Kinetic theory & the behaviour of gases

Thursday 25th March

Learning outcomesuse a particle model to describe solids, liquids, gases & changes of stateexplain gas pressure and thermal expansion in terms of kinetic theorydescribe how a barometer measures atmospheric pressure estimate the height of the atmosphere using a physical modelrecall and use the gas laws to make quantitative predictionsrelate the gas laws & Absolute Zero of temperature to the behaviour of ideal gasesconvert temperatures between Kelvin, Celsius and other scales introduce microscopic atoms and molecules through reasoning based on careful observation of macroscopic behaviour

How do we know?

What evidence is there for the existence of atoms?

Evidence for atoms

crystals – regularity of surfaces, cleaving

mixing different liquids

change of volume: solid -> gas, liquid -> gas

air occupies space and has mass

diffusion: solid into solid, solid into liquid, gas into gas

Brownian motion

All things are made of atoms

“If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words?

“I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms - little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.

“In that sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.” Richard Feynman

Starting points - teaching challenges

Atoms and molecules are far too small to be glimpsed

by even the most highly-powered optical

microscope.

Diagrams of particle arrangements are often static.Dynamic animations showing the random thermal motion of

particles, at all temperatures and in all states of matter,

usefully overcome the misconceptions which static

diagrams can foster.

Students find it difficultto appreciate that gas pressure acts equally in all directions

to account for the consequences of pressure differences

to convert between units of volume (cm3, m3)

Starting points - mmisconceptions

Students often think that particles

– have the properties of bulk matter (particles change in size as

the temperature changes, particles can melt and solidify)

– have air in between them

– have thoughts and intentions e.g. ‘they prefer to move to

places that are less crowded’.

There is much confusion about the nature of particle

motion in solids, liquids and gases.

Many states of mattersuperfluids, liquid crystals, solid solutions, plasmas, aerogels, foams, thin films, colloids, immiscible liquid mixtures, gas dissolved in a liquid, condensed matter, biopolymers…..

… solid, liquid, gas

An ideal gas

huge number of point molecules (occupy negligible volume) in continual

random motion (and so ‘kinetic’)

colliding elastically with each other and with container walls

no forces between the molecules, except in collision

time in collisions very small compared to time between collisions

distance travelled between collisions (‘mean free path’) depends on gas

density

average speed of molecules depends on gas temperature

in a gas composed of different molecules, the average molecular Ek is the

same for all, so those with larger mass have smaller speed

Gases: bulk properties

quantity symbol SI unit

pressure p Pa (Nm-2)

volume V m3

temperature T kelvin, K

density kgm-3

pressure = force applied over a unit area.

SPT Forces, episode 8 Pressure

A

Fp

Gas pressure

bombardment of

the container walls

change of momentum

with each collision

mv - (-mv) = 2mv t

vm

t

mvF

)(

Deriving gas pressure relationshipNewton's Laws and Collisions Applying Newton's Laws to an ideal gas under the assumptions of kinetic theory allows the determination of the average force on container walls. This treatment assumes that the collisions with the walls are

perfectly elastic.

Under the assumptions of kinetic theory, the average force on container walls has been determined to be

The average force and pressure on a given wall depends only upon the components of velocity toward that wall. But it can be expressed in terms of the average of the entire translational kinetic energy using the assumption that the molecular motion is random.and assuming random speeds in all directions

Then the pressure in a container can be expressed as

Expressed in terms of average molecular kinetic energy:

This leads to a concept of kinetic temperature and to the ideal gas law.

The expression for gas pressure developed from kinetic theory relates pressure and volume to the average molecular kinetic energy. Comparison with the ideal gas law leads to an expression for temperature sometimes referred to as the kinetic temperature.

This leads to the expression

The more familiar form expresses the average molecular kinetic energy:

It is important to note that the average kinetic energy used here is limited to the translational kinetic energy of the molecules. That is, they are treated as point masses and no account is made of internal degrees of freedom such as molecular rotation and vibration. This distinction becomes quite important when you deal with subjects like the specific heats of gases. When you try to assess specific heat, you must account for all the energy possessed by the molecules, and the temperature as ordinarily measured does not account for molecular rotation and vibration. The kinetic temperature is the variable needed for subjects like heat transfer, because it is the translational kinetic energy which leads to energy transfer from a hot area (larger kinetic temperature, higher molecular speeds) to a cold area (lower molecular speeds) in direct collisional transfer.

The speed of gas moleculesWays of estimating an average speed in airfrom the speed of sound (340 ms-1 at s.t.p.)thought experiment: a molecule falls freely from the top of the

atmosphere

Direct measurement: Zartman (1931) experiment to find the distribution of molecular speeds in a beam emitted from an oven opening. Average speed N2 at room temperature ~ 500 ms-1

asv

u

asuv

2

0

222

The size of a molecule

oil film experiment.

Devised by John William Strutt, Lord Rayleigh, who also

explained why the sky looks blue, and many other things!

Nobel Prize-winner 1904.

The size of atoms

‘If an apple were magnified to the size of Earth, the atoms in it would each be about the size of a regular apple.’ Richard Feynman

AFM showing atoms within hexagonal

graphite unit cells. Image size 2 nm × 2 nm.

An empirical lawRelationship between pressure and volume at constant temperature

re-plot to show inverse proportionality

Boyle’s law: pV = constant

Another Law – Extrapolating from data

Another empirical law…

Charles’ law: TV

In oC, a linear relationship.

Direct proportionality if temperature scale is redefined. (mjp)

T in kelvins, whereK = oC - 273

Relationship between temperature and volume at constant pressure

Other gas laws

pressure law: (T in kelvins)

All 3 relationships combined:

where n is number of moles, gas constant R = 8.31 J K-1

In a possibly more useful or meaningful form

Tp nR

T

pVconstant

2

22

1

11

T

Vp

T

Vp

Air pressure.

http://www.youtube.com/watch?v=IRPvvJA8I_8

Setting up a water barometer – well worth a go!

Or…

http://www.rmets.org/weather/observing/make-barometer.phphttp://www.home-weather-stations-guide.com/make-your-own-barometer.html

http://www.practicalphysics.org/go/Experiment_883.html

Height of the atmosphere

weight of a column

pressure of a column

pressure of air and water columns are equal, so

VgmgW

ghA

Ahg

A

Vg

A

Fp

wa

wa

wwaa

hh

hh

Hydraulic machines

Hydraulic machines exploit these facts:pressure is the same throughout a fluid (at same height).

liquids are incompressible.

How do hydraulic systems work?

Kinetic theory – a chronologyc.420BC – atomic theory (Democritus: matter ultimately uncut-able)

1662 – Boyle’s law

1738 – Bernoulli Hydrodynamica (molecular collisions -> gas

pressure)

1787 – Charles’ law

1798 – atomic theory of heat

1827 – Brownian motion

1834 – ideal gas law

1849 – kinetic theory

k = 1.38 x 10-23 J K-1 (Boltzmann constant)

kTvm2

3

2

1

cmacroscopicmicroscopi

2

Gas properties

PhET simulation Gas propertiesPump gas molecules to a box and see what happens as you change the volume, add or remove heat, change gravity, and more. Measure the temperature and pressure, and discover how the properties of the gas vary in relation to each other.

Phase diagrams

A phase diagram (p - T) shows boundaries between phases of matter. – At the triple point, all 3 phases

co-exist. – Beyond the critical point, there

is no distinction between gas and liquid phases.

PhET: phase change simulation

Thermometer Scales

Upper fixed pointLower fixed pointFundamental interval

Linear, non linear, calibration and ranges