Condensed Matter
What is it?
Ideal Gas
The ideal gas model is used to help understand thermodynamics. The model is based on a few assumptions
- Large Number of Particles in the system
- The system follows the rules of Newtonian Mechanics
- Perfect Elastic Collisions
- Random Motion
Although the theory is based on assumption it describes systems with good accuracy. It isn't perfect however and breaks down at low temperatures and high pressures, but so what, nobody's perfect. Before we go any further it is worth going over some of the important equation for a gas.
If you put a gas in a box it will exert a pressure on the sides of the box. This is due to the gas particles hitting the sides with their random motion and providing a force. Force is defined as the rate of change of momentum so for 1 particle hitting the wall the force will be
for the time interval Δt. The reason for the factor of 2 is to take into account the change in momentum of the particle. In a perfectly elastic collision the particle will go from having momentum in one direction to having the same amount of momentum in the opposite direction.
To work out the force from all of the particles all we have to do is add up the forces from the individual particles. However not all of the particles in the box are going to hit the wall at once, so we need to work out the probability of a particle hitting the wall in the time interval Δt. In order for a particle to hit the wall it would need to be within a length of vΔt of the total length of the box, L.
The factor of 1/2 comes from the fact that, taking into account only one dimension of motion i.e only one set of the box walls, then on average only half of the particle will be travelling in the direction of the wall we are interested in. We can now use this probability to sum up the forces from all of the particles like so
This simplifies down nicely to
Notice that the particles mass and the length of the box have been taken out of the summation. This is because L is a constant and we are assuming that m is the same for all of the gas particles. So all we have to do is sum up the velocity squared for all of the particles.
Before we do that however we can make a quick change. We are more interested in pressure than force and we know the two are related by area so we can use the force equation to work out the pressure of the gas.
Now to work out the sum of v2. It isn't too hard to do and just requires some logical thinking. If you add up the speed of a bunch of particle and divide by how many you have you will get an average value of the speed, so it makes sense that if you add up the v2's and divide by the number of particles you will get an average value of v2. So the sum of the v2's is just the average value times N, the number of particles. However because the sample is in the real world there are 3 dimensions of motion to take into account, so we just divide by 3. Simple. So our value for the sum of v2 is just
The angled brackets or chevrons denote that the value inside them is an average value, and in this case is called the Mean Square Speed. We can now put this value back into equation 1 and replace the LA on the bottom for V, as a length times an area is a volume.
It makes things a bit easier at this point if we take the volume term to the other side to go with the pressure term like so
This now gives equation 2 a form in which it makes it easier to solve for ‹v2›.To solve it we need to introduce possibly the most important equation when dealing with ideal gases, The Ideal Gas Law
Where
- P is pressure, usually in Pascals.
- V is volume, usually in cubic meters.
- T is temperature, usually in Kelvin.
- n is the number of moles of the gas.
- N is the number of molecules.
- k is the Boltzmann constant, 1.380x10-23 J K-1.
- R is the gas constant, 8.314 J mol-1K-1.
The law works with any consistent set of units, provided that the temperature scale starts at absolute zero, and the appropriate gas constant is used.
The law comes from the combination of Boyle's law, Charles' law, Gay-Lussac's law and Avogadro's law. Which of the two forms of the equation you use depends on what you know. If you know how many moles of the gas you have then you use PV=nRT, whereas if you know how many molecules you have you use PV=NkT.
You can see that equation 3 has a PV term like the one in equation 2, so we can equate them and rearrange for ‹v2›
Now that we have a way of calculating ‹v2›, we can also work out the Internal energy of the gas. Since the only energy in the system comes from the motion of the particle the Internal energy will just be the sum of the kinetic energies.
So subbing in out two values for the value of the mean square speed we get two eqautions to work out the energy of the gas
Now, although these derivations we have made assumptions and simplifications, but I can assure you these equations are in agreement with what is observed during experimentation.
Speed
Possibly annoyingly, there are a few different measurements of particle speeds that are important in the study of gasses and other particle distributions.
Although I have kept using ‹v2›, a more useful value is the square root of this, and is called the Root Mean Square (RMS) Speed.
Another important value is the average speed, which is found by summing v for all of the particles
And there is the most probable speed vm (or sometimes vp)
This last value is obtained by differentiating the distribution of speeds to find the maximum. The distribution of particle speeds in a gas is given by the Maxwell Distribution.
The Maxwell Distribtion
The Maxwell Distribution was calculated in the 1850's by James Clerk Maxwell (the same Maxwell who did the Maxwell Equations). The distribution of speeds in a sample follows is described by the follwing equation
The shape of this distribution and the positions of the different speed values are shown below
As you can see the graph crosses the (0,0) origin, this means that in a sample of gas there are no particles with a velocity of 0. A velocity of 0 would mean a temperature of 0 and an energy of 0, which would violate the uncertainty principle.
