Transcript Document 7586396

Chapter 16
Temperature
and the
Kinetic Theory of Gases
Overview of Thermodynamics


Extends the ideas of temperature and internal
energy
Concerned with concepts of energy transfers
between a system and its environment
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And the resulting variations in temperature or
changes in state
Explains the bulk properties of matter and the
correlation between them and the mechanics
of atoms and molecules
Temperature
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We associate the concept of temperature with
how hot or cold an objects feels
Our senses provide us with a qualitative
indication of temperature
Our senses are unreliable for this purpose
We need a reliable and reproducible way for
establishing the relative hotness or coldness
of objects that is related solely to the
temperature of the object

Thermometers are used for these measurements
Thermal Contact

Two objects are in thermal contact with
each other if energy can be exchanged
between them
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The exchanges can be in the form of heat
or electromagnetic radiation
The energy is exchanged due to a
temperature difference
Thermal Equilibrium

Thermal equilibrium is a situation in
which two object would not exchange
energy by heat or electromagnetic
radiation if they were placed in thermal
contact

The thermal contact does not have to also
be physical contact
Zeroth Law of
Thermodynamics

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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Let object C be the thermometer
Since they are in thermal equilibrium with
each other, there is no energy exchanged
among them
Zeroth Law of
Thermodynamics, Example

Object C (thermometer) is placed in contact with A until
it they achieve thermal equilibrium
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Object C is then placed in contact with object B until
they achieve thermal equilibrium
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The reading on C is recorded
The reading on C is recorded again
If the two readings are the same, A and B are also in
thermal equilibrium
Temperature


Temperature can be thought of as the
property that determines whether an
object is in thermal equilibrium with
other objects
Two objects in thermal equilibrium
with each other are at the same
temperature

If two objects have different temperatures, they
are not in thermal equilibrium with each other
Thermometers

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A thermometer is a device that is used
to measure the temperature of a system
Thermometers are based on the
principle that some physical property of
a system changes as the system’s
temperature changes
Thermometers, cont
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The properties include
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The volume of a liquid
The length of a solid
The pressure of a gas at a constant volume
The volume of a gas at a constant pressure
The electric resistance of a conductor
The color of an object
A temperature scale can be established on
the basis of any of these physical properties
Thermometer, Liquid in Glass
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A common type
of thermometer is
a liquid-in-glass
The material in
the capillary tube
expands as it is
heated
The liquid is
usually mercury
or alcohol
Calibrating a Thermometer
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A thermometer can be calibrated by
placing it in contact with some
environments that remain at constant
temperature
Common systems involve water

A mixture of ice and water at atmospheric
pressure

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Called the ice point or freezing point of water
A mixture of water and steam in equilibrium

