MATTER: VERY SIMPLE

Chapter 13

Maxwell1831-1879

Boltzmann1844-1906

Amadeo Avogadro (1776-1856)

1818-1889

kTnNNkTnRTpV A

Equations

2

3

1vNmpV

kTE2

3 pVnRTNkTU

2

3

2

3

2

3

Revision Task

Use the paper in front of you Summarize everything you can

remember about Gases from before summer

For example: Equations, definitions, scientists etc…

One law for all gases

2

34

1

volume V

Boyle’s law

pressure p

compress gas:pressure p increasesconstant temperature T

2

34

1

number N

Amount law

pressure p

add more molecules:pressure p increasesconstant temperature T

2

34

1

Pressure law

pressure p

heat gas:pressure p increasesconstant volume V

T/K

45.1

volume V

Charles’ law

heat gas:volume V increasesconstant pressure p

T/K

45.1

p1/V

pN

pT

VT

Combine therelationships into one

pN/V

pVNor

pVintroduceconstant k:

pVNkT

combine:

combine:

Combine unknown Nand k into measurablequantity R

Number of moleculesN not known

Nk can be measured:Nk = pV/T

For one mole, defineR = NAk

For n moles:pV = nRT

pVT

k = Boltzmann constantNA = Avogadro number(number of molecules per mole)R = molar gas constant = 8.31 J K –1 mol–1

measured from pV/T for one mole

When NA could be measured:

Avogadro number NA = 6.02 1023 particles mol–1

R = molar gas constant = NAk = 8.31 J K–1 mol–1

Boltzmann constant k = 1.38 10–23 J K –1 mol–1

combine:

4

5

0

1

2

3

105

N m–2

Kinetic Theory

A stripped down view of the kinetic theory

general ideas ideas about a particle in a box

F = – pt

force equal torate of changeof momentum

p = – 2mv

t = 2cv

F = mv2

c

P = mv2

abc

P = mv2

V

P = Nmv2

V

P = Nmv2

V13

P = Nmv2

V13

particle mass, m ,speed, v, travelling2c between hits

A = a×b

box dimensions:a×b×c

V = a×b×c

P = FA

definition ofpressure

N particles

13

in each direction

randommovement

gas molecules movingat many speeds,special average, v2

Speeds of Molecules

Kinetic energy of gas molecules and the Boltzmann constant

compare these

Kinetic model

pV = Nmv2

Gas laws

pV = NkT

One molecule Many molecules

mv2 = kT12

32

kinetic energy permolecule = kT3

2

Nmv2 = NkT12

total kinetic energy ofmolecules = NkT3

2

32

mv2 = 3kT

total kinetic energy ofone mole ofmolecules U = RT

32

Internal kinetic energyof m olecu les o f onem ole a t T = 300 K

Boltzmann constant k

random thermalenergy of onemolecule is of orderkT

13

U = RT32

U = 3 .7 kJ m ol–1

for one moleN = NAR = NAkR = 8.31 J K–1 mol–1

Speed of a nitrogen molecule

Assume warm room temperature T = 300 K

mass of 1 mole of N2 = 28 10–3 kg mol–1

Avogadro constant NA = 6 1023 particles mol–1

Boltzmann constant k = 1.38 10–23 J K–1

kinetic energy of a molecule

from dynamics from kinetic model

mv212

kT32v2 = 3 kT

m

mass m of N2 molecule calculate speed

v = 500 m s–1 approximately

Air molecules (mostly nitrogen) at room temperature go as fast as bullets

m = mass of 1 mole of N2

Avogadro constant NA

m =

m = 4.7 10–26 kg

v2 = 3 1.4 10–23 J K–1 300 K

4.7 10–26 kg

v2 = 2.7 105 J kg–1 [same as (m s–1)2]28 10–3 kg mol–1

6 1023 mol–1

Hints for 90S1. A histogram is a bar chart where the area of a particular bar is related to

the total number in that range. As the range intervals are the same here this is not a problem.

2. Which is most common?

3. Work out the average of (100 + 100 + 100 + 100 + 100 + 150 + 150 + 150 + 150 + 150 + 150 + 150 + … + 600 + 600 + 600) / 150. This method is tedious and error-prone. Is there a quicker way?

4. Square each speed, multiply by the number of molecules with that speed, add all 11 categories up and divide by the total number of molecules. Don’t forget to take the square root of the total at the end.

10. The ideal equation is:

Density = mass / volume so eliminate the volume term.

13. You should show that the faster hydrogen and helium molecules travel at speeds comparable with the escape speed. What happens to these faster molecules? What happens to the slower ones that remain?

Energy and Temperature

Transfers of energy to molecules in two ways

Hit the molecules yourself

molecules speeded up piston pushed in

work done = force distance

Let other molecules hit them

cool gas or other material hot wall

thermal transfer = mc

when both ways are used:

Engine designers arrange to transfer energy by way of heating and by way of doing work

change in internal energy U

work done W plus thermal transfer Q=

H ere the p is ton strikes m olecu les and g ives extram om entum and so extra k inetic energy.

energy transferred = work W done

H ere the m olecu les in the hot w a ll h it o ther m olecu leshard and on average g ive them extra k ine tic energy.

energy transferred = energy Q transferred thermally

U = W + Q

Energy per particle

Looking at the kinetic energy of a single particle

Starting points

From the kinetic theory From experimental work with gases

P: pressure of the gasV: volume of the gasN: number of moleculesm: mass of one molecule

v2: mean square velocity

P: pressure of the gasV: volume of the gasn: number of molesT: temperature of gasR: universal gas constant

Note similaritiesrearrange

P = 3Nmv2

V1

PV = 3 Nmv21

PV = 2N

( 2 mv2 )13

rearrange

PV = nRT

Ek = 1

mv2

2

PV = 2N

( 2 mv2 )13

N

PV (mol–1) = 2 (Ek)3

Work with one mole

PV (mol–1) = RT

n

(Ek) = RT23rearrange

Ek = RT3

2

Ek = kT3

2

define, k, Boltzmann’s constant =RN

Boltzmann’s constant,k = 1.38 10–23 J K–1

For onemolecule

This is the energy for onemolecule, no matter whatit is, depending only ontemperature, and auniversal constant.

Something measured aboutthe Universe, turning out tobe important

.. ..