Transport Properties
Transport Properties are the properties of a material that relate to its ability to transport specific quantities, stuff like heat or mass. You can only get transport when the amount of 'stuff' varies from one place to another, if its the same all over then nothing is going to move. You need non-equilibrium for transport.
The 3 main transport properties that I will be attempting to explain are
- Diffusion - The Transport of Mass
- Viscosity - The Transport of Velocity
- Thermal Conductivity - The Transport of Energy
The transport properties are all related to how the particle in the material move and behave, so before we jump into the transport properties we will first need some more information, specifically information about Mean Free Paths and collision times.
Free Paths and Collisions
The average distance a particle travels between collisions is called The Mean Free Path. Assume you have a particle with a radius of σ
You can clearly see that you will only get a collision every time the distance between the centres of two particles is 2σ. Knowing this we can simplify the situation to make things easier for ourselves. We will now assume that we have one particle of radius 2σ and all of the other particles are stationary points. This isn't the case however, but it works for the calculations.
In a time of Δt the particle will move a distance of vΔt. This will sweep out a volume of π(2σ)2vΔt. If there are n stationary points in this volume then there will be
collisions per Δt. To find the total we just do what we've been doing a lot of, sum over all of the particles, and we get a Collision Frequency of
collisions per unit time, per unit volume. We have divided by 2 as we are assuming each collision involves 2 particles, and the extra n comes from the fact that we have summed all of the v's which is n‹v›
If we wanted to find the distance between the collisions all we have to do is divide the distance it travels by the number of collisions it would have in that distance. This gives us a Mean Free Path, λ, of
It turns out that in getting to this formula one of our assumptions was wrong. The calculation of the mean free path should take into account that all of the particles in the gas are moving. TO do this we need to replace the speed of the particle with their relative speed as we are using the viewpoint that we are sitting on one particle moving and the rest are stationary. After many tedious calculations we find that all we have to do to make the equation accurate is add a factor of √2. So the equation should be
It is worth noting that this equation only holds for Ideal Gases that follow the Maxwell Distribution, so at low pressure or high temperature it will not hold.
Viscosity
Viscosity is a measure of a gas or liquids resistance to shear forces, forces parallel to a surface. Viscosity is defined as
but this isn't a very nice formula so we will have to come up with a better one.
In order to calculate viscosity we will use the 6 streams method. The 6 streams refer to the positive and negative parts of the 3 axis of dimension. The speed particle will have in the gas will depend on where they are. Their speed will depend on their temperature which will vary from place to place.
Let's assume that in the diagram above the particles have a fixed speed of vx. The number of particles arriving at A in a time of Δt will be
This comes from the fact that in the time Δt particles will travel a distance of vΔt, so any particles in the volume AvΔt will be able to reach A. n is the number of particles per unit area and the 1/6 comes from the fact that only 1/6th of the particles in that volume will be moving down. The average velocity of the particles will be vx plus any change due to their velocity before. This can be expressed as
λ is the mean free path lenght and (dvx/dy) is just the term that describes how the velocity varies with position. The change in momentum at A will just be the momentum of the particle times the number of particles at A. The number of particles at A was calculated by equation 6 and the momentum can be obtained by timesing equation 7 by m, so we get
The 2 comes from the fact that particles can reach A from above and below, and the t term was taken to the other side so that dp/dt, the rate of change of momentum, can be rewritten as force, so
Using the definition of viscosity given in equation 4, we can get the Equation of Viscosity as
Which can also be written as
By substituting λ, as given by equation 4. Unlike some of the other equations that only give close estimates, this equation gives an exact numerical value. From these equations you can see that Viscosity, η, is independent of pressure
Thermal Conductivity
Thermal conductivity is the transport of heat energy and is represented as a greek kappa, κ. κ is defined as
Thermal Conductivity can be analysed the same way as we did for viscosity. The only difference being that we swap the momentum term for an energy term. So instead of equation 8 we get
where the momentum term of mλ(dvx/dy) has been replaced by the energy term (dE/dy). (dE/dy) is the energy gradient of the gas, is tells you how the energy varies as you move around. This can be converted to a temperature gradient with a sneaky little maths trick like so
Which we can sub straight back into equation 11 to get
Combining this with our definition of thermal conductivity in equation 10 we get an Equation for Thermal Conductivity
We know what λ equals from equation 4, so we know that κ is independent of pressure, and also n
Diffusion
For diffusion you really need 2 different gases, one gas on its own can't really diffuse. However if you have 2 gases things can get complicated as they will have different masses and diameters. As we dont like things to be complicated (unless we're doing quantum mechanics) we will just assume that the two gases have the same mass and same diameters. For things like air, or isotopes this works really well.
