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Advanced level chemistry kinetics notes: Consequences of the Maxwell-Boltzmann distribution

Doc Brown's Advanced A Level Chemistry

Kinetics-Rates Part 5

In previous courses 'kinetics' will have been described as 'rates of reaction'. This page explains the advanced particle collision theory with reference to the Maxwell–Boltzmann distribution of particle kinetic energies and using the distribution curves–graphs to explain the effect of increasing temperature and the theory of catalysed reactions.

5.1 More advanced particle theory to help explain the kinetic effects of temperature change and catalysis

5.1a The statistical distribution of particle kinetic energies

• THE PARTICLES IN A LIQUID/SOLUTION/GAS HAVE A RANGE OF KINETIC ENERGIES

• AND THIS HAS CONSEQUENCES WHEN CONSIDERING KINETIC FACTORS SUCH AS THE VARIATION OF THE SPEED OF REACTION AT DIFFERENT TEMPERATURES.

• In any gas or liquid the particles are in total random motion in all directions with a huge range of kinetic energies (or velocities).

• At room temperature there are about 1028 particle collisions per cm3 every second which means on average an individual particle undergoes over 109 (1000 million) collisions per second!

• This is a 'strange' world where dimensions are incredibly small and 'event' times are correspondingly short!

• In fact the lifetime of an 'intermediate' in a reaction mechanism might be as little as 10–9 of a second!

• BUT, what we really need to consider is the Maxwell–Boltzmann distribution of molecular velocities.

• The distribution of the translational kinetic energy (KE) of the particles is derived from the statistical mathematics of Maxwell–Boltzmann and the KE distribution curve for a given 'population' of molecules is shown in the graph above.

• The average KE is just to the right of the peak because, although there is a lower limit of zero for KE, theoretically there is no upper limit, it just depends on how hot the gas/liquid is.

• The peak of the curve equals the most probable KE in this unsymmetrical distribution.

• Although there is virtually no chance of a particle having a KE of zero because of the collision frequency, though there is always a chance of a small proportion of the particle population having a KE way above the average AND it is this small fraction of high KE molecules which collide with enough KE to break open bonds i.e. to allow the reactants to overcome the activation energy and form products.

• Reminder: The activation energy is the minimum kinetic energy the particles must have, so that on collision, bonds are broken and products formed i.e. a fruitful collision!

• An understanding of the statistical nature and shape of the Maxwell–Boltzmann particle KE distribution graph is crucial to a higher level understanding of the effect of (i) temperature change, and (ii) a catalyst, on the speed of a chemical reaction, especially as little as 1 collision in 104 to 1011 can leads to product formation! Most collisions are NOT 'fruitful'!

• Computer simulations of kinetic particle theory – Maxwell Boltzmann Distribution of particle speeds/KE's

• Some results from this are shown below for a range of molecular masses showing typical distributions of speeds..

• The molecules will have the same distribution of kinetic energies but the average speed and distribution of speeds are quite different.

• The smaller the molecular mass the higher the average speed and the wider the distribution of speeds.

• Although not strictly relevant to these pages on chemical kinetics, these graphs explain why 'lighter' molecules diffuse faster than 'heavier' ones, despite having the same average and range of kinetic energies at the same temperature.

Just an ad hoc comment on the atmospheres of planets (NOT required for A level chemistry, I just found it interesting!)

• Planets - atmospheric composition and molecular velocities

• The following comments are based on the data table below.

• As well as explaining relative diffusion rates, the differences in molecular speeds (for the same average kinetic energy and same temperature), also partly explain why the atmospheres of the planets, dwarf planets and moons differ so much.

• A small planet like Mercury, a dwarf planet like Pluto and our own Moon, have relatively small masses and therefore a relatively weak pull of gravity. Therefore gravity is insufficient to hold back atmospheric molecules which will have a great enough 'escape velocity' to be lost into space. This applies to all gaseous molecules, so bye bye atmosphere over millions of years!

