chapter 11 ideal gases
TRANSCRIPT
L E A R N I N G O U T C O M E S
NUMBER LEARNING OUTCOME
i L e a r n a n e w S I b a s e q u a n t i t y , t h e a m o u n t o f
s u b s t a n c e .
ii Understand the difference between macroscopic and properties of a
substance. Know what is the meaning of state variables.
iii Understand the importance of Brownian motion to the kinetic model.
iv L e a r n t h e i d e a l g a s e q u a t i o n a n d u s e i t . U n d e r s t a n d
w h a t i d e a l g a s e s a r e .
v W h a t i s i n t e r n a l e n e r g y ?
vi L e a r n t h e a s s u m p t i o n s o f t h e k i n e t i c t h e o r y f o r
i d e a l g a s e s a n d l e a r n h o w g a s e s e x e r t p r e s s u r e .
vi D e r i v e e q u a t i o n s t h a t r e l a t e a m a c r o s c o p i c p r o p e r t y ,
t e m p e r a t u r e a n d a m i c r o s c o p i c o n e , a v e r a g e s p e e d .
THE MOLETHE MOLE• The amount of substance is one of the 6• The amount of substance is one of the 6
S.I. base quantities.
• Units of amount of substance is the mole(symbol = mol).
• 1 mol of a substance is defined as theamount of that substance that has aequal number of elementary particles tothe number of atoms in 0.012 kg ofcarbon -12.
THE MOLETHE MOLE• Avogadro’s constant is equal to the number
�
• Avogadro’s constant is equal to the numberof atoms in 0.012 kg of carbon – 12.
• The Avogadro’s constant, �� is equal to�. ��� ���.
• The amount of matter (number of moles), n(in mol) can be calculated from the mass ofsubstance, m (in g) by using � �
�, where M
= molar mass (in g / mol).
• The molar mass, M of a substance is the mass(in g) of 1 mol of a substance.
THE MOLETHE MOLE• We can use Avogadro’s constant• We can use Avogadro’s constant
and the number of moles to findthe number of elementaryparticles present in a sample of asubstance.
• How? Use � ,wherethe number of elementary
particles.
M I C R O S C O P I C v s .
M A C R O S C O P I C
M I C R O S C O P I C v s .
M A C R O S C O P I C
• Substances (solid, liquids or gases) are
is made up of
• Substances (solid, liquids or gases) are made up of the elementary units of the substance.
• For example, gaseous ��� is made up of elementary ��� molecules.
• Microscopic properties are properties ofthe elementary particles that make upthe substance.
M I C R O S C O P I C v s .
M A C R O S C O P I C
M I C R O S C O P I C v s .
M A C R O S C O P I C
• For example, a sample of gaseous ��� would• For example, a sample of gaseous ��� wouldhave molecules, and each of its moleculeswould have momentum, velocities (or speeds),mass and kinetic energy.
• It is difficult to measure the microscopicproperties of all the elementary particles in asubstance due to the large number ofelementary particles that make up thesubstance.
M I C R O S C O P I C v s .
M A C R O S C O P I C
M I C R O S C O P I C v s .
M A C R O S C O P I C
Macroscopic properties are properties
specific temperature,
• Macroscopic properties are properties of the substance of the whole.
• For example, a sample of gaseous ���would have a specific temperature, pressure, volume, mass, density and number of moles.
• Macroscopic properties define the state the of the substance.
STATE VARIABLESSTATE VARIABLES• The state variables are the variables• The state variables are the variables
that define the state of a matter.
• The state variables we will encounter arepressure, temperature, volume, densityand amount of substance.
• State variables are related to oneanother via an equation of state.
BROWNIAN MOTION
• Robert Brown, an English botanist, put forward his• Robert Brown, an English botanist, put forward hisobservation of tiny pollen grains floating on waterundergoing a constant , random , haphazard motion,even though the water appeared still when viewed underthe microscope.
• This movement is now known as Brownian motion.
