14 MATTER: VERY HOT AND COLDAverage energy per particle

• Know how to compute average particle energy when temperature is known

• Describe processes such as boiling and evaporation in terms of energy exchange between particles

Temperatures everywhere….Find the temperature:· wood catches fire spontaneously· mercury becomes superconducting· methane is stored as a liquid for transport round the world· typical domestic freezer compartment· porcelain is fired in a kiln· helium nuclei first formed in the early Universe· the dark side of the Moon· the human body at which you should suspect a fever· inside meat at which the meat is judged 'done' when roasted in the oven

States of matter

Temperature

moretightlybound

Do not worry about the shape of the valleys: only the depth

Consider.....concentrating on a typical particle

one particle trappedin an energetic boxby its neighbours

but the particle is always vibrating -always attempting to escape from itsenergetic box

1. Particles continually attempt to break out of a trap made by their

neighbours.2. These breakouts succeed when a particle randomly acquires enough

energy to do so.3. The frequency of breakouts also

depends on the frequency of attempts to breakout.

So what is the link between states of matter and

temperature???

States of matter

Temperaturemore

tightlybound

Do not worry about the shape of the valleys: only the depth

Increasing temp – increases both average energy of particles and the frequency of breakout attempts!!!

What was the equation for the internal energy of a monoatomic gas?

U = 3/2kTWhere does the 3/2 come from?

The number of degrees of freedomWhat about a metal…. Do software activity

130S

As most particles have a small number of degrees of freedom we often ignore them

and just use kT – average energy per particle.

Physicists often measure the energy of charged particles in the unit electron volt

(eV). This is the work done when an electron is moved through a potential

difference of 1 volt.

Key SummaryConnections to kT

kT

multiply byNA

1000

6.02 × 1023

103particles mol–1

energy per mole of particleskJ mol–1

Average energy per particleJ

divide by e

1.6 × 10–19 C

average energy per particle in eV

divide by k

1.38 × 10–23 J K–1

absolute temperatureK

kT – average energy per particle

The “magic ratio”, ε/kT

Q1. Reviewing work from Chapter 13, state the expressions for the internal energy of an ideal gas, and for the internal energy of a metallic solid (individual molecule units).Why is the factor preceding the kT term different in each case?Why is kT a valid estimate of the average energy, even when the pre-factor may be a lot bigger than 1?

Q2. Using kT as a measure of the average internal energy of a particle, calculate its value at temperatures of 300 K and 1000 K in units of :

(a) Joules per particle(b) electron Volts per particle(c) kJ mol-1

Temperature associated with photons

Photons of electromagnetic radiation may be regarded as particles whose average energy varies with the temperature of the source producing them.

The curves on the right show how the distribution of energy from a hot object varies with temperature.

What is the relationship between peak wavelength and temperature?How does this explain the colour changesseen when a lump of iron is heatedstrongly?

The magic ratio and the Boltzmann factor

• Determine experimentally the specific energy per molecule required to evaporate water

• Combine vaporisation energy and average energy per molecule (kT) to determine the “magic ratio” for vaporisation

• From the magic ratio, calculate the Boltzmann factor for vaporisation of water, explaining its meaning

Given an electric kettle, a balance and a stopwatch, can you devise a method that will allow you to measure the energy needed to vaporise one mole of water? Hint: if a kettle is boiled with the lid off, the power switch will not trip to stop the boiling.

What is the “magic ratio”?When trying to determine if a process will happen at a particular temperature, we need to know:(a) The specific energy required for the process

to take place (for example the energy required to vaporise a mole of water)

(b) The average energy per particle at the temperature concerned (approximately equal to kT, or RT if molar units used)

The “magic ratio”, ε/kTQ1. Reviewing work from Chapter 13, state the expressions for the internal energy of an ideal gas, and for the internal energy of a metallic solid (individual molecule units).Why is the factor preceding the kT term different in each case?Why is kT a valid estimate of the average energy, even when the pre-factor may be a lot bigger than 1.

Q2. Using kT as a measure of the average internal energy of a particle, calculate its value at temperatures of 300 K and 1000 K in units of :

(a) Joules per particle(b) electron Volts per particle(c) kJ mol-1

Determining the magic ratio for the evaporation of water

Q1. What was the starting and finishing mass of the kettle, water and lid? What mass of water was evaporated? How many moles of water is this?Q2. What was the power rating of the kettle in Watts?Q3. How long was the water boiled for?Q4. How much energy was supplied to evaporate the water?Q5. Use your answers to Q4 and Q1 to work the energy needed

to evaporate one mole of water (ε).Q6. Calculate the average energy per mole (R=8.3 J K-1 mol-1)Q7. Use your answers to determine the magic ratio ε/RTQ8. Calculate the Boltzmann factor for evaporation of water at

its boiling point. Comment on your answer.

