14 MATTER: VERY HOT AND COLD Average energy per particle

1. 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

2. 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

3. States of matterTemperature

4. Consider..... So what is the link between states of matter and temperature??? 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.

5. States of matterTemperature Increasing temp – increases both average energy of particles and the frequency of breakout attempts!!!

6. What was the equation for the internal energy of a monoatomic gas? U = 3/2kT Where does the 3/2 come from? The number of degrees of freedom What 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.

7. 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.

8. Key Summary kT – average energy per particle

9. 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

10. 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 changes seen when a lump of iron is heated strongly?

11. 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.

12. What is the “magic ratio”? When trying to determine if a process will happen at a particular temperature, we need to know: • The specific energy required for the process to take place (for example the energy required to vaporise a mole of water) • The average energy per particle at the temperature concerned (approximately equal to kT, or RT if molar units used)

13. 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

14. 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/RT Q8. Calculate the Boltzmann factor for evaporation of water at its boiling point. Comment on your answer.

15. 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.

16. 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. • Investigate the statistical origin of the Boltzmann factor

17. The ergodic hypothesis If 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.

18. 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 190S The variable CHANCE can take values between 1 and 10. It represents throwing a die with number of sides = CHANCE If 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?

21. The Boltzmann factor The 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 fBat 300 K and 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.

22. Key Summary kT – average energy per particle

23. 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.

24. 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.