Thermal & Kinetic Lecture 12 Entropy and the 2nd law LECTURE 12 OVERVIEW Recap…. Entropy and the 2nd law Reversible and quasistatic changes Thermal equilibrium and entropy A definition of temperature: relating S and T

Last time…. Thermal equilibrium – uniform sharing of energy amongst oscillators is most probable distribution by far. S = k ln (W) The 2nd law.

Entropy S = k ln(W1W2)  S = k ln(W1)+ k ln (W2) 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.

Entropy and the second law of thermodynamics The 1st law of thermodynamics states DU = Q + W 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)

? 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. S = k ln(W). Why would a decrease in temperature produce a decrease in S? ? Adiabatic wall

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!)

Reversibility 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) Returning to one of our statements of the 2nd law: So, what precisely is meant by reversible?

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

Reversibility Could also have a system with a container that allows heat in freely (a container with diathermal walls). Q 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

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.

? Quasistatic and reversible changes A reversible process involving a large change in the properties of the system proceeds via a series of quasistatic steps. Reversing the process step-by-step would produce the same initial state. Processes which don’t involve quasistatic states are irreversible. …but are quasistatic processes always reversible? ? When might a quasistatic process not be reversible?

Defining temperature What can we say about the value of here?

Defining temperature At equilibrium: …but q2 = 100 – q1 and therefore: Thus, at equilibrium: ….but what physical property do these derivatives represent? NB Don’t confuse q (no. of quanta) with Q (heat) and don’t confuse the number of microstates W with work (also W)

The relationship of S and T At thermal equilibrium, the temperature of the blocks is the same. Hence, the rate of change of entropy with respect to energy must be related to temperature. Which line corresponds to higher temperature? ? 2 ANS: Line 2 However, line 2 represents a smaller gradient. 1

Definition of temperature We define temperature in terms of the rate of change of entropy with respect to energy. Size of one quantum of energy will vary from system to system, so use following definition: Definition of temperature. Subtle but very important point: more correct to write expression in terms of a partial derivative, eg:

Temperature and entropy Why is a partial derivative required? We will return to a discussion of this point when we cover heat and work in Section IV. For now: Partial derivative means that variables other than S and U are held constant when we determine the derivative. Eg. if volume of a gas remains constant then no work is done. Turns out (for reasons to be discussed later on) that for reversible processes work does not produce an entropy change. Only energy transfer in the form of heat produces an entropy change.

Entropy change associated with small input of heat energy Input of thermal energy means that more energy states are accessible. Hence, entropy will increase. There’s another simple expression that relates the change in entropy to the amount of heat transferred: ONLY VALID FOR A REVERSIBLE PROCESS.