• This motion is only possible if the water molecules are in astate of rapid and random motion. These watermolecules randomly collide with the pollen grains fromall directions causing the pollen grains to experience thisBrownian motion.
BROWNIAN MOTION BROWNIAN MOTION
Figure 12a, Chapter 17 : Atoms, Molecules and Atomic Processes, page 6; PHYSICS
2000 ; E.R. HUGGINS; Moose Mountain Digital Press, New Hampshire 2000.
Diagram shows
how a simple set
up can be used to
show Brownian
motion.Laser beam can be
replaced by another
coherent source of
light.
BROWNIAN MOTION
Figure 6.14: Observing Brownian Motion, Page 148, Chapter 6: Thermal Physics ,
International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder
Education, United Kingdom, 2008.
T H E I D E A L G A S E Q UAT I O N
If this hypothetical experiment was
carried out using a gas, what would
be observed are:
I. The volume of the gas, V, is
directly proportional to the
amount (number of moles) of
the gas, n (if temperature, T and
pressure, p are kept constant).
II. The pressure of the gas, p is
inversely proportional to the
volume, V of the gas (for
constant T and n).
Figure 18.1, page 591: Chapter 18: THERMAL PROPERTIES OF MATTER; SEARS AND ZEMANSKY’S
UNIVERSITY PHYSICS WITH MODERN PHYSICS; Young, Hugh D. and Freedman, Roger A., Addison Wesley,
San Francisco, 2012.
T H E I D E A L G A S E Q UAT I O N
III. The pressure, p of the gas is
directly proportional to the
thermodynamic temperature
of the gas, T, for constant V
and n.
• The thermodynamic
temperature of a gas is the
temperature of the gas
expressed in Kelvin.
• For conversion, use
� ��� � � ��� �
��. �
Figure 18.1, page 591: Chapter 18: THERMAL PROPERTIES OF MATTER; SEARS AND ZEMANSKY’S
UNIVERSITY PHYSICS WITH MODERN PHYSICS; Young, Hugh D. and Freedman, Roger A., Addison Wesley,
San Francisco, 2012.
T H E I D E A L G A S E Q UAT I O N
• All of those previous
relationships can be summed
up into one equation
�� � ���
• This equation is known as the
ideal gas equation
• �, the proportionality
constant, is the universal gas
constant
� �. �� !"�"
• What are the units of p, V, n
and T?
Figure 18.1, page 591: Chapter 18: THERMAL PROPERTIES OF MATTER; SEARS AND ZEMANSKY’S
UNIVERSITY PHYSICS WITH MODERN PHYSICS; Young, Hugh D. and Freedman, Roger A., Addison Wesley,
San Francisco, 2012.
IDEAL GASESIDEAL GASES
• Ideal gases are gases that precisely
obey the ideal gas equation at all
temperatures, volumes and
pressures.
• Real gases obey this law only at low pressures
and high temperatures, when they are furthest
apart and moving the fastest. However, we can
use this equation for approximate calculations.
T H E I D E A L G A S E Q UAT I O N
• T h e i d e a l g a s e q u a t i o n c a n b e• T h e i d e a l g a s e q u a t i o n c a n b em a n i p u l a t e d t o o b t a i n o t h e rf o r m s o f i t .
�� � #�$ �
��� , o r
% ���
��, w h e r e % � d e n s i t y ,
o r ,� �
� �
� �� �
� �w h e r e
t h e s u b s c r i p t s 1 a n d 2r e p r e s e n t d i f f e r e n t s t a t e s .
T H E I D E A L G A S E Q UAT I O N
� ��
�• Recall that � �
�
��.Hence,
�� � ��� ��
���� � �
�
���, or
�� � �&� , where & � �
��� . � �
�"��/�
• k is known as the Boltzmann constant.