• Planet Mars has a greater mass and great gravity pull than Mercury or Pluto and does retain a very thin atmosphere of the relatively heavy atmospheric molecule carbon dioxide [Mr(CO2) = 44]. Other relative molecular masses quoted in ().

• Planet Earth has a strong enough gravity pull to retain nitrogen (28), oxygen (32), water (18) and carbon dioxide (44) in its atmospheres, and Venus retains carbon dioxide, but both retain very little hydrogen (2) or helium (4) which can move fast enough to escape!

• Things are very different with the four giant gas planets, Jupiter, Saturn, Uranus and Neptune. These are so massive that their gravity pull is so strong, that despite the highest possible molecular speeds of hydrogen and helium, (see diagram above for Mr = 2), these 'light' mass moving molecules cannot escape the atmosphere of these massive planets.

Seven data sets of seven other planets, one dwarf planet and our Moon relative to Earth

 Relative: MERCURY VENUS EARTH MOON MARS JUPITER SATURN URANUS NEPTUNE PLUTO Mass 0.0553 0.815 1 0.0123 0.107 317.8 95.2 14.5 17.1 0.0025 Diameter 0.383 0.949 1 0.2724 0.532 11.21 9.45 4.01 3.88 0.186 Density 0.984 0.951 1 0.605 0.713 0.240 0.125 0.230 0.297 0.380 Gravity 0.378 0.907 1 0.166 0.377 2.36 0.916 0.889 1.12 0.071 Escape Velocity 0.384 0.926 1 0.213 0.450 5.32 3.17 1.90 2.10 0.116 Atmos. Press./atm 10-14 90 1 3 x 10-15 0.006 >1000 >1000 >1000 >1000 3 x 10-4 Major gases of the  atmosphere Extremely thin atmosphere, ~no H2 or He mainly CO2, ~no H2 or He Mainly N2 + O2, ~zero H2, tiny % of He Tiny traces of various gases Very thin - mainly CO2 Very dense, mainly H2 and some He Very dense, mainly H2 and some He Very dense, mainly H2 and some He and CH4 Very dense, mainly H2 and some He and CH4 Mainly N2 with some CH4 and CO

You can see that as the mass of the moon or planet increases, so does the pull of gravity and the necessary molecular escape velocity increases too!

5.1b Effect of increasing temperature with reference to activation energy of a reaction

A reminder of reaction profiles of uncatalysed reactions to set the scene.

Effect of increasing temperature on the KE profile

• WHY DOES THE RATE/SPEED OF A REACTION INCREASE WITH TEMPERATURE?

• WHAT HAS THIS TO DO WITH THE DISTRIBUTION OF PARTICLE KINETIC ENERGIES AND THE ACTIVATION ENERGY?

• When the temperature is raised the added 'heat energy' shows itself in the form of increased particle kinetic energy. In the graph above, two distribution curves are shown for a lower/higher temperatures, T1/T2, and it is assumed that the area under the whole curve is the same for both temperatures i.e. the same number/population of molecules.

• Looking at the diagram above, by comparing lower temperature T1 with higher temperature T2, you can see that as the temperature increases, the peak for the most probable KE is reduced, and more significantly with the rest of the KE distribution, moves to the right to higher values so more particles have the highest KE values.

• Now, if we consider an activation energy Ea, the minimum KE the particles must have to react via e.g. bond breaking, the fraction of the population able to react at T1 is given by the blue area.

• However, at the higher temperature T2, the fraction with enough KE to react is given by the combined blue area plus the red area.

• Therefore, because of the shift in the distribution at the higher temperature T2, a greater fraction of particles has the minimum KE to react and hence a greater chance of a fruitful collision happening i.e. reactant molecule bonds breaking en route to product formation.

• We are talking about an increase in frequency of a fruitful collision leading to the increase in the formation of products in a given time i.e. the increase in speed/rate of a reaction.

• In the diagram, for the sake of argument, a temperature rise from T1 to T2 results in the fraction of particles with a KE of >=Ea being doubled (area blue==>blue +  red).

