Thermal Physics Tutorial 5  -  The Boltzmann Factor

(Extension for Pre-U students)

 Contents

Boltzmann Factor

We have seen how the Kinetic Theory of Molecules is based on statistical assumptions.  Molecules have a range of energies which can be due to their translational kinetic energy, or their vibrational energy (or both).  Molecules can lose or gain energy due to collisions.   In a large population of molecules, the proportion of molecules that have a certain value of energy will be constant, even if individual molecules gain or lose energy.  Molecules have a probability of having a certain energy.  Like all other probabilities, the probability must be within the range 0 to 1.  If the probability is 1, that means that every molecule has a particular energy (which is very unlikely).

Suppose nε is the number of molecules with a particular energy, ε.  The total number of molecules is N.  Therefore:

Proportion of molecules of energy ε = Probability of finding a molecule with energy ε = nε ÷ N

This probability is called the Boltzmann Factor, given by the equation:

The terms are shown below:

The Boltzmann Factor is shown:

 The Boltzmann Factor must not be confused with the Boltzmann Constant, k.

 The average translational kinetic energy of a molecule was found to be 6.0 × 10-21 J.  (a)  If the temperature is 30 oC, calculate the Boltzmann Factor. (b)  Work out the number of molecules that are likely to have that energy, if there are 10 moles of molecules.

Boltzmann Factor and Activation Processes

In a chemical reaction, there is an activation energy.  This means that the reactants have to have a certain energy before the reaction can happen.  The idea is shown on this graph:

The reactants have to be lifted to the top of the energy hill.  Then they can roll down the energy hill to make the products.

In many chemical reactions, we need to raise the energy of the reactants to initiate the reaction.  We do this by increasing the temperature.

Suppose we have a reaction that has an activation energy of 5.0 kJ mol-1.  There is 1 mole of reactants.  We can work out the average energy per reactant molecule:

Energy = 5000 J ÷ 6.0 × 1023 = 8.33 × 10-21 J

We can use this to calculate the probability of having molecules of this energy at room temperature (21 oC):

Work out the the natural logarithm of the Boltzmann Factor:

ln (nε/N) = -(8.33 × 10-21 J ÷ (1.38 × 10-23 J K-1 × 294 K)) = -(8.33 × 10-21 J ÷ 4.057 × 10-21 J) = -2.05

The Boltzmann factor is given by:

nε /N = e-2.05 = 0.13

This would indicate that there would be a good chance of this reaction proceeding, especially as heat will be released as the reaction proceeds.  However this does not take into account the probability of the collisions that are needed for a reaction, or that the collisions are in the right orientation.  The Arrhenius Equation gives a better prediction:

The terms are:

• K = rate constant of the reaction;

• A = pre-exponential factor, which depends on the reaction itself.

You are not expected to know this equation, but you can see its similarity to equation for the Boltzmann Factor.

 Identify the Boltzmann Factor in the equation above.

Other activation processes include:

• Evaporation of liquids;

• Viscosity of fluids;

• Creep in ductile solid materials;

• Semi-conduction;

• Ionisation.

The structure of a viscous fluid is that of a liquid, in that the molecules go about in small groups.  The larger the small groups are, the more viscous the fluid is.  There is a small, but finite, probability that an individual particle gets enough energy to break free of the group so that the fluid can flow.

Although glass seems solid, it is actually a very viscous fluid.  The glass in a window that is a couple of hundred years old is noticeably thicker at the bottom that at the top:

This can be a serious problem in many engineering applications.  If a material is subjected to a continuous stress, deformation can occur even though the stress itself may be lower than the yield stress.  It is more marked in ductile materials like copper and lead.  Lead is used as a roof covering on many old buildings like churches.  The stress caused by the weight of the lead can pull the lead down the roof:

The church organ in York Minster (one of Europe's largest cathedrals) is currently having a major overhaul costing £2 million, in which the instrument has been completely dismantled.  One problem identified is that many of the metal pipes have been subject to creep.  They have shortened slightly and bulged at the bottom.  This has caused them to go slightly out of tune.  Since the pipes are shorter, their fundamental frequency will have slightly increased.  To a musician their notes would be sharp.  The pipes will be restored to their original dimensions so that they play true.

Creep is dependent on temperature, and materials like lead will creep at environmental temperatures.  For many materials, creep happens at about 35 % of the Kelvin melting point.

