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Chapters 17 &19
Temperature, Thermal Expansion and The Ideal
Gas Law
Units of Chapter 17 & 19
• Temperature and the Zeroth Law of Thermodynamics
• Temperature Scales
• Thermal Expansion
• Heat and Mechanical Work
• Specific Heats
• Latent Heats
17-3 Temperature, heat, Thermal Equilibrium and the
Zeroth Law of Thermodynamics
Definition of heat: Heat is the energy transferred between objects
because of a temperature difference.
Objects are in thermal contact if heat can flow between them. E.g. If
a hot object is brought into thermal contact with a cold object, heat
will be exchanged or thermal energy will transfer from one object to
another.
When the transfer of heat between objects in thermal contact
ceases, they are in thermal equilibrium and the objects are at the
same Temperature
Definition of Temperature:
Temperature is a measure of how hot or cold something is.
A hot oven is said to be at HIGH temperature and ice of a frozen
lake is said to be at LOW temperature
A more correct definition: Temperature is a measure of the average
kinetic energy of all particles in an object.
The zeroth law of
thermodynamics:
If object A is in thermal
equilibrium with object B, and
object C is also in thermal
equilibrium with object B, then
objects A and C will be in
thermal equilibrium if brought
into thermal contact.
That is, temperature is the only
factor that determines whether
two objects in thermal contact
are in thermal equilibrium or not.
17-3 Temperature, heat, Thermal Equilibrium and the
Zeroth Law of Thermodynamics
17-2 Temperature Scales
The Celsius scale:
Water freezes at 0 Celsius.
Water boils at 100 Celsius.
The Fahrenheit scale:
Water freezes at 32 Fahrenheit .
Water boils at 212 Fahrenheit .
Converting from Celsius to Fahrenheit:
Converting from Fahrenheit to Celsius:
32C)(F)(59 TT
32F)(C)(95 TT
17-2 Temperature Scales
The pressure in a gas is proportional to its temperature. The
proportionality constant is different for different gases, but they all reach
zero pressure at the same temperature, which we call absolute zero:
Absolute zero forms the basis of a temperature scale known as
Absolute Scale or Kelvin Scale.
The Kelvin scale is similar to the Celsius scale, except that the Kelvin
scale has its zero at absolute zero.
Conversion between a Celsius temperature and a Kelvin temperature:
15.273C)((K) TT
17-2 Temperature Scales
The three temperature scales compared:
17-4 Thermal Expansion Most substances expand when heated and contract when cooled.
However the amount of expansion and contraction depends on the
materials.
Most solids generally expand in Length, Area and Volume as
temperature increases. This can be understood as an increase in the
amplitude of vibration of the atoms or molecules about their positions.
Linear Expansion
Experiments show that the change in length, ( L) is directly proportional
to the change in temperature ( T) and also proportional to the original
length (L0) of the object.
i.e L T, and L L0
We can write the proportionality as an equation:
The proportionality constant is called the coefficient of linear expansion.
Some typical
coefficients of
thermal expansion:
17-4 Thermal Expansion We can write: Final length = Original length + Change in length:
) 1( 000
0
TLTLLL
LLL
Area and Volume Expansion:
The expansion of Area and Volume of a flat substance is derived from
the linear expansion in two and three dimensions respectively:
Definition of Coefficient of
Volume Expansion, :
17-4 Thermal Expansion
TATAA 002 Where: = 2 :
Coefficient of area
expansion TVTVV 003 Where: = 3 :
Coefficient of volume expansion
TVTVV 003
SI unit for : K-1 = ( C)-1
For liquids and gases, only the
coefficient of volume expansion
is defined:
The Table shows some typical
coefficients of volume expansion:
17-4 Thermal Expansion
Exercise 1: A steel railway track has a length of 30.000 m when the
temperature is 0 C. What is the length on a hot Melbourne day when
the temperature is 40 C?
Exercise 2: The steel bed of a suspension bridge is 200 m long at
20 C. If the extremes of the temperature to which it might be
exposed are 30 C to 40 C. What total range of change in length
must the expansion joints accommodate? (i.e. How much will it
contract and expand?)
17-4 Thermal Expansion
Exercise 3: A 70 L steel gas tank of a car is filled to the top with
gasoline at 20 C. The car sits in the sun and the tank reaches a
temperature of 40 C. How much gasoline do you expect to overflow
from the tank?
Exercise 4: A copper ball with a radius of 1.6 cm is heated from an
initial temperature of 22 C to a final temperature of 680 C. Find the
change in the volume of the ball and the final radius of the ball.