• For reactions with an activation energy in the range 50–100 kJmol–1 (i.e. most reactions), this results in an approximately  doubling of the reaction rate for every 10o rise in temperature i.e. where T2 = T1 + 10, because if you double the number of particles of KE >= Ea, you therefore double the chance of a fruitful collision and hence double the rate of reaction.

• So, a relatively small change in temperature e.g. 10o rise, can have quite a 'statistically' dramatic effect on the small, but significant population of the highest KE molecules, hence a significant change in reaction rate.

• The last point accounts for why a plot of rate versus temperature shows an 'exponential' or 'accelerating' curve upwards. Almost all reaction rates increase by a factor of 1.5 to 4.5 on doubling the temperature, but it does depend on the actual activation energy, so the "10o temperature rise effect" is a very rough rule of thumb when we say it "doubles the rate"!

• 2nd minor factor note:

• The rise in temperature does lead to an increase in collision frequency, and hence an increase in the possibility of a 'fruitful' collision and so increasing the speed of a reaction.

• However, this effect on the rate of reaction, is proportionally much smaller by a factor of 100–200x, compared to the increase in reaction speed due to the increase in the proportion of high KE molecules on increasing the temperature as described above.

• Computer simulations of kinetic particle theory – Maxwell Boltzmann Distribution of particle speeds/KE's

• Some results from this are shown below for a range of temperatures.

• The graph shows the Maxwell - Boltzmann distribution of kinetic energies for 300 to 500 K at 50o intervals.

• You can clearly see the proportion of particles with >= activation energy rises dramatically with increase in temperature, hence the equally dramatic increase in the speed of a reaction with rise in temperature.

5.1c The effect of a catalyst

A reminder of reaction profiles of uncatalysed and catalysed reactions to set the scene.

• HOW DOES A CATALYST AFFECT THE ACTIVATION ENERGY? AND HOW DOES THIS AFFECT THE REACTION KINETICS?

• A catalyst speeds up a reaction, but it must be involved 'chemically', however temporarily, in some way, and is continually changed and reformed as the reaction proceeds.

• Catalysts work by providing an alternative reaction pathway of lower activation energy, e.g. it can assist in endothermic bond breaking processes (see section on catalytic mechanisms for some examples).

• If you consider the KE distribution curve above, at a fixed temperature, the green area shows the molecules which have sufficient KE to react and overcome the activation energy Ea1 for the un–catalysed reaction.

• However, in the presence of a catalyst, the lower activation energy Ea2, allows a much greater proportion of the molecules to have enough energy to react at the same temperature.

• More particles in a given instance in time have the minimum kinetic energy to overcome activation energy barrier.

• Again, we are talking about an increase in frequency of a fruitful collision leading to the increase in the formation of products in a given time i.e. increasing the speed/rate of a reaction.

• This is shown by the combined green area plus the purple area and this increased fraction of molecules (increased area) considerably increases the chance of a 'fruitful' collision leading to product formation, so speeding up of the reaction.

• Above is another graphical comparison of the effect for a reaction of an uncatalysed activation energy and a catalysed activation energy for the same reaction AND their relative effect on the proportion of molecules with the minimum kinetic energy to react.

• Irrespective of the temperature, the lower activation energy pathway facilitated by the catalyst, significantly increases the proportion of molecules with sufficient kinetic energy to allow a fruitful collision to produce reaction products.

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Advanced Level Theoretical Physical Chemistry of chromium – A level Revision notes to help revise for GCE Advanced Subsidiary Level AS Advanced Level A2 IB Revise AQA GCE Advanced Level Chemistry OCR GCE Advanced Level Chemistry Edexcel GCE Advanced Level Chemistry Salters AS A2 Chemistry CIE Chemistry, WJEC GCE AS A2 Chemistry, CCEA/CEA GCE AS A2 Chemistry revising courses for pre–university students (equal to US grade 11 and grade 12 and AP Honours/honors level courses) how to explain the particle collision theory using the Maxwell–Boltzmann distribution of particle kinetic energies and using the distribution curves–graphs to explain the effect of increasing temperature and the theory of catalysed reactions.

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