In early day turbines, the forces acting on rapidly rotating blades could cause distortion by creep.  If one blade fails as a result, it can shear off all of the others ("having a haircut"), which is highly expensive.  In large power station turbines, the turbine rotors are very heavy.  If the turbines are turned off, the shaft can sag due to creep, becoming slightly banana shaped.  This can be prevented by slowly turning the shaft using a barring motor.

A general equation for creep is shown below.  You are not expected to know it.

The meanings of the main components are shown:

The point of this diagram is to show the presence of the Boltzmann Factor.  In the equation, Q is activation energy of the creep.  The term ε is "epsilon", the physics code for strain.  The other terms need not concern us here.

Boltzmann Constant and Semi-Conduction

Semi-conduction is described in Electricity Tutorial 6.  Before you do this section, you may wish to review semi-conduction.  We know that in a bipolar junction, there is a depletion layer that require there to be a certain voltage to overcome it.  In the model in Electricity Tutorial 6, we talked about the depletion layer disappearing in forward bias.  That is the usual model that is used by electronic engineers.  However there is more to it, but it's not that complex.

The depletion layer disappears because the electrons have sufficient energy to jump into the conduction band and cross it.  The electron energy required is:

This is entirely consistent with energy being the product of charge, e (from the electron, 1.602 × 10-19 C) and the bias voltage, Vb.  When the bipolar junction just conducts with a very small current, ib, the equation is:

We can see the Boltzmann Factor in the equation:

The term a is a proportionality constant, a property of the semiconductor.

 A bipolar junction just starts to conduct at 0.45 V.  The diode is at 25 oC, and the proportionality constant, a, is 40 A. (a)  Calculate the Boltzmann Factor. (b)  Calculate the current, ib.

The Boltzmann factor is the probability that an electron will jump across the depletion layer.  If the temperature is increased, the bigger the chance of an electron having the energy to jump over.  This explains why semiconductors conduct a bigger current as their temperature rises.

Boltzmann Factor and Thermionic Emission

The most common way of accelerating electrons is to use an electron gun, in which current passes through a filament, which glows just like in a light bulb (Particle Physics Tutorial 4).  The filament is connected to a source of electrons (the negative terminal of a high voltage source), called the cathode.  Electrons are boiled off by a process of thermionic emission.  They are attracted to a positively charged anode.   Most hit the anode and go back to the source.  Some go through a small hole in a narrow beam. (This was called the cathode ray.)

Since they are attracted by the very high potential difference, the electrons accelerate.  Once they get to the anode (and pass out of the little hole), the electrons are moving very fast.  All their energy is kinetic.

Thermionic emission is another activation process which works using the ideas of probability.  It is quite hard to boil electrons off tungsten as it has a relatively high work function (see Quantum Physics Tutorial 2).  The work function of tungsten is about 4.5 eV.  This would reduce the probability of an electron being boiled off.  To improve thermionic emission, the filament is coated with thorium, with a work function of 3.45 eV.

Thermionic emission can occur when an electron acquires energy of more than 3.45 eV.  In the diagram below, we can see the model of a metal being a fixed lattice of positive ions in a sea of free moving electrons.  One electron has gained enough energy to leap out of the metal.

Experiment shows that the cathode current, Ic, depends on the Boltzmann factor.  This is consistent with the probability of an electron escaping the metal, given by:

If the Kelvin temperature is low, then the probability is low.  This explains why the filament has to be hot.  Once the electrons are free of the filament, they are accelerated to the anode, as we saw above.  The current is given by the following proportionality expression:

Experiment shows that the current density, J, (current per unit area) is given by:

The proportionality constant is in two parts:

• The square of the Kelvin temperature;

• The Richardson Constant, A.

The equation above describes Richardson's Law, named after Owen Willans Richardson (1879 - 1959).  The Richardson constant surprisingly does not depend on the material; it is a universal constant:

A = 1.2 × 106 A m-2 K-2

 A filament in an electron gun operates normally at a temperature of 1000 oC.  The tungsten filament has a coating of thorium which has a work function of 3.45 eV. (a)  Show that the probability of an electron being boiled off is about 2 × 10-14. (b)  Calculate the current density.  Give the correct unit. (c)  The hole in the anode through which the electrons travel is 2.0 mm across.  Calculate the anode current. (d)  Explain without a calculation why the thermionic emission will not happen at room temperature.

The Boltzmann Factor is important in Statistical Mechanics.  You will study this at university.