Thermal & Kinetic Lecture 16 Entropy and the 2 nd Law

1. Thermal & Kinetic Lecture 16 Entropy and the 2nd Law Overview Distribution of energy at thermal equilibrium Entropy and the 2nd law Changes in entropy

2. Last time… Distribution of energy quanta amongst oscillators. Considering most probable distribution of energy. Microstates and macrostates.

3. Thermal equilibrium Take 2 blocks, one containing six atoms, the other containing three atoms. Total energy of the two blocks: 100 quanta. The blocks are adiabatically isolated – no heat in or out.

4. ? To the nearest integer, where on the x-axis is the maximum located? Why? Distribution of energy ANS: 67 (100 quanta, split 2:1)

5. Thermal equilibrium Width of distribution (FWHM) is proportional to N . For a mole of a solid or a gas the odds are overwhelmingly in favour of having a uniform energy distribution. Largest number of microstates (by far!) are associated with the most probable distribution. Even for a small collection of atoms we’re dealing with very large numbers of possible microstates – much more convenient to use logarithms. (More fundamental reasons for appearance of logs which we’ll see later on…) Reconsider previous histograms using logarithms…………………

6. ? To the nearest integer, where on the x-axis is the maximum of ln (W1W2) located? Thermal equilibrium and entropy ANS: 67

7. If Block 2 were three times the size of Block 1, what value on the x-axis would correspond to the maximum value of ln (W1W2)? • 25 • 50 • 75 • 100

8. …and now we finally get to a definition of entropy! S =k ln(W) ? What are the units of entropy? Entropy No. of accessible states Boltzmann’s constant Thermal equilibrium and entropy ANS: JK-1

9. Entropy Entropy is fundamentally a measure of the number of possible microstates available to a system. S = k ln(W1W2)  S = k ln(W1)+ k ln (W2) We can get the total entropy of a system by adding up the entropies of the individual parts of the system. At thermal equilibrium, the most probable energy distribution is that which maximises the total entropy.

10. Entropy and the second law of thermodynamics DU = Q + W The 1st law of thermodynamics states This is simply a rewording of the conservation of energy principle. The 2nd law – which deals with entropy and not energy – can be written in a number of different, but equivalent, forms. For now, we’ll focus on the following definitions: “The entropy of a thermally isolated system increases in any irreversible process and is unaltered in a reversible process”. (p. 79, Thermal Physics, CB Finn) “If a closed system is not in equilibrium, the most probable consequence is that the entropy will increase.” (p. 354, Matter & Interactions, Vol I, Chabay and Sherwood)

11. ? S = k ln(W). Why would a decrease in temperature produce a decrease in S? Entropy and the second law of thermodynamics A more succinct statement of the 2nd law is as follows: A closed system will tend towards maximum entropy. We need to be careful with the wording here…..! (Lots of ‘abuse’ of 2nd law) The system is thermally isolated (closed) and heat flows from body A to body B. A B Q Net entropy of system goes up but entropy of body A goes down. Adiabatic wall

12. Entropy and the second law of thermodynamics As the most probable energy distribution is that which maximises entropy (of closed system) then entropy will tend to increase. However, we’re not only concerned with energy distributions – the molecules in the ‘milk drop simulation’ we discussed in a previous lecture also move towards the highest entropy distribution. Microstates in this case can be considered as the spatial distributions of the molecules. Molecules spread to occupy the available space uniformly (because the largest number of microstates is associated with this distribution!)

13. Returning to one of our statements of the 2nd law: “The entropy of a thermally isolated system increases in any irreversible process and is unaltered in a reversible process”. (p. 79, Thermal Physics, CB Finn) Reversibility So, what precisely is meant by reversible?

14. Reversibility In a reversible process the system must be capable of being returned to its original state. The surroundings must be unchanged. AND… Gas contained in adiabatic enclosure. Piston completely frictionless – no energy dissipated in form of heat. Small mass dm, placed on piston. Infinitesimal change in pressure. If we now remove the mass dm, the gas expands back to its original volume and the temperature returns to its original value. REVERSIBLE PROCESS

15. Q Reversibility Could also have a system with a container that allows heat in freely (a container with diathermal walls). Let external temperature increase by a tiny amount dT. Energy flows in through the walls. If we now slowly cool surroundings back to original temperature the gas will contract back to its original volume. A reversible process is a process which may be exactly reversed to bring the system back to its original state with no other change in the surroundings. A reversible process is an idealisation

16. Quasistatic changes Must a reversible process therefore only involve an infinitesimal change in the properties of the system? What if we want to reversibly change the pressure by a large amount? We can make a large total change as long as we do it in very small steps. The system remains in equilibrium at all times: a quasistatic process. If we push the piston down very rapidly this won’t be the case: finite temperature and pressure gradients, turbulence.