Chapter 10 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 microscopic

properties of a substance. Know what is the meaning of state variables.

iii 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 .

iv Understand the importance of Brownian motion to the kinetic model.

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 .

vii 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 S.I. base quantities.

Units of amount of substance is the

mole (symbol = mol).

THE MOLETHE MOLE

1 mol of a substance is defined as

the amount of that substance that

has a equal number of particles to

the number of atoms in 0.012 kg of

carbon -12.

THE MOLETHE MOLE Avogadros constant is equal to

Avogadros constant is equal tothe number of atoms in 0.012 kgof carbon 12.

0.012 kg of carbon 12 has 6.022 1023 atoms.

Hence, Avogadros constant, is equal to .

THE MOLETHE MOLE Hence, 1 mole of a substance is the

Hence, 1 mole of a substance is theamount of substance that has anumber of particles equal to .

The number of moles, , can becalculated from the mass of substance,m by using the equation

, where

M = molar mass of the substance.

THE MOLETHE MOLE

The molar mass, M of a

substance is the mass (in g) of 1

mol of a substance.

THE MOLETHE MOLE We can use Avogadros constant We can use Avogadros constant

and the number of moles to findthe number of particles presentin a sample of a substance.

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

molecules.

Microscopic properties are properties ofthe elementary particles that make upthe substance.





For example, a sample of gaseous For example, a sample of gaseous would have molecules, and each of itsmolecules would have momentum, velocity(or speed), mass and kinetic energy.

It is difficult to measure the microscopicproperties of all the elementary particles ina substance due to the large number ofparticles that make up the substance.





Macroscopic properties are properties Macroscopic properties are properties of the substance of the whole.

For example, a sample of gaseous

would have a 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.

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

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 ZEMANSKYS

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.

IV. 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 ZEMANSKYS

UNIVERSITY PHYSICS WITH MODERN PHYSICS; Young, Hugh D. and Freedman, Roger A., Addison Wesley,

San Francisco, 2012.


.

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?


The ideal gas equation can be

The ideal gas equation can bemanipulated to obtain otherforms of it.

I. pV #$

, or

II. &'

(), where * density, or,

III.

where the subscripts 1 and

2 represent different states.


Recall that

.Hence,

o +, #$

, or

o -,

o 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

ZEMANSKYS 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 ,



INTERNAL ENERGY

Each of the gas molecule has a certain

amount of energy.

This energy is the kinetic energy,

associated with its movement, and the

potential energy due to the forces that

exist between the gas molecules.

INTERNAL ENERGY

In addition all the molecules will have

different kinetic energies as some are

moving faster and some slower and also

different amount of potential energies

as this energy is dependent on the

position of the molecule in a given

container.

INTERNAL ENERGY

We can now say that the kinetic energies

of each of the molecules are randomly

distributed, and the potential energy of

each of the molecule also follows a

random distribution.

INTERNAL ENERGY

When we add the kinetic energies and

potential energies of all the gas

molecules, we remove the random

nature of the energies.

What we get is known as the internal

energy of the gas.

INTERNAL ENERGY

Definition: The internal energy of a

substance is the sum of the random

distribution of kinetic and potential

energies of all the molecules associated

with the system.

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 ,



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 have

macroscopic and microscopic properties.

We can relate the macroscopic and

microscopic properties of gases.

However, we must make some

assumptions about the atoms/

molecules of the gases first.


G A S E S


G A S E S

These assumptions are known as the kinetic

theory of ideal gases.

We will look at these assumptions first, and

then derive a very important relationship

between a microscopic property, 1 2 3

(average kinetic energy of one molecule)

with a macroscopic (state) property , T

(temperature).


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

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.

45


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 = 2745 The time taken for one gas molecule to

move from one wall to the other and

back , t 9

:;

Hence, the force exerted by one gas

molecule on the wall, < =:;

>

9


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 ?@@A BC

D

The pressure exerted by the gas

molecules on one face of the wall,

?@@A

D

BC

D

BC


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 square of the speeds, 45

Our equation now becomes + E= :;

>

F


I D EA L G A S E S


I D EA L G A S E S

The molecules are also equally likely to move

in the x, y and z directions.

We obtain B

We can rearrange the above equation to yield

B


I D EA L G A S E S


I D EA L G A S E S

The equation above is one equation that

relates the macroscopic properties of +, ,

with microscopic properties of 7,1 4 3.

We will now move on to relate the

macroscopic quantity of with 2.

This will be done by using the equations

- and B

.


I D EA L G A S E S


I D EA L G A S E S

Recall 2-

B where 2-

the average kinetic energy of a gas

molecule.

Hence, +,

HI

J

7 4

HI KL .


I D EA L G A S E S


I D EA L G A S E S

We already know that +, IM$

Hence,

2- -, or 2-

-

We now have an equation to help us

calculate the average kinetic energy

of a gas molecule by just knowing its

temperature.


I D EA L G A S E S


I D EA L G A S E S

+,

Some analysis of the equation(s):

+, 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


I D EA L G A S E S


I D EA L G A S E S

p, I K

Some analysis of the equation(s) (contd):

We can use p,

HI KL 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

K J7 4

HM$

Some analysis of the equation(s) (contd):

From KL J

7 4

H

M$, we can get

4 HLO

=, and 4

HLO

=, where

B root mean squared (r.m.s.) of

the 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 (contd).

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 (contd).

H O M E W O R K

1. Winter 2008, Paper 4, question 5.1. Winter 2008, Paper 4, question 5.

2. Winter 2009, Paper 41, question 2.

3. Summer 2010, Paper 41, question 2.






H O M E W O R K


10.Winter 2012, Paper 43, question 1.