Called the steam point or boiling point of water
Celsius Scale
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The ice point of water is defined to be
0oC
The steam point of water is defined to
be 100oC
The length of the column between these
two points is divided into 100 equal
segments, called degrees
Problems with
Liquid-in-Glass Thermometers
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An alcohol thermometer and a mercury
thermometer may agree only at the
calibration points
The discrepancies between
thermometers are especially large when
the temperatures being measured are
far from the calibration points
Gas Thermometer
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The gas thermometer offers a way to
define temperature
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Also directly relates temperature to internal
energy
Temperature readings are nearly
independent of the substance used in
the thermometer
Constant Volume
Gas Thermometer
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The physical change
exploited is the variation
of pressure of a fixed
volume gas as its
temperature changes
The volume of the gas
is kept constant by
raising or lowering the
reservoir B to keep the
mercury level at A
constant
Constant Volume
Gas Thermometer, cont
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The thermometer is calibrated by using
a ice water bath and a steam water bath
The pressures of the mercury under
each situation are recorded
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The volume is kept constant by adjusting A
The information is plotted
Constant Volume
Gas Thermometer, final
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To find the
temperature of a
substance, the gas
flask is placed in
thermal contact with
the substance
The pressure is
found on the graph
The temperature is
read from the graph
Absolute Zero
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The thermometer
readings are virtually
independent of the gas
used
If the lines for various
gases are extended, the
pressure is always zero
when the temperature is
–273.15o C
This temperature is
called absolute zero
Absolute Temperature Scale
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Absolute zero is used as the basis of
the absolute temperature scale
The size of the degree on the absolute
scale is the same as the size of the
degree on the Celsius scale
To convert: TC = T – 273.15
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TC is the temperature in Celsius
T is the Kelvin (absolute) temperature
Absolute
Temperature Scale, 2
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The absolute temperature scale is now
based on two new fixed points
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Adopted in 1954 by the International
Committee on Weights and Measures
One point is absolute zero
The other point is the triple point of water
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This is the combination of temperature and
pressure where ice, water, and steam can all
coexist
Absolute
Temperature Scale, 3
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The triple point of water occurs at 0.01o
C and 4.58 mm of mercury
This temperature was set to be 273.16
on the absolute temperature scale
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This made the old absolute scale agree
closely with the new one
The unit of the absolute scale is the kelvin
Absolute
Temperature Scale, 4
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The absolute scale is also called the
Kelvin scale
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The triple point temperature is 273.16 K
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Named for William Thomson, Lord Kelvin
No degree symbol is used with kelvins
The kelvin is defined as 1/273.16 of the
temperature of the triple point of water
Some Examples of
Absolute Temperatures
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This figure gives some
absolute temperatures at
which various physical
processes occur
The scale is logarithmic
The temperature of
absolute cannot be
achieved
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Experiments have come
close
Energy at Absolute Zero
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According to classical physics, the kinetic
energy of the gas molecules would become
zero at absolute zero
The molecular motion would cease
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Therefore, the molecules would settle out on the
bottom of the container
Quantum theory modifies this and shows
some residual energy would remain
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This energy is called the zero-point energy
Fahrenheit Scale
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A common scale in everyday use in
the US
Named for Daniel Fahrenheit
Temperature of the ice point is 32oF
Temperature of the steam point is 212oF
There are 180 divisions (degrees)
between the two reference points
Comparison of Scales

