PHYS20352 Thermal and Statistical Physics

Official online notes for this course are very complete, and so very little note taking should really be needed in lectures. However, steps can be missing in some derivation online, and so it can be beneficial to attend if they're not written up here.

Aims

 * 1) To develop the ideas of classical thermodynamics
 * 2) To deepen the appreciation of the link between the microscopic properties of individual atoms or other particles and the macroscopic properties of many-body systems formed from them
 * 3) To demonstrate the power of statistical methods in physics

Learning Outcomes
On completion successful students will be able to:


 * 1) Demonstrate an understanding of the first and second laws of thermodynamics, and of the concept of entropy;
 * 2) Explain and derive the fundamental thermodynamic relation;
 * 3) Use the formalism of thermodynamics, including the thermodynamic potentials and Maxwell's relations, and apply these tools to simple systems in thermal equilibrium;
 * 4) Explain the basic concepts of statistical mechanics, including the derivation of the general formula for entropy in terms of the ensemble probability distribution function;
 * 5) Explain the statistical origin of the second law of thermodynamics; and
 * 6) Construct a partition function for a system in thermal equilibrium and use it to obtain thermodynamic quantities of interest.

Thermodynamic basics
A thermodynamic system is a system with an order of or above 1023particles, which is a figure based on Avagadro's Number, NA, 6.0221415 × 1023. These systems can be analysed statistically rather than per-particle, with the result that the state and behaviour of the system can be defined in just a few variables, such as temperature, pressure and volume. These are called the thermodynamic variables.

There are two types of variables: An easy way to figure out which is which is to consider two identical systems, say gases contained within two cubes, and then 'attach' the cubes together on one face, removing the boundary between them. Variables that would change here are extensive, such as the volume (which doubles). Variables that don't are intensive, such as the temperature or the pressure.
 * 1) Extensive variables, meaning they are dependant on the number of particles in the system.
 * 2) Intensive variables, meaning they are independent of the number of particles in the system.

State Functions
A state function is any function of the thermodynamic variables. A state function is either extensive of intensive. Examples of state functions would be the internal energy,


 * $$E = E(T,V,P)\,$$,

or entropy,


 * $$S = S(T,V,P)\,$$.

Thermodynamic Equilibrium
A system is in equilibrium if it does not change with time. In thermodynamics all states considered are treated as if in equilibrium, and a change of state is supposed to take place through successive states of equilibrium.

Equation of state
The equation of state of a system relates the thermodynamic variables with the state of the system. For example, the equation of state for an ideal gas is


 * $$PV=Nk_BT\,$$,

or more generally,


 * $$f(P,V,T) = 0\,$$.

These equations of state can only be found through experimentation or statistical mechanics.

Thermodynamic Transformations
A TD transformation is where a system goes from one state to another. A transformation is quasi-static is if the conditions change slowly enough so that at any instant the system can still be considered in equilibrium.

A transformation is considered reversible if the following conditions are met:
 * It is Quasi-static.
 * There are no frictions present (Remember hysteresis in EM, a magnet could be permanently magnetised, an example of a 'friction').

Work and Heat
The work done on a system is defined as
 * $$W=-P\Delta V\,$$,

the negative is because if you were to compress it (i.e. the change in volume was negative) you'd be doing work on it, so the work would be positive. The work done by a gas is also defined in lowercase,


 * $$w=-W=P\Delta V\,$$.

Heat is the energy absorbed by the system from a change in temperature. With no work being done on the system, the heat energy absorbed from a change in temperature is


 * $$Q=C\Delta T\,$$,

C is the heat capacity, and can take on different values depending on the process that heats the, such as CP for constant pressure and CV for constant volume. Lowercase q is again defined to be the heat lost by the system, i.e. the opposite of Q,


 * $$q=-Q=-C\Delta T\,$$.

First Law of Thermodynamics
If we consider the conservation of energy in the cases above, then the work done on the system added to the heat energy absorbed by the system will be the change in energy for the entire system, so it follows that
 * $$\Delta E=W+Q\,$$,

which is the first law of thermodynamics. For an infinitesimal change in energy, you get


 * $$ dE = \delta W + \delta Q \,$$,

where $$\delta$$ means the variable is process dependant, whereas dE is an exact derivative (Note that in lectures the process dependant delta is usually a d with a strike through, like $$\hbar$$).

The energy is a state function, it only depends on the current state of the system. However, work and heat are not state dependant, they depend entirely on how you get to that state, i.e. the process. If you take an ideal gas for an example, by looking at it's equation of state (PV = nRT) it can be seen that it has 3 variables, P, V and T. If you select two of these to take on a particular value, then the third is defined by them. This shows that an ideal gas only has two degrees of freedom and can therefore be represented on a graph of two of these three variables.

You can imagine a P-V diagram with two points, each point describes a certain state of the gas. You can then draw any number of different paths from the first point to the second. Using the equation for work above you can ascertain that


 * $$\delta W = -PdV\,$$,

Integrating this over the change in volume from state one to state two and stating P=P(V) you get


 * $$W = -\int P(V)dV \,$$,

P(V) can be any function, and so does not have to depend on the start point and end point alone as a state function does. The same can be shown for Q using the same logic and the corresponding equation for Q.

Since an ideal gas is a function of two variables,


 * $$E=E(P,V)=E(P,T)=E(V,T)\,$$,

you can take the total derivative of E, for example, for E(P,V) we get,


 * $$dE = \frac{\partial E}{\partial P} dP + \frac{\partial E}{\partial V} dV$$.

Heat Capacity
If we consider a gas going through a quasi-static reversible transformation we can write


 * $$dE = \delta Q + \delta W\,$$

and


 * $$\delta W = -PdV \,$$.

Subbing the second into the first and rearranging for dQ we can show


 * $$\delta Q = dE + PdV \,$$.

If we divide this by dT we find the total derivative of heat, Q, with respect to temperature, T,


 * $$\frac{\delta Q}{\delta T} = \frac{\partial E}{\partial T} + P \frac{\partial V}{\partial T}$$.

This differential is the definition of Heat Capacity, i.e. how much heat energy is needed to rise the temperature. If we say this process was constant under either pressure or volume, we obtain two formulae for the Heat Capacity at constant pressure and volume, CP and CV,


 * $$C_P = \left( \frac{\delta Q}{\delta T} \right)_{P} = \left(\frac{\partial E}{\partial T}\right)_P + P \left(\frac{\partial V}{\partial T}\right)_P$$


 * $$C_V = \left( \frac{\delta Q}{\delta T} \right)_{V} = \left(\frac{\partial E}{\partial T}\right)_V$$

where a subscript P or V indicates constant pressure or volume.

Official Materials
The online materials for this course can be found on teachweb or can all be downloaded as a zip file. Individual files are also listed here:
 * Chapter 1, Summary (Lectures 1 to 4)
 * Chapter 2, Summary (Lectures 5 to 12)
 * Chapter 3, Summary (Lectures 13 to 15)
 * Chapter 4, Summary (Lectures 16 to 22)
 * Zip file of all materials.

Past Papers

 * 2011 Exam / Solutions / Feedback
 * 2010 Exam / Solutions / Feedback
 * 2009 Exam / Solutions / Feedback
 * 2008 Exam / Solutions
 * 2007 Exam / Solutions
 * 2006 Exam / Solutions
 * 2005 Exam / Solutions