Behaviour of gases depends on the following properties of the gases:
Pressure
Volume
Temperature
Mass
Number of Molecules
Gases are the easiest state of matter to describe, as all ideal gases
exhibit similar behavior.
An ideal gas is one that is thin (dilute) enough, and far away enough
from condensing, that the interactions between molecules can be
ignored.
Real Gases: The behavior of real gases is generally quite well
approximated by that of an Ideal gas at low pressure (or low density),
and at room temperature (or when T is not close to Liquefaction
point).
17-6 The Ideal Gas Laws and Absolute Temperature
We can describe the way the
Pressure, P, of an ideal gas
depends on:
Temperature, T
Number of molecules, N, and the
Volume, V, from a few simple
observations:
17-7 The Ideal Gas Law
(i) If the volume of an ideal gas is
held constant, (as in the constant
volume gas thermometer), we find
that the pressure varies linearly
with absolute temperature: (P T )
(ii) If the volume and temperature of a gas are
kept constant, but more gas is added (such
as in inflating a tire or basketball), the
pressure will increase: (i.e. P N )
17-7 The Ideal Gas Law
(iii) Finally, if the temperature and the number of
molecules are held constant and the volume
decreases, (such as sitting on a ball), the
pressure increases. That is the pressure varies
inversely with volume: (P 1/V or PV = constant)
Combining all three observations, we can write a mathematical
expression fro the Pressure of a gas:
where k is called the Boltzmann constant:
17-7 The Ideal Gas Law
Rearranging gives us the equation of state for an ideal gas:
Instead of counting molecules, we can count moles.
A mole is the amount of substance that contains as many
atoms or molecules as there are atoms in 12 g of carbon-12.
Experimentally, the number of atoms or molecules in one mole is
given by Avogadro’s number:
17-7 The Ideal Gas Law
Avogadro’s number and the Boltzmann constant can be combined to
form the Universal Gas Constant, R, defined as:
Therefore, n moles of a gas will contain N = nNA molecules.
Substituting this into the ideal gas equation:
nRTkTnNNkTPV A
K)J/(mol :unit SI
)Kmol/()atmL( 0821.0
K)J/(mol 314.8
)J/K1038.1)(molmolecules/1002.6( 2323kNR A
17-8 Problem Solving with the Ideal Gas Law
The ideal gas law is an extremely useful tool.
We often refer to “Standard Conditions” or Standard Temperature
and Pressure (STP) which means:
T = 273 K (0 C) and P =1.00 atm = 1.013 105 N/m2 =101.3 kPa
When using the ideal gas law: the Temperature,T, must be given in
Kelvin (K) and the pressure, P, must always be the Absolute
pressure, not gauge pressure.
In many situations, it is not necessary to use the value of R at all.
For example, many problems involve a change in pressure,
temperature, and volume of a fixed amount of gas.
In this case:
constantnRT
PV
nRTPV
Since n and R remain constant, we can let P1 , V1 and
T1 denote the initial variables and P2 , V2 and T2 denote
the variables after the change (final conditions), then
we can calculate the unknown variable using: 2
22
1
11
T
VP
T
VP
Boyle’s law: is consistent with the ideal
gas law.
For a fixed quantity of gas, the volume
of the gas is inversely proportional to
the absolute pressure at constant
temperature. (V 1/P, or PV = constant)
17-8 Problem Solving with the Ideal Gas Law
These
curves of
constant
temperature
are called
isotherms.
Charles’s law: is also consistent with
the ideal gas law.
The volume of a fixed quantity of gas
is directly proportional to the absolute
temperature if the pressure is kept
constant. (V T, or V/T = constant)
2
2
1
1
T
V
T
V
2211 VPVPFixed number of molecules, N;
Fixed temperature, T
Fixed number of molecules, N;
Fixed pressure, P
17-8 Problem Solving with the Ideal Gas Law
Exercise 4: Determine the volume of 1.00 mole of any gas, assuming
it behaves like an ideal gas at STP.
Exercise 5: A person’s lungs can hold 6.0 L (1L = 10-3 m3) of air at a
body temperature of 310 K and atmospheric pressure of 101 kPa.
Given that the air is 21% oxygen, find the number of oxygen
molecules in the lungs.
17-8 Problem Solving with the Ideal Gas Law
Exercise 6: How many moles of air are in an inflated basketball?
Assume that the pressure in the ball is 171 kPa, the temperature is
293 K, and the diameter of the ball is 30.0 cm.
Exercise 7: An automobile tyre is filled to a gauge pressure of
200kPa at 10 C. After a drive of 100km, the temperature within the
tyre rises to 40 C. What is the new pressure within the tyre at this
temperature?
