)
We get a second equation by measuring Ts, the boiling point of
water.
Ts
P
= lim s .
i
Ti
→ 0 Pi
THERMOMETRY
11
This measurement has been made by many experimenters using
many different gases. Typical data for such experiments are
graphed in figure 1.5. The accepted value is
Ts
= 1.36609 ± 0.00004 .
(1.3)
Ti
Solving equations (1.2) and (1.3) yields
Ti = 273.16
and
Ts = 373.16 .
As thermometers became more precise and the temperatures
measured became colder, the use of Ti as a standard became a
problem. The ice point, defined as the temperature of a mixture
of ice and water saturated by air at a pressure of 1 atmosphere, is
reproducible only to an accuracy of ±0.01°C. The major source of
uncertainty in this measurement is the amount of air in the water.
To eliminate this uncertainty, the use of the ice and steam points
in the definition was replaced with a single fixed reference point
whose value was chosen to reproduce approximately the earlier
scale. The General Conference of Weights and Measures in 1954
chose the triple point of water, 273.16, as the fixed point. The
373.8
O2
373.6
T (K)
373.4
N2
373.2
H2
He
373
0
20
40
60
PTP (kPa)
80
100
120
Figure 1.5: Graph of measurements made in determining the
temperature of a sample using four different gases. For this case, the
sample is at Ts.
12
TEMPERATURE
triple point of water is the temperature at which ice, water, and
water vapor can coexist. This temperature is easily reproducible
to within ±0.0001°C. Figure 1.6 shows a simple triple point cell
used to maintain the thermometer bulb at TTP. The cell is filled
with pure water and sealed. The central well is filled with a
mixture of salt and ice until a fraction of the water in the cell has
frozen. The mixture is then replaced with the thermometer bulb.
Making a single temperature measurement requires the following procedure:
1. Fill the bulb with the chosen thermometric gas, generally hydrogen.
2. Allow the gas and bulb to reach equilibrium with a
triple point cell. Take repeated measurements of the
pressure.
3. Move the cell or thermometer so that the bulb can be
placed in the sample.
4. Allow the gas and bulb to reach equilibrium with the
sample. Take repeated measurements of the pressure.
5. Decrease the amount of gas in the bulb and repeat
steps 2-4.
vapor
ice
water
Figure 1.6: Simple triple point cell. The cell is symmetric about a
vertical axis, thus forming a central well.
THERMOMETRY
13
6. Extrapolate P/PTP to PTP = 0. The temperature of the
system is then given by
T = 273.16K lim P .
PTP → 0 PTP
See example 1.1 for one way in which the extrapolation can be
done. A single high-precision temperature measurement with an
ideal gas thermometer typically takes many months.
Using this new definition to measure the ice point and steam point
experimentally, one finds Ti=273.15 and Ts=373.125 on the ideal
gas temperature scale.
1.5 The International Practical
Temperature Scale (IPTS-68)
It has been our ultimate goal to define a temperature scale which
is not dependent on an arbitrary choice of thermometric property.
We will see, in Chapter 5, that such a scale can be defined
theoretically and is called the Absolute Thermodynamic Temperature Scale or, more simply, the Kelvin scale. To make such a
theoretical temperature scale useful, one must find empirical
temperature scales which coincide with it over the desired range
of temperatures. We will show, in Chapter 5, that the ideal gas
temperature scale coincides exactly with the Kelvin scale over the
range of temperatures for which the ideal gas scale is defined.
We do not use the ideal gas thermometer as our empirical
temperature scale, however, because a single precision measurement with a gas thermometer can take many months. To meet the
need for a practical way to measure temperatures with high
precision, the Seventh General Conference of Weights and Measures adopted an international practical temperature scale. This
scale has been revised several times to take advantage of improving technology and better data. The currently accepted scale is
called the International Practical Temperature Scale of 1968
(IPTS-68).
This scale was named in
honor of William
Thomson, Lord Kelvin
(1824-1907) the British
physicist who suggested
the use of an ideal gas
thermometer and the
triple point of water to
define a temperature
scale.
14
TEMPERATURE
Example 1.1
A student uses a low-precision constant volume gas thermometer to measure the
temperature of a water bath. Use a least square fit to a line to calculate the
temperature and uncertainty in temperature from the following data (in kPa):
PTP
40.0
60.0
80.0
101.0
120.0
P
51.2
76.5
102.2
128.8
152.6
Solution:
To use the definition of ideal gas temperature
P ,
PTP
we plot P/PTP versus PTP. We then fit a line to the data using a least square fit. That
line will have the equation
P =m P + b ,
TP
PTP
where m is the slope and b is the intercept. The limit as PTP → 0 is then b. First
calculate P/PTP.
T = 273.16K lim
PTP → 0
PTP
40.0
P/PTP 1.2800
60.0
1.2750
80.0
1.2775
101.0
1.2752
120.0
1.2717
A computer program was used to calculate the slope and intercept and their
uncertainties. The slope was (-8.1356±.0003) kPa-1 and the intercept was 1.2824±.002.
Therefore,
T = 273.16K (1.2824±.002) = 350.3K ± 0.5K
1.285
1.280
P/ PTP
1.275
20
40 PTP
60 (kPa)
80
100
120
THE INTERNATIONAL PRACTICAL TEMPERATURE SCALE
Table 1.2: Temperatures of primary fixed points of the International
Practical Temperature Scale of 1968.