E X A M P L E S
Example 18.1, page 593: Chapter 18: THERMAL PROPERTIES OF MATTER; SEARS AND
ZEMANSKY’S UNIVERSITY PHYSICS WITH MODERN PHYSICS; Young, Hugh D. and
Freedman, Roger A., Addison Wesley, San Francisco, 2012.
E X A M P L E S
Example; Section 6.3 “The Gas Laws”, Page 145, Chapter 6: Thermal Physics ,
International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder
Education, United Kingdom, 2008.
E X A M P L E S
Exercises; Section 6.3 “The Gas Laws”, Page 146, Chapter 6: Thermal Physics ,
International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder
Education, United Kingdom, 2008.
E X A M P L E S
Exercises; Section 6.3 “The Gas Laws”, Page 146, Chapter 6: Thermal Physics ,
International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder
Education, United Kingdom, 2008.
INTERNAL ENERGYE a c h o f t h e g a s m o l e c u l e h a s• E a c h o f t h e g a s m o l e c u l e h a sa c e r t a i n a m o u n t o f e n e r g y.
• T h i s e n e r g y i s t h e k i n e t i ce n e r g y, a s s o c i a t e d w i t h i t sm o v e m e n t , a n d t h e p o t e n t i a le n e r g y d u e t o t h e f o r c e st h a t e x i s t b e t w e e n t h e g a sm o l e c u l e s .
INTERNAL ENERGYI n a d d i t i o n a l l t h e m o l e c u l e s• I n a d d i t i o n a l l t h e m o l e c u l e sw i l l h a v e d i f f e r e n t k i n e t i ce n e r g i e s a s s o m e a r e m o v i n gf a s t e r a n d s o m e s l o w e r a n da l s o d i f f e r e n t a m o u n t o fp o t e n t i a l e n e r g i e s a s t h i se n e r g y i s d e p e n d e n t o n t h ep o s i t i o n o f t h e m o l e c u l e i n ag i v e n c o n t a i n e r.
INTERNAL ENERGY
• W e c a n n o w s a y t h a t t h e
k i n e t i c e n e r g i e s o f e a c h o f
t h e m o l e c u l e s a r e r a n d o m l y
d i s t r i b u t e d , a n d t h e
p o t e n t i a l e n e r g y o f e a c h o f
t h e m o l e c u l e a l s o f o l l o w s a
r a n d o m d i s t r i b u t i o n .
INTERNAL ENERGYW h e n w e a d d t h e k i n e t i c• W h e n w e a d d t h e k i n e t i ce n e r g i e s a n d p o t e n t i a le n e r g i e s o f a l l t h e g a sm o l e c u l e s , w e r e m o v e t h er a n d o m n a t u r e o f t h ee n e r g i e s .
• W h a t w e g e t i s k n o w n a s t h ei n t e r n a l e n e r g y o f t h e g a s .
INTERNAL ENERGY
• D e f i n i t i o n : “ T h e i n t e r n a l
e n e r g y o f a s u b s t a n c e i s t h e
s u m o f t h e r a n d o m
d i s t r i b u t i o n o f k i n e t i c a n d
p o t e n t i a l e n e r g i e s o f a l l t h e
m o l e c u l e s a s s o c i a t e d w i t h
t h e s y s t e m .”
K I N E T I C T H E O R Y O F I D E A L
G A S E S
K I N E T I C T H E O R Y O F I D E A L
G A S E S• We also know that samples of gases• We also know that samples of gases
have macroscopic and microscopicpropert ies .
• We can relate the macroscopic andmicroscopic propert ies of gases.However, we must make someassumptions about the atoms/molecules of the gases f irst .
• These assumptions are known as thekinetic theory of ideal gases .
K I N E T I C T H E O R Y O F I D E A L
G A S E S
K I N E T I C T H E O R Y O F I D E A L
G A S E S
• The assumptions are:
Section 6.4 “A microscopic model of a gas”, Page 149, Chapter 6: Thermal Physics ,
International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder
Education, United Kingdom, 2008.