Celsius and Kelvin have the same size
degrees, but different starting points
TC = T – 273.15
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Celsius and Fahrenheit have difference
sized degrees and different starting
points
9
TF  TC  32 F
5
Comparison of Scales, cont
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To compare changes in temperature
5
TC  T  TF
9
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Ice point temperatures
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0oC = 273.15 K = 32oF
steam point temperatures
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100oC = 373.15 K = 212oF
Thermal Expansion
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Thermal expansion is the increase in the
size of an object with an increase in its
temperature
Thermal expansion is a consequence of the
change in the average separation between
the atoms in an object
If the expansion is small relative to the
original dimensions of the object, the change
in any dimension is, to a good approximation,
proportional to the first power of the change
in temperature
Thermal Expansion, example
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As the washer is heated, all
the dimensions will increase
A cavity in a piece of
material expands in the
same way as if the cavity
were filled with the material
The expansion is
exaggerated in this figure
Linear Expansion
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Assume an object has an initial length L
That length increases by L as the
temperature changes by T
The change in length can be found by
L = a Li T
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a is the average coefficient of linear
expansion
Linear Expansion, cont
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This equation can also be written in
terms of the initial and final conditions of
the object:
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Lf – Li = a Li(Tf – Ti)
The coefficient of linear expansion has
units of (oC)-1
Linear Expansion, final
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Some materials expand along one
dimension, but contract along another
as the temperature increases
Since the linear dimensions change, it
follows that the surface area and
volume also change with a change in
temperature
Thermal Expansion
Volume Expansion
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The change in volume is proportional to the
original volume and to the change in
temperature
V = Vi b T
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b is the average coefficient of volume
expansion
For a solid, b  3 a
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This assumes the material is isotropic, the same in all
directions
For a liquid or gas, b is given in the table
Area Expansion
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The change in area is proportional to
the original area and to the change in
temperature
A = Ai g T
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g is the average coefficient of area
expansion
g=2a
Thermal Expansion, Example
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In many situations,
joints are used to allow
room for thermal
expansion
The long, vertical joint is
filled with a soft material
that allows the wall to
expand and contract as
the temperature of the
bricks changes
Bimetallic Strip
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Each substance has
its own
characteristic
average coefficient
of expansion
This can be made
use of in the device
shown, called a
bimetallic strip
It can be used in a
thermostat
Water’s Unusual Behavior
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As the temperature
increases from 0o C to
4o C, water contracts
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Above 4o C, water
expands with increasing
temperature
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Its density increases
Its density decreases
The maximum density
of water (1 000 kg/m3)
occurs at 4oC
Gas: Equation of State
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It is useful to know how the volume, pressure
and temperature of the gas of mass m are
related
The equation that interrelates these quantities
is called the equation of state
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These are generally quite complicated
If the gas is maintained at a low pressure, the
equation of state becomes much easier
This type of a low density gas is commonly
referred to as an ideal gas
Ideal Gas – Details
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A collection of atoms or molecules that
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Move randomly
Exert no long-range forces on one another
Are so small that they occupy a negligible
fraction of the volume of their container
The Mole
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The amount of gas in a given volume is
conveniently expressed in terms of the
number of moles
One mole of any substance is that
amount of the substance that contains
Avogadro’s number of molecules
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Avogadro’s number, NA = 6.022 x 1023
Moles, cont
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The number of moles can be
determined from the mass of the
substance: n = m / M
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M is the molar mass of the substance
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Commonly expressed in g/mole
m is the mass of the sample
n is the number of moles
Gas Laws
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When a gas is kept at a constant
temperature, its pressure is inversely
proportional to its volume (Boyle’s Law)
When a gas is kept at a constant pressure,
the volume is directly proportional to the
temperature (Charles’ Laws)
When the volume of the gas is kept constant,
the pressure is directly proportional to the
temperature (Guy-Lussac’s Law)
Ideal Gas Law
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The equation of state for an ideal gas
combines and summarizes the other gas laws
PV = n R T
This is known as the ideal gas law
R is a constant, called the Universal Gas
Constant
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R = 8.314 J/ mol K = 0.08214 L atm/mol K
From this, you can determine that 1 mole of
any gas at atmospheric pressure and at 0o C
is 22.4 L
Ideal Gas Law, cont
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The ideal gas law is often expressed in
terms of the total number of molecules,
N, present in the sample
P V = n R T = (N / NA) R T = N kB T
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kB is Boltzmann’s constant
kB = 1.38 x 10-23 J / K
Ludwid Boltzmann
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1844 – 1906
Contributions to
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Kinetic theory of gases
Electromagnetism
Thermodynamics
Work in kinetic
theory led to the
branch of physics
called statistical
mechanics
Kinetic Theory of Gases
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Uses a structural model based on the
ideal gas model
Combines the structural model and its
predictions
Pressure and temperature of an ideal
gas are interpreted in terms of
microscopic variables
Structural Model Assumptions
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The number of molecules in the gas is
large, and the average separation
between them is large compared with
their dimensions
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The molecules occupy a negligible volume
within the container
This is consistent with the macroscopic
model where we assumed the molecules
were point-like
Structural
Model Assumptions, 2
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The molecules obey Newton’s laws of
motion, but as a whole their motion is
isotropic
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Any molecule can move in any direction
with any speed
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Meaning of isotropic
Structural
Model Assumptions, 3
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The molecules interact only by short-range
forces during elastic collisions
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This is consistent with the ideal gas model, in
which the molecules exert no long-range forces on
each other
The molecules make elastic collisions with
the walls
The gas under consideration is a pure
substance
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All molecules are identical
Ideal Gas Notes
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An ideal gas is often pictured as
consisting of single atoms
However, the behavior of molecular
gases approximate that of ideal gases
quite well
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Molecular rotations and vibrations have no
effect, on average, on the motions
considered
Pressure and Kinetic Energy
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Assume a container
is a cube
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Edges are length d
Look at the motion
of the molecule in
terms of its velocity
components
Look at its
momentum and the
average force
Pressure and
Kinetic Energy, 2
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Assume perfectly
elastic collisions
with the walls of the
container
The relationship
between the
pressure and the
molecular kinetic
energy comes from
momentum and
Newton’s Laws
Pressure and
Kinetic Energy, 3

The relationship is
2  N   1 ___2 
P     mv 
3  V  2


This tells us that pressure is
proportional to the number of molecules
per unit volume (N/V) and to the
average translational kinetic energy of
the molecules
Pressure and
Kinetic Energy, final
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This equation also relates the macroscopic
quantity of pressure with a microscopic
quantity of the average value of the molecular
translational kinetic energy
One way to increase the pressure is to
increase the number of molecules per unit
volume
The pressure can also be increased by
increasing the speed (kinetic energy) of the
molecules
A Molecular Interpretation
of Temperature

We can take the pressure as it relates to the
kinetic energy and compare it to the pressure
from the equation of state for an idea gas
2  N   1 ___2 
P     mv   NkBT
3  V  2