Diffusion, is defined by Ficks Law as
Where j is the amount of stuff that will flow through a small area during a small time interval, D is the diffusion coefficient, the thing we want to work out, and (dn/dy) is the concentration gradient, it just tells you how the amount of stuff varies with position.
Like we did with thermal conduction, we will use the results of viscosity to help our working, only this time we will replace the momentum gradient with a concentration gradient. So we get
Which we can just stick straight into equation 13 to get an Equation for Diffusion
Relations
We now have 3 equations for the transport properties, describing Viscosity (η), Thermal Conductivity (κ) and Diffusion (D).
As you can see these equations have some common features and can be combined to obtain relationships between η, κ and D. By combining the equations for D and η we get
where ρ is the density. And by combining the equations for κ and η we get
((dE/dT) was replaced for Cv/Na) where Cv is the specific heat at constant volume, Na is Avogadro's number and M is the molar weight.
Experimental values for equation 15 vary from 1.3 to 1.5 and values of equation 16 vary from 1.5 and 2.5. This is because real gases do not behave like ideal gases.
Now let's see if we can apply this theory to electrons.
Classical Electrons
In 1900 the kinetic theory was applied to the problem of conduction in metals. It was assumed (by now you might be getting the impression that assuming things is all that scientists do) that in the absence of a magnetic field the electrons would move freely in the metal and because they had mass they would behave like an ideal gas and obey the kinetic theory. Lets see how far this assumption got them.
Density
How many electrons would you expect in a metal? The number of free electrons should relate to conductivity as we are assuming the electrons are doing the conducting. But conductivity has a huge range, spanning 25 orders of magnitude. This is the largest range for a physical parameter in the universe. The first estimate will be one free electron per atom. That seems ok doesn't it? This gives about 1028-1029 electrons per m3 for metals.
Conductivity
If we are assuming the electrons are an ideal gas then the average velocity will be 0 (If it were anything else the sample would be moving across the room). If we apply an electric field to the sample then the electrons will be accelerated in the direction of the field. This will then give them a drift velocity. It is this drift velocity, vd that causes a current. If we have one electron in the field then it will follow Newtons law
If the field is of strength E, then the force on the particle will be eE, and if we replace acceleration for the rate of change of velocity we get
If we once again sum over all of the particles in the sample we get
‹vd› is the average drift velocity in the direction of the field. According to this equation the velocity will increase with time, so a current should increase in a steady field. This isn't what happens, however we can explain this because the electrons will collide with atoms, slowing them down and randomising their motion. So the electrons will only be accelerated for a time, τ. If we integrate equation 17 with respect to time we get the Electron Drift Velocity
We also know the current density of the electrons, J. Its just the number of them, n, times their charge, e, times their velocity, ‹vd›,
Combining equations 18 and 19 we get a relationship between current density and field
Which is Ohms Law! Looks like we're on the right track. We can use this then to get an expression for conductivity, σ (You will have to bear with me here. I know in the first section I introduced σ as radius but it is also used as conductivity. It could get a bit confusing, especially since they are going to appear in the same formula soon. I will try my best to make it clear which is which), as σ=J/E the Electrical Conductivity is
Between collisions an electrons speed in increased by ‹vd›, so its kinetic energy in increased by 1/2m‹vd›2. This extra energy represents the heating of the wire the the current is going through. This heating is proportional to E2, a property that has been proved experimentally.
We have a correct law and a measured effect out of the theory. So far so good.
Now let's try and get a value for τ. Electrons are very small, so they will only collide when they get within an atomic radius σ. Using the same working we did here we get an expression for the electron mean free path
As you know, time is equal to distance over speed so τ is just
I dont know why it is ‹1/vd› as opposed to 1/‹vd›, but it just is (probably something to do with some horribly complex maths). We can find the value of ‹1/vd› from the Maxwell distribution and it is
So τ becomes
So if we now combine equations 21 and 22 and put them into equation 20 we get an expression for Electrical Conductivity
where the κ2 in the denominator is atomic radius. The fact that there is an e2 term show we are on the right lines. It shows that electrical conductivity is independent of charge sign i.e. Electrical Conductivity is always Positive. This equation also shows that the electrical conductivity, κ, is proportional to T -1/2
Thermal Conductivity
Good electrical conductors are the best thermal conductors. However thermal conductivity doesn't have the same huge range as electrical conductivity. Thus free electrons cannot be responsible for all of the thermal conductivity of a metal. To work out the thermal conductivity κ for the electrons we will just use equation 12 that we came up with for an ideal gas here, with the temperature gradient replaced by the specific heat of electrons
Substituting in equation 21 for the mean free path of electrons we get
Values for the transport coefficients are only accurate to within a factor of 2, so we can forget the multiplying constants of the order 1, and we get
This is our equation for the Thermal Conductivity of Electrons. Note that from equation 22, the τ term contains a T -1/2, which combines with the T in the numerator to give κ a T1/2 dependence.