Experimental work has shown that heat is another form of energy.
James Joule used a device
similar to this one to measure the
mechanical equivalent of heat:
As the mass falls, it turns the
paddles in the water, which
results in increase in water
temperature.
Thus Joule was able to show that
mechanical energy (P.E.) is
converted to heat
19-1 Heat and Mechanical Work
One kilocalorie (kcal) is defined as the amount of heat needed to raise
the temperature of 1 kg of water by 1 C (i.e. from14.5 C to 15.5 C)
Joule used his experiments to find the mechanical equivalent of heat:
1 kcal = 4.186 kJ
Heat, (Q) is the energy transferred from one object to another
because of temperature difference
19-1 Heat and Mechanical Work
Exercise 8: Working off the extra calories
A 74 kg man eats too much ice cream on the order of 305 C. How
many stairs of height 20.0 cm must he climb to work of the ice cream?
In studies of Nutrition, A different calorie is used.
(Calorie with a capital C): 1 C = 1 kcal
The Heat Capacity of an object is the amount of heat added to it
divided by its rise in temperature:
Q is positive if ΔT is positive; that is,
if heat is added to a system.
Q is negative if ΔT is negative; that is,
if heat is removed from a system.
19-3 Specific Heats
Specific Heat (c)
The quantity of heat required to change the temperature of a given
material is proportional to the mass, m of the material and to the
temperature change, T .
The specific heat, c, of any substance is defined as the amount of heat
required to increase the temperature of 1kg of the substance by 1 C
It can be rearranged as:
Q = mc T
Here are some
specific heats of
various materials:
19-3 Specific Heats
An isolated system is a closed system in which no heat energy is
exchanged across its boundaries with the surroundings
We use conservation of energy to figure out the final equilibrium
temperature when two substances at different temperature are mixed
and allowed to come to equilibrium within an isolated system.
That is different parts of the system are at different temperatures.
Heat flow from the part at higher temperature to the part at lower
temperature within the system. Heat lost by one part of the system
equals heat gained by the other part. i.e. Q = 0
Heat Lost = Heat Gained
This is the basis for Calorimetry Technique: A calorimeter is a
lightweight, insulated flask containing water. When an object is put
in, it and the water come to thermal equilibrium. If the mass of the
flask can be ignored, and the insulation prevents any heat exchange
with the surroundings:
1. The final temperatures of the object and the water will be equal.
2. The total energy of the system is conserved.
This allows us to calculate the specific heat of the object.
19-4 Calorimetry- Problem Solving using conservation of energy
When two phases coexist, the temperature remains constant even
if a small amount of heat is added.
Instead of raising the temperature, the heat goes into changing the
phase of the material – melting ice, for example. i.e. certain
amount of energy is used in this change of phase
Figure shows temperature as a function of heat added to bring 1.0
kg of ice at 20 C to steam above 100 C
19-5 Latent Heats
The heat required to convert from one phase to another is called
the latent heat.
The latent heat, L, is the heat that must be added to or removed
from one kilogram of a substance to convert it from one phase to
another. During the conversion process, the temperature of the
system remains constant.
Heat involved in a change of phase depends on the Latent Heat
and also on the total mass of the substance. That is:
19-5 Latent Heats
The latent heat of fusion, (LF ), is the heat (required/released) to
change from (solid to liquid/liquid to solid) phase
Latent heat of vaporization, (LV ), is the heat (required/released) to
change from (liquid to gas/gas to liquid) phase.
Table shows latent heats of fusion and vaporisation for various
substances
19-5 Latent Heats
19-5 Problem Solving using conservation of energy
Exercise 9: The Cup Cools the Tea
If 200 cm3 of tea at 95 C is poured into a 150 g glass cup at 25 C.
What will be the common final temperature, T, of the tea and cup at
equilibrium assuming no heat flows to the surrounding?
Exercise 10: Unknown Specific Heat determined by Calorimetry
An Engineer wishes to determine the specific heat of a new metal
alloy. A 0.150 kg sample of the alloy is heated to 540 C. It is then
quickly placed in 0.400 kg of water at 10 C contained in a 0.200 kg
aluminum calorimeter cup. The final temperature of the system is
30.5 C. Calculate the specific heat of the alloy.
19-5 Problem Solving using conservation of energy
Exercise 11: Will all Ice Melt
Determine the final equilibrium temperature and phase (state) of the
final mixture when 10 g of steam at 100 C is added to 80 g of ice
20 C.