Fixed Points
T68 (K)
t68 (°C)
triple point of hydrogen
13.81
-259.34
boiling point of hydrogen at 25/76 atm
17.042
-256.108
boiling point of hydrogen at 1 atm
20.28
-252.87
boiling point of neon at 1 atm
27.102
-246.048
triple point of oxygen
54.361
-218.789
triple point of argon
83.798
-189.352
boiling point of oxygen at 1 atm
90.188
-182.962
boiling point of water at 1 atm
373.15
99.975
melting point of zinc at 1 atm
692.664
419.514
melting point of silver at 1 atm
1235.08
961.93
melting point of gold at 1 atm
1337.58
1064.43
IPTS-68 consists of a table of primary and secondary fixed point
temperatures, a thermometer and thermometric property to be
used for different ranges of temperature, and a method for
interpolating between the fixed point temperatures. These are
summarized in tables 1.2 and 1.3.
Temperature Range
Thermometer
Interpolation Method
13.81K to 273.15K
platinum resistance
thermometer
polynomial fit to
tabulated function
273.15K to 903.89K
platinum resistance
thermometer
quadratic fit to triple
point of water and
primary fixed points
903.89K to 1337.58K
platinum and
90% Pt-10%Rh
thermocouple
quadratic fit to
903.89K and melting
points of silver and
gold
above 1337.58K
optical pyrometer
Planck radiation law
Table 1.3: Thermometers, thermometric properties, and methods of
interpolation for the International Practical Temperature Scale.
15
16
TEMPERATURE
The Kelvin scale and thus IPTS-68 are now used as the basis of the
definition of the Celsius and Fahrenheit temperature scales.
These definitions are summarized as
tc = T - 273.15K 1°C
K
9°F
and
t f = tc
+ 32° F .
5°C
For the remainder of this book, we will use the Kelvin scale and
give temperatures in °C. Remember, however, that this usage
assumes an identification, which we have not yet shown, between
the empirical temperature scales of IPTS-68 and the thermodynamic temperature scale, which we haven’t yet defined.
PROBLEMS
Problems
1.1
The normal boiling point for helium is the lowest of any
substance at -268.93°C. Tungsten has the highest melting
point of any metal at 3410°C. What are the corresponding
temperatures on the Kelvin, Fahrenheit, and Réaumur
scales?
1.2
The Rankine temperature scale is an absolute temperature
scale with the degree size chosen to be equal to 1°F. The
defining equations are equation (1.3) and
T s - Ti = 180 .
Find Ts and Ti on the Rankine scale.
1.3
At what temperature do Celsius and Fahrenheit temperatures have the same numerical values? At what temperature do Kelvin and Fahrenheit temperatures have the same
numerical values? At what temperature do Kelvin and
Rankine (see problem 1.2) temperatures have the same
numerical values?
1.4
The data in the following table were taken from a constant
volume gas thermometer. Determine the temperature of the
sample and its uncertainty by plotting Ps/PTP versus PTP
and using a least square fit to extrapolate to zero pressure.
PTP in torr 100.0
200.0
300.0
400.0
500.0
Ps in torr 233.9
471.7
714.7
962.7
1215.4
1.5
The resistance of a platinum resistance thermometer is
found to be 7.26Ω at the ice point, 9.14Ω at the steam point,
and 16.17Ω at the normal melting point of zinc. Use these
data to determine the coefficients of a quadratic fit. Plot the
resistance versus temperature for t=0°C to 450°C. The
resistor is placed in a sample and allowed to come to equilibrium. The resistance is measured as 12.24Ω. What is the
temperature of the sample?
17
18
TEMPERATURE
1.6
The electromotive force produced by a thermocouple can be
described by the equation in t (Celsius temperature)
φ = a + bt + c t2 .
a) Find the expression for φ as a function of T (Kelvin
temperature).
b) Define a centigrade temperature based on φ. That is, the
scale should be linear in φ and tφ=0 at t=0°C and tφ=100 at
t=100°C.
c) Define an absolute temperature based on φ,
φ
T φ ≡ 273.16
.
φ TP
Give Tφ as a function of t and the parameters a, b, and c.
1.7
What is the ratio of the pressure of a gas at the normal
melting point of zinc to that at the triple point of water?
What must be the maximum uncertainty in your pressure
measurements if you are to get an answer with as many
significant figures as given in Table 1.2?
1.8
A thermocouple has been calibrated and the following voltages have been measured:
50°C
1.43 mV
100°C
2.94 mV
200°C
5.93 mV
300°C
8.94 mV
When immersed in a sample, the thermocouple produces
2.00 mV. Determine the temperature of the sample by fitting
the data exactly to a cubic equaiton. Get a second value for
the temperature by determining the quadratic equation
which best fits the data. Which estimate of the temperature
is better?
PROBLEMS
1.9
An optical pyrometer is used to measure temperatures above
the normal melting point of gold, called the gold point. The
sample and a black body at the gold point temperature are
each sighted with the same optical pyrometer. When viewed
through a filter (λ=6000Å), the sample appears as bright as
the blackbody only when crossed polarizers are used to dim
the sample by 50%. Use equation (3.4) to determine the
temperature of the sample.
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