Wiedemann-Franz Law
If we assume for the moment that all of the thermal conductivity in a metal is due to the electrons, then the ratio of conductivities would be simply
Thus if all or most of the thermal conductivity was due to electrons then this ratio should be the same for all metals (at a set T).
Experimental Tests
We know, and have checked experimentally, that it is the electrons that are the things that cause the conduction. So that part of our theory is correct. What about the values of conductivities? Do the predicted values match with those found experimentally?
The Wiedemann-Franz Law (equation 25) predicts the ratio of conductivities as
Experimentally it was found to be
This is a satisfactory agreement; however it can be improved with a more detailed set of calculations, which keep a better track of constants than we did. But out rough theory definitely fits.
What about the temperature dependence of σ and κ? Experimentally the electrical resistivity, ρ, (inverse of conductivity) goes as
At room temperatures ρ1(T) varies linearly with T, and at low temperatures ρ1(T) varies as T5. According to equation 23 σ∝T -1/2 so ρ∝T1/2, but this doesn't match with experiment. Also, from equation 24 κ∝T1/2, however experimentally κ is found to be virtually independent of T.
And it was all going so well.
Specific Heat of Electrons
Using the Maxwell distribution you get that electrons have an average thermal energy of 3/2kT, this means they have a specific heat of 3/2nk per unit volume.
At room temperature the specific heats of metals and insulators are about the same and are independent of T. This suggests that Electrons contribute very little to the Specific Heat of a Metal. This is only possible if the electron density is low.
So let's lower the density, see what happens. If we lower it by about 0.01 so we get a density of about 1027 we get a relaxation time of about 10-12s. This gives us a speed for the electrons of about 105m/s, This may seem a bit big but remember that electrons are a lot lighter than your average gas particles. This speed gives us a mean free path of 10-7m. This is very large. We would expect them to only be able to get an atomic spacing before hitting something, usually an atom. All these numbers are consistent with An Electron Density higher than the Atomic Density. But we made a point of setting the density almost 100 times lower than the atomic density at the start.
Once again we have a problem.
But don't worry, it isn't all bad. The theory contains a lot of correct physics. The things responsible for the conduction are the free electrons, and we got Ohms Law and the Wiedemann-Franz Law out of it. It would be a huge coincidence if those laws had just fallen out of the theory randomly, so we're at least partly correct. The problem comes from the specific heats and the temperature dependence of the conductivities. Both of which arise from the Maxwell Distribution. Maybe electrons don't follow this distribution. Instead we will have to employ the deep dark magics of Quantum Mechanics to solve the problem.
This page was written by The Rev and edited by Tom Glossop.
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Page 1
•What is it?
•Ideal Gas
•Speed
•Maxwell Distribution
•Transport Properties
•Free Paths
and Collisions
•Viscosity
•Thermal
Conductivity
•Diffusion
•Relations
•Classical Electrons
•Density
•Conductivity
•Thermal
Conductivity
•Wiedemann-Franz
Law
•Experimental Tests
•Specific Heat
Page 2
•Quantum Electrons
•The Sommerfeld
Model
•Infinite Square
Well
•Degeneracy
•Density of
States
•Distribution at
Higher Temps
•Specific Heat 2
•Conductivity 2
•Thermal
Conductivity 2
•Wiedemann-Franz
Law 2
•Pauli
Paramagnetism
•The Hall Effect
Page 3
•Crystallography
and Energy Bands
•The Lattice
•Basis Vector
•Typical Structures
•Cubic Cell
•Packing Fraction
•Bloch Waves
Page 4
•Diamond Structure
•Energy Bands
•Calculation of
nc(T) and pv(T)
•Law of Mass
Action
•Intrinsic
Conduction
•Fermi Level
Position
•Conductivity
•Extrinsic
Semiconductors
•Doping
•N-Type
Semiconductors
•P-Type
Semiconductors
•P-N Junction
•Capacitance
•I-V Behavior
•LED's
•Superconductivity
•Magnetic
Properties
•Type 1
•Type 2
•Theoretical Model
Model