1. Ch 22 pp. 552-568 Lecture 2 – The Boltzmann distribution

2. If I speak of Heat and I asked you… What is it? If I say a substance is at a certain temperature.. What exactly am I measuring?

3. Do you remember what classical entropy is?

4. Macroscopic (i.e. thermodynamic) properties can be related to microscopic (mechanical) properties using statistical mechanics (and viceversa) Equilibrium thermodynamic properties are averages over microscopic states (e.g. energy) Summary of lecture 1 Thermodynamic energy, reversible heat and work are related to the (microscopic) energy levels E i and the occupation numbers N i or distributions P i

5. In principle, we can calculate energy levels from quantum mechanics, though in practice this is often very difficult or impossible; in order to obtain thermodynamic properties, we must also know the distribution There are in general many distributions that result in the same average energy. However, there exists a distribution that is much more probable than all the others (most probable distribution). It is postulated that this is the distribution that characterizes a system at equilibrium. This most probable distribution is of central importance. The Boltzmann distribution function

6. We will show that the heat absorbed by a system during reversible change is related to the changes in the number of molecules in each energy level As heat is absorbed, the number of molecules in higher energy levels increases, while that of those in lower energy levels decreases. The reversible heat change is the part of the total energy that is related to changes in the distribution of the molecules among the energy levels. We will also show that work is instead related to changes in the energy levels themselves Work and Heat

7. If I speak of Heat and I asked you… What is it?

8. Work and Heat Consider a gas in a contained with a movable wall (piston). If we move the piston by a distance dx in the direction of an external force F x, the work done on the gas by the surrounding is: dw = F x dx The total force against the piston exerted by all molecules is: where F ix is the force exerted by each molecule in the E i energy level

9. Work and Heat If the work is carried out reversibly, then the forces exerted by the molecules must balance the external force, so that the work done by a reversible process is: The force can be equated to the negative derivative of an energy. For an ideal gas, the force F ix is related to the change in its energy E i due to the change in the dimension of the container da

10. Work and Heat By substituting into the previous equation (dx is simply the opposite of da) The work done on the system for a reversible process is related to the changes in the energy levels due to the change in the dimension of the container This is true for any system, not just ideal gases, though real gases changes the size of the system affects molecular interactions as well and therefore energy levels and distributions

11. Work and Heat If we now write the differential expression for the change of energy with changes in energy levels and occupation numbers and substitute the expression for the reversible work: Since the first law of thermodynamics states that:

12. Thus, the heat absorbed by a system during reversible change is related to the changes in the number of molecules in each energy level As heat is absorbed, the number of molecules in higher energy levels increases, while that of those in lower energy levels decreases. The reversible heat change is the part of the total energy that is related to changes in the distribution of the molecules among the energy levels. Work and Heat

13. The heat absorbed by a system during a reversible change is related to the changes in the number of molecules in each energy level Work and Heat The work done on a system during a reversible change is related to the changes in the energy levels

14. A distribution reflects a microscopic arrangement of the system. There can be, and in general there are, many distributions that result in the same average energy. The number of microscopic arrangements that result in the same distribution is called the degeneracy of the distribution The number of ways W of arranging N distinguishable molecules among n energy levels such that N 1 have energy E 1, N 2 have energy E 2 N 3 have energy E 3 etc. is given by Degeneracy of a distribution

15. Example – Poker - The degeneracy or weight of a distribution W may be thought of as a probability, in the same sense that the probability of drawing a five card poker hand is related to the number of occurrences of that hand in a deck of 52 cards. There are 3,744 possible combinations of five cards in a 52 card poker deck that yield “full house” hands, but there are only four ways to draw royal flush hands. It is common to state that one has a greater chance or a higher probability of being dealt a full house than a royal flush. Degeneracy of a distribution

16. Example - In a system containing 10 molecules, it is possible that 9 have zero energy (the lowest energy level) and one has 10 times the average energy, but this is not likely. It is much more probable that a wide range of energy values is represented. Degeneracy of a distribution

17. Example - Consider three distributions of 10 molecules over seven equally spaced energy levels. The level spacing is defined as E 0. Assume E 1 =0 Degeneracy of a distribution

18. Distribution A: Distribution B: Distribution C: Degeneracy of a distribution For all 3 distributions:

19. Distributions A, B, and C have equal average energies, but they have very unequal degeneracy: Degeneracy of a distribution There is only one way to put all molecules in energy level 3, but 113,400 ways to generate distribution A and 151,200 to generate distribution C (there are two ways of assigning two molecules to two different levels, but only one way to assign both molecules to the same level).