Therefore, the temperature is a direct
measure of the average translational
molecular kinetic energy
A Microscopic Description
of Temperature, cont
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Simplifying the equation relating
temperature and kinetic energy gives
___
2
1
3
mv  kBT
2
2

This can be applied to each direction,
1 ___2 1
mv x  kBT
2
2
with similar expressions for vy and vz
A Microscopic Description
of Temperature, final
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Each translational degree of freedom
contributes an equal amount to the
energy of the gas
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In general, a degree of freedom refers to
an independent means by which a
molecule can possess energy
A generalization of this result is called
the theorem of equipartition of
energy
Theorem of
Equipartition of Energy
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The theorem states that the energy of a
system in thermal equilibrium is equally
divided among all degrees of freedom
Each degree of freedom contributes
½ kBT per molecule to the energy of the
system
Total Kinetic Energy
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The total translational kinetic energy is just N
times the kinetic energy of each molecule
Etotal

 1 ___2  3
3
 N  mv   NkBT  nRT
2
2
 2
This tells us that the total translational kinetic
energy of a system of molecules is
proportional to the absolute temperature of
the system
Monatomic Gas


For a monatomic gas, translational kinetic
energy is the only type of energy the particles
of the gas can have
Therefore, the total energy is the internal
energy:
3
Eint  n RT
2

For polyatomic molecules, additional forms of
energy storage are available, but the
proportionality between Eint and T remains
Root Mean Square Speed

The root mean square (rms) speed is
the square root of the average of the
squares of the speeds


Square, average, take the square root
Solving for vrms we find
v rms

___
2
3 kBT
3 RT
 v 

m
M
M is the molar mass in kg/mole
Some Example vrms Values
At a given
temperature,
lighter
molecules
move faster,
on the
average,
than heavier
molecules
Distribution of
Molecular Speeds

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The observed speed
distribution of gas
molecules in thermal
equilibrium is shown
NV is called the
Maxwell-Boltzmann
distribution function
Distribution Function

The fundamental expression that describes
the distribution of speeds in N gas molecules
is
3
 mo  2 2  mv 2 2kBT
NV  4 N 
 v e
 2 kBT 

mo is the mass of a gas molecule, kB is
Boltzmann’s constant and T is the absolute
temperature
Average and
Most Probable Speeds

The average speed is somewhat lower
than the rms speed
8kBT
kBT
v 
 1.60
 mo
mo

The most probable speed, vmp is the speed
at which the distribution curve reaches a
peak
2k T
kT
v mp 
B
mo
 1.41
B
mo
Speed Distribution
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The peak shifts to the right
as T increases

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This shows that the average
speed increases with
increasing temperature
The width of the curve
increases with temperature
The asymmetric shape
occurs because the lowest
possible speed is 0 and the
upper classical limit is
infinity
Speed Distribution, final
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v rms  v  v mp
The distribution of molecular speeds
depends both on the mass and on
temperature
The speed distribution for liquids is
similar to that of gasses
Evaporation


Some molecules in the liquid are more energetic
than others
Some of the faster moving molecules penetrate
the surface and leave the liquid



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This occurs even before the boiling point is reached
The molecules that escape are those that have
enough energy to overcome the attractive forces
of the molecules in the liquid phase
The molecules left behind have lower kinetic
energies
Therefore, evaporation is a cooling process
Atmosphere


For such a huge volume of gas as the
atmosphere, the assumption of a
uniform temperature throughout the gas
is not valid
There are variations in temperature


Over the surface of the Earth
At different heights in the atmosphere
Temperature and Height

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At each location, there is a
decrease in temperature
with an increase in height
As the height increases, the
pressure decreases
The air parcel does work on
its surroundings and its
energy decreases
The decrease in energy is
manifested as a decrease
in temperature
Lapse Rate



The atmospheric lapse rate is the
decrease in temperature with height
The lapse rate is similar at various
locations across the surface of the earth
The average global lapse rate is about –
6.5o C / km

This is for the area of the atmosphere
called the troposphere
Layers of the Atmosphere

Troposphere
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Tropopause

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The lower part of the atmosphere
Where weather occurs and airplanes fly
The imaginary boundary between the troposphere
and the next layer
Stratosphere


Layer above the tropopause
Temperature remains relatively constant with
height