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S
• Ideal gas particles / molecules also exert
pressure on the inner walls of the
container.
• Question: How and why?
• We can use the kinetic theory of ideal
gases to explain this.
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S USING SIMPLE KINETIC MODEL TO EXPLAIN PRESSURE EXERTED
BY GASES
• Gas particles / molecules are in a state of continuous, random motion.
• As a result, the gas particles / molecules are constantly involved in
collisions with the inner walls of the container
• The collisions are assumed to be elastic.
• The momentum of the colliding particles / molecules change and the
particles / molecules have lower momentum after collision.
• This change of momentum over a short time produces a force acting on
the particular area of the surface of the container.
• This force acting per unit of area is the pressure that is exerted by the gas
particles on the inner surface of the container.
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S We will try to derive an equation for• We will try to derive an equation forthe pressure exerted by gas moleculeson the walls of a cubic container oflength = L.
• We assume that all molecules have thesame speed in the x - direction, bothbefore and after collision.
• The mass of each gas molecule = m
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S • The change in momentum
of the one of the gas
molecules = 2)*+• The time taken for one gas
molecule to move from
one wall to the other and
back , t ��-
./
• Hence, the force exerted
by one gas molecule on
the wall, 0 � 1./
2
-
Figure 19.14, page 614: Chapter 19: THE IDEAL GAS; Physics for Scientists and Engineers ,
Volume 1; 3rd edition Ohanian, Hans C. and Markert, John T., W.W Norton and Company Inc,
New York, 2007.
*+
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S • We assume there are N
gas molecules, hence
034356 � 71./
2
-
• The pressure exerted by
the gas molecules on one
face of the wall,
8 � 9:;:<=
-2�
>1./2
-?�
>1./
2
@
*+
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S • We assumed all molecules
have the same speed, but
some molecules move
slower, others faster.
Hence use the average
speed, *+�
• Our equation now
becomes 8 � >1 ./
2
@
• The molecules are also
equally likely to move in
the x, y and z directions.
*+
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S
• We obtain � � � A�
�• We can rearrange the
above equation to yield
�� � � A�
*+
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S
8B � 7C$
8B �
• We have earlier shown that 8B � 7C$
and now we have obtained 8B �
>1 .2
D
• E& �
� A� where E& � the
average kinetic energy of a gas
molecule.
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S
8B � �
D7
F
�) *� �
�
D7 GH
8B � 7C$
• But 8B � �
D7
F
�) *� �
�
D7 GH .
• We already know that 8B � 7C$
• Hence, �
� E& � �&�, or E& �
�&�
• We now have two equations to help us
calculate the average kinetic energy of a gas
molecule.
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S
8∆B
Some analysis of the equation(s):
• 8∆B actually gives us the work done on or by the gas. This work done will change the kinetic energy of an ideal gas, but not the potential energy of an ideal gas.
• In other words, when we change the internal energy of an ideal gas, we are changing only the average kinetic energy of each molecule
• We can use p∆B � �
D7∆ GH to calculate the change
in the average kinetic energy of an ideal gas molecule.
K I N E T I C T H EO RY O F
I D EA L G A S E S
K I N E T I C T H EO RY O F
I D EA L G A S E S
G � F) * �
DC$
Some analysis of the equation(s):
• From GH �F
�) *� �
D
�C$, we can get
*� �DHK
1, and *� �
DHK
1, where
A� �root – mean – square speed.
E X A M P L E S
Example; Section 6.4 “A Microscopic Model of a Gas”, Page 152, Chapter 6: Thermal
Physics , International A/AS Level Physics, by Mee, Crundle, Arnold and Brown,
Hodder Education, United Kingdom, 2008.
E X A M P L E S
Exercises; Section 6.4 “A Microscopic Model of a Gas”, Page 152, Chapter 6: Thermal
Physics , International A/AS Level Physics, by Mee, Crundle, Arnold and Brown,
Hodder Education, United Kingdom, 2008.