20. If a very large number of distributions {P i } can indeed result in the same average property (e.g. ) how can we proceed to relate mechanical properties to thermodynamic properties? There is one distribution for which the degeneracy is much, much larger than the rest. This degeneracy is called W max ; the distribution corresponding to the largest degeneracy is called the most probable distribution. The identity of this distribution is of fundamental importance. The Boltzmann distribution

21. Let us consider the plot of W as a function of all distributions of N ideal gas molecules that result in the same average energy The Boltzmann distribution

22. The most probable distribution {N 1,N 2,…N n } (or equivalently {P 1,P 2,…P n }) corresponds to the maximum W max in the plot of the function The Boltzmann distribution Subject to two constraints: Total energy and particle numbers are conserved

23. The conclusion of this exercise in calculus is that the most probable distribution, the distribution that characterizes a system at equilibrium is the so-called Boltzmann distribution because he first derived it: The Boltzmann distribution q is the sum over molecular states and is called the molecular partition function (note that the text uses Z for the partition function).

24. The Boltzman distribution is one of the most important concepts in statistical thermodynamics, because it provides us with the probability of finding a system within a particular energy state E i The Boltzmann distribution

25. The derivation given above is only valid for the case where molecules in the system do not interact (ideal gas) If molecules interact (non-ideal gas), we can no longer talk about the average energy being defined in terms of averages over single molecule energies The microscopic energies are now functions of interactions between all the molecules in the system. The Boltzmann distribution - warnings

26. In this case an analogous derivation is performed which involves a large collection, or ensemble, of isolated systems, which are in thermal contact Thus the systems have the same N, V, and T. The thermodynamic energy is now the average of the system energies over the ensemble. The Boltzmann distribution - warnings

27. Near 0K, only the ground level is populated: all molecules (or all systems) are in the lowest-energy level As the temperature is raised, other levels become progressively more populated As the temperature becomes very high, all energy levels become equally populated because the exponential factors all approach 1 Properties of the Boltzmann distribution

28. The magnitude of k b T is a significant characteristic of a system. If the energy difference between levels is small compared to k b T, many energy levels are populated If the spacing is large (compared to k b T), only the lowest energy level is populated. Properties of the Boltzmann distribution

29. Now that the distribution is available, we can calculate any thermodynamic property, if we know the energy levels (which, remember, in general is very difficult). For example, the thermodynamic, internal energy is the average energy: Thermodynamic properties from the Boltzmann distribution

30. Thermodynamic properties of a system can be obtained from the distribution function. For example Thermodynamic properties from the Boltzmann distribution For an isolated system composed of N non-interacting particle, it can be shown that

31. Starting from: Thermodynamic properties from the Boltzmann distribution Take the derivative of q with respect to T at constant V Use the fact that:

32. Calculation of the molecular partition function and average energy for the translational motion of an ideal gas in a one-dimensional box of length a In Lecture 1 we have provided the equation for the quantized energy levels for a particle in a box: Use of the ‘partition’ function

33. In deriving this expression, we have assumed that the energy levels are spaced very close together (corresponding to a system of large mass, or a macroscopic, classical system), so the summation can be replaced by an integral, which in turn can be evaluated using the expression: Use of the ‘partition’ function

34. To a high degree of approximation, the energy of a molecule in a particular state is the simple sums of various types of energy (translational, rotational, vibrational, electronic, etc.): Why the name ‘partition’ function?

35. It can easily be shown that for an ideal gas in a three- dimensional box the partition function is just the cube of the one dimensional partition function: Equipartition principle From which it can be shown that: Equi-partition Principle: the energy is equally partitioned among all translational degrees of freedom.

36. Do you remember what classical entropