Faculty of Engineering
Engineering Physics I (BE121) Course
Fall 2012
Chapter 3
Part 1
Temperature and thermal properties
3.1
Thermal equilibrium and the zeroth law of thermodynamics:
Temperature:
It is a measure of the average internal molecular kinetic energy of an object.
We often associate the concept of temperature with how hot or cold an object feels
when we touch it. Thus, our senses provide us with a qualitative indication of
temperature.
We are all familiar with the fact that two objects at different initial
temperatures eventually reach some intermediate temperature when placed in
contact with each other.
Thermal equilibrium:
It is a situation in which two objects would not exchange energy by heat or
electromagnetic radiation if they were placed in thermal contact.
Let us consider two objects A and B, which are not in thermal contact, and a
third object C, which is our thermometer. We wish to determine whether A and B
are in thermal equilibrium with each other. The thermometer (object C) is first
placed in thermal contact with object A until thermal equilibrium is reached, From
that moment on, the thermometer’s reading remains constant, and we record this
reading. The thermometer is then removed from object A and placed in thermal
contact with object B. The reading is again recorded after thermal equilibrium is
reached. If the two readings are the same, then object A and object B are in thermal
equilibrium with each other. If they are placed in contact with each other, there is
no exchange of energy between them. We can think of temperature as the property
that determines whether an object is in thermal equilibrium with other objects.
The zeroth law of thermodynamics (the law of equilibrium):
If objects A and B are separately in thermal equilibrium with a third object C,
then A and B are in thermal equilibrium with each other.
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Faculty of Engineering
Engineering Physics I (BE121) Course
Fall 2012
Temperature Scales
In order to establish temperature scale it is necessary to make use of fixed
points. Three such points are defined below.
The ice point
is the temperature at which pure ice can exist in
equilibrium with water at standard atmospheric
pressure
The steam point
is the temperature at which pure water can exist
in equilibrium with its vapour at standard
atmospheric pressure
Triple point
of water is that unique temperature at which pure
ice, pure water and pure water vapour can exist
together in equilibrium
3.2
Thermometers and the temperature scales:
Thermometers are devices that are used to measure the temperature of a system. All
thermometers are based on the principle that some physical property of a system
changes as the system’s temperature changes. Some physical properties that change
with temperature are:
a) The volume of a liquid.
b) The volume of a gas at constant pressure.
c) The pressure of a gas at constant volume.
d) The dimensions of a solid.
e) The electric resistance of a conductor.
f) The color of an object.
A temperature scale can be established on the basis of any one of these physical
properties.
Temperature scales:
SCALE
Celsius (TC)
Kelvin (T)
Fahrenheit (TF)
ICE POINT OF
WATER
0 C
273 K
32 F
2
STEAM POINT OF
WATER
100 C
373 K
212 F
Faculty of Engineering
Engineering Physics I (BE121) Course
Fall 2012
The Celsius temperature (TC) is shifted from the absolute (Kelvin)
temperature (T) by 273. The two scales differ only in the choice of the zero point.
The relationship between the Celsius temperature (TC) and absolute temperature (T)
scales is:
T  TC  273
A common temperature scale in everyday use in the United States is the
Fahrenheit scale. This scale sets the temperature of the ice point of water at 32°F
and the temperature of the steam point at 212°F. The relationship between the
Celsius temperature (TC) and Fahrenheit temperature (TF) scales is:
5
TC  TF  32 
9
Example:
On a day when the temperature reaches 50°F, what is the temperature in
degrees Celsius and in Kelvin?
Solution
In degree Celsius:
5
TF  32
9
5
TC  50  32   10  C
9
TC 
In degree Kelvin:
T  TC  273
T  10  273  283K
Example:
A pan of water is heated from 25°C to 80°C. What is the change in its
temperature on the Kelvin scale and on the Fahrenheit scale?
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Solution
TC  80  25  55  C
The change on the Kelvin scale can be found from:
T  TC  273
T  TC  55K
The change on the Fahrenheit scale can be found from:
5
TF  32  TC  5 TF
9
9
9
9
TF  TC   55  99 o F
5
5
TC 
3.3 Thermal expansion of solids and liquids:
Our discussion of the liquid thermometer makes use of one of the best known
changes in a substance: as its temperature increases, its volume increases. This
phenomenon, known as thermal expansion, has an important role in numerous
engineering applications.