Independent learning....

Develop magic ratio concept by considering the different temperature regimes 300 K, 1500 K, 6000 K, 6 000 000 K, 4 K referring to p119-121.

For each case, note the value of kT in suitable units, and the energy required for a specific process.

For example, at 1500 K, kT = 10 kJ mol-1, which is comparable to hydrogen bond strengths (/kT = 2) but not ionic bonds (/kT = 40). Thus proteins etc. destroyed at 1500 K, but ceramics unaffected.

Origin of the Boltzmann factor

• Investigate the statistical origin of the Boltzmann factor

Starter:Q1. What happens to the magic ratio for a process as the temperature is increased? Q2. What value does the magic ratio tend towards with increasing T?Q3. How does a particle, initially with less energy than that required for a specific process, gain the energy needed for the process to happen?Q4. In a fixed time interval, are all particles in a sample equally likely to gain the required energy? Explain your answer.

The ergodic hypothesisIf you take a “snapshot” of a set of particles at one instant of time and look at the number of

particles with each energy value, you get exactly the same distribution as you get if you

followed one particle over a period of time and measured how long it spends in each

energy level. or, put simply,MANY PARTICLES AT

ONCE LOOKS THE SAME AS ONE

PARTICLE OVER A LONG PERIOD OF

TIME.

To investigate energy distribution in a system of particles, do Activity 170S and Activity 190S. In both models, particle gain and lose energy in a random manner by colliding with other particles. The probability of gaining energy in a collision is modelled with a simple random number generator.

Recording results from Activity 190SThe variable CHANCE can take values between 1 and 10.It represents throwing a die with number of sides = CHANCEIf the number thrown = 1, the particle gains energy and moves up to the next energy level.If any other value is thrown, the particle drops down an energy level.When CHANCE = 10, there is only a 1 in 10 chance of a particle gaining energy in each time step.When CHANCE = 1, there is a 1 in 2 chance of a particle gaining energy in each time step.Which situation corresponds to a system at high temperature?

Carry out Activity 190S for a specified value of CHANCE. When the simulation completes, use the measurement tool to record the lengths of the bars in the histogram in Animation 2. Work out the population ratios using pop2/pop1, pop3/pop2, pop4/pop3 etc.What do you find?

Climbing a ladder of energy by chance

One particle may acquire energy several timesin succession

With many particles, a fraction f acquire energy at each step of ladder

fraction f3

fraction f2

fraction f

gets lucky three times

gets lucky twice

gets lucky once energy

energy

energy

By chance, particles may get extra energy from the random thermal motion of particles in thesurroundings

Climbing a ladder of energy by chance

One particle may acquire energy several timesin succession

With many particles, a fraction f acquire energy at each step of ladder

fraction f3

fraction f2

fraction f

gets lucky three times

gets lucky twice

gets lucky once energy

energy

energy

By chance, particles may get extra energy from the random thermal motion of particles in thesurroundings

The Boltzmann factorThe Boltzmann factor (fB) is a measure of the fraction of particles in a system that have energies >= the “activation energy” (ε) at absolute (Kelvin) temperature T. It makes use of the “magic ratio”, the ratio of activation energy to average particle energy.

fB = e- ε/kT , where k is Boltzmann’s constant.

The term kT is a measure of the average energy per particle at temperature T.

The equation for fB can also be written in molar units, in which kT is replaced by RT.

For evaporation of a liquid, ε = 7 x 10-20 J. Calculate ε /kT and hence fB at 300 Kand 1000 K. Repeat the calculation for another liquid with ε = 7 x 10-20 J. Boltzmann constant k = 1.4 x 10-23 JK-1

Comment on your results in each case.

Key SummaryConnections to kT

kT

multiply byNA

1000

6.02 × 1023

103particles mol–1

energy per mole of particleskJ mol–1

Average energy per particleJ

divide by e

1.6 × 10–19 C

average energy per particle in eV

divide by k

1.38 × 10–23 J K–1

absolute temperatureK

kT – average energy per particle

Investigating an activated process

• Determine how the conductivity of a semiconductor varies with temperature

• Hence determine the “activation energy” for ionisation of semiconductor atoms

Starter: Write down expressions for the resistance and conductance of a conductor of length L and cross sectional area A.

Applications of the Boltzmann law

• Solve problems involving the Boltzmann law

• Explain the consequences of the Second Law of Thermodynamics

Starter: Write down the expression for the magic ratio, identifying what the terms in the numerator and denominator are. Write down the expression for the Boltzmann factor. Explain carefully what the numerical value of the Boltzmann factor means, in terms of a collection of particles AND in terms of a single particle.