Chapter 11 Ideal Gases

CAMBRIDGE A – LEVEL

PHYSICS

IDEAL GASES

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



• 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.





• 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.





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.


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

� ��

��. �





• 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?




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 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 .


� ��

�• 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 ,



E X A M P L E S

Exercises; Section 6.3 “The Gas Laws”, Page 146, Chapter 6: Thermal Physics ,



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


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 .


G A S E S


G A S E S

• The assumptions are:

Section 6.4 “A microscopic model of a gas”, Page 149, Chapter 6: Thermal Physics ,



K I N E T I C T H EO RY O F

I D EA L G A S E S


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.


I D EA L G A S E S


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.


I D EA L G A S E S


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


I D EA L G A S E S


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.

*+


I D EA L G A S E S


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

@

*+


I D EA L G A S E S


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.

*+


I D EA L G A S E S


I D EA L G A S E S

• We obtain � � � A�

�• We can rearrange the

above equation to yield

�� A�

*+


I D EA L G A S E S


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.


I D EA L G A S E S


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.


I D EA L G A S E S


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.


I D EA L G A S E S


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.

E X A M P L E S

May/June 2008 Paper 4, Question 2.

E X A M P L E S

May/June 2008 Paper 4, Question 2 (cont’d).

E X A M P L E S

May/June 2009 Paper 4, Question 2.

E X A M P L E S

May/June 2009 Paper 4, Question 2 (cont’d).

H O M E W O R K

1. Winter 2008, Paper 4, question 5.


3. Summer 2010, Paper 41, question 2.





Chapter 11 Ideal Gases

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