Thermal expansion is a consequence of the change in the average separation
between the atoms in an object.
If thermal expansion is sufficiently small relative to an object’s initial
dimensions, the change in any dimension is proportional to the first power of the
temperature change. Suppose that an object has an initial length L i along some
direction at some temperature and that the length increases by an amount L for a
change in temperature T. We define the average coefficient of linear expansion
() as:
L
L
T
 
i
 L  L T
(1)
i
 L  L  L T  T 
f
i
i
4
f
i
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Engineering Physics I (BE121) Course
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Where Li is the initial length at temperature Ti and Lf is the final length at
temperature Tf .If the coefficient of linear expansion () is positive, the length
increases with the increase of temperature.
Because the linear dimensions of an object change with temperature, it
follows that surface area and volume change as well. The change in volume is
proportional to the initial volume Vi and to the change in temperature according to
the relationship:
 V   V T
i
V  V   V T  T 
f
i
i
f
i
Where  is the average coefficient of volume expansion. For an isotropic solid, the
average coefficient of volume expansion is three times the average linear expansion
coefficient ( =3). Similarly, the change in area of a rectangular plate is given by:
 A  2A T
i
 A  A  2A T  T 
f
i
i
5
f
i
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Table 1
Average Expansion Coefficients for Some Materials near Room Temperature
Example
A segment of steel railroad track has a length of 30.000 m when the temperature is
0.0oC.
(A) What is its length when the temperature is 40.0oC?
Solution
Use Equation 1 and the value
of the coefficient of linear
expansion from Table 1:
Find the new length of the
track:
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Example:
A poorly designed electronic device
has two bolts attached to different
parts of the device that almost touch
each other in its interior as in Figure.
The steel and brass bolts are at
different electric potentials, and if they
touch, a short circuit will develop,
damaging the device. The initial gap
between the ends of the bolts is 5.0 µm
at 27oC. At what temperature will the
bolts touch? Assume the distance
between the walls of the device is not
affected by the temperature change.
Solution
LiSteel  0.01m
LiBrass  0.03m
Brass=1910-6 °C-1
Ti  27 0 C
d=5m
Steel=1110-6 °C-1
L   Steel LiSteel T   Brass LiBrassT
  Steel LiSteel   Brass LiBrass T
T 
L
 Steel LiSteel   Brass LiBrass
T f  Ti 
L
 Steel LiSteel   Brass LiBrass
T f  Ti 
L
 Steel LiSteel   Brass LiBrass
The total change in length of the two bolts must equal the length of the initial gap
between the ends.
 𝐿 = 𝑑 = 5 𝑚
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T f  Ti 
L
 Steel LiSteel   Brass LiBrass
 27 
11  10
6
5  10 6
 0.01  19  10 6  0.03
 

 34.35 C
0
3.4 Heat and internal energy:
It is important that we make a major distinction between internal energy and
heat.
Internal energy: it is all the energy of a system that is associated with its
microscopic components, atoms and molecules. Internal energy includes kinetic
energy and potential energy of molecules.
Heat: it is defined as the transfer of energy across the boundary of a system
due to a temperature difference between the system and its surroundings. When you
heat a substance, you are transferring energy into it by placing it in contact with
surroundings that have a higher temperature. When you cool a substance, you are
transferring energy from it by placing it in contact with surroundings that have a
lower temperature.
Units of heat:
The heat is measured in calorie (cal), which is defined as the amount of energy
transfer necessary to raise the temperature of 1 g of water from 14.5°C to 15.5°C.
In SI unit of heat is measured in joule, where:
1 cal = 4.186 J
The heat capacity (C):
The heat capacity of a particular sample of a substance is defined as the amount of
energy needed to raise the temperature of that sample by 1°C.
From this definition, we see that if energy (Q) produces a change (T) in the
temperature of a sample, then:
Q  CT
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Faculty of Engineering
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The unit of the heat capacity is J/kg.
The specific heat (c):
The specific heat of a substance is the heat capacity per unit mass. We can define it
as the amount of energy needed to raise the temperature of 1kg of that sample by
1°C.
Thus, if energy Q transfers to a sample of a substance with mass (m) and the
temperature of the sample changes by (T), then the specific heat of the substance
is:
c 
Q
mT
Specific heat is essentially a measure of how thermally insensitive a substance is to
the addition of energy. The greater a material’s specific heat, the more energy must
be added to a given mass of the material to cause a particular temperature change.
From this definition, we can relate the energy (Q) transferred between a
sample of mass (m) of a material and its surroundings to a temperature change T
as:
Q  mcT
From the definition of the heat capacity and specific heat, we can write:
C  mc
Note:
When the temperature increases, Q and T are taken to be positive, and
energy transfers into the system. When the temperature decreases, Q and T are
negative, and energy transfers out of the system.
Calorimetry
One technique for measuring specific heat involves heating a sample to some
known temperature 𝑇𝑥 , placing it in a vessel containing water of known mass and
temperature 𝑇𝑤 < 𝑇𝑥 , and measuring the temperature of the water after
equilibrium has been reached. This technique is called calorimetry and devices in
which this energy transfer occurs are called calorimeters.
9
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Figure shows the hot sample in the
cold water and the resulting energy
transfer by heat from the hightemperature part of the system to the
low-temperature part. If the system of
the sample and the water is isolated,
the principle of conservation of energy
requires that the amount of energy
𝑄ℎ𝑜𝑡 that leaves the sample (of
unknown specific heat) equal the
amount of energy 𝑄𝑐𝑜𝑙𝑑 that enters the
water. Conservation of energy allows
us to write the mathematical
representation of this energy statement
as
In a calorimetry experiment, a hot
sample whose specific heat is unknown
is placed in cold water in a container that
isolates the system from the environment
𝑸𝒄𝒐𝒍𝒅 = − 𝑸𝒉𝒐𝒕
Example:
A 0.05 kg ingot of metal is heated to 200 °C and then dropped into a beaker
containing 0.4 kg of water initially at 20 °C. If the final equilibrium temperature of
the mixed system is 22.4°C, find the specific heat of the metal. The specific heat of
water is 4200 J/kgC.
Solution
Qlost  Q gained
mm c m Tmetal  mw c w Tw
0.05  c m  200  22.4  0.4  4200  22.4  20
c m  454.05 J / kg C
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Latent heat (L):
A substance often undergoes a change in temperature when energy is
transferred between it and its surroundings. There are situations, however, in which
the transfer of energy does not result in a change in temperature. This is the case
whenever the physical characteristics of the substance change from one form to
another; such a change is commonly referred to as a phase change. Two common
phase changes are from solid to liquid (melting) and from liquid to gas (boiling).
All such phase changes involve a change in internal energy but no change in
temperature.
If a quantity Q of energy transfer is required to change the phase of a mass m
of a substance, the ratio 𝐿 = 𝑄/𝑚 characterizes an important thermal property of
that substance. The quantity L is called the specific latent heat of the substance.
From the definition of latent heat, the energy required to change the phase of a
given mass m of a pure substance is:
𝑄 = 𝑚𝐿
Specific Latent heat of fusion Lf is the energy needed to change 1 Kg of
solid into liquid at constant temperature.
Specific Latent heat of vaporization Lv is the energy needed to change 1 Kg
of liquid into gas at constant temperature.
To understand the role of latent heat in phase changes, consider the energy
required to convert a 1.00-g cube of ice at 230.0°C to steam at 120.0°C. the Figure
indicates the experimental results obtained when energy is gradually added to the
ice. The results are presented as a graph of temperature of the system of the ice
cube versus energy added to the system.
11
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Fig. a plot of temperature versus energy added when 1.00 g of ice initially at 30.0°C is converted to steam at 120.0°C.
Example
A mass of 24 g of ice at –15 °C is taken from a freezer and placed in a
beaker containing 200 g of water at 28 °C. Data for ice and for water are
given in Fig.
Assuming that the beaker has negligible mass, calculate the final
temperature of the water in the beaker?
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Solution
Heat taken from hot water is used to;
1-warm ice
2-melt ice
3-warm 24g of water obtained from melting ice to temperature T
(𝑚𝑐𝛥𝑇)200𝑔 𝑤𝑎𝑡𝑒𝑟 = (𝑚𝑐𝛥𝑇) 𝑖𝑐𝑒 + (𝑚𝑙)𝑖𝑐𝑒 + (𝑚𝑐𝛥𝑇)24𝑔 𝑤𝑎𝑡𝑒𝑟
(200×10-3×4.2×103× (28-T) = (24×10-3×2.1×103×15)
+ (24×10- 3×330×103) + (24×10-3×4.2×103×T)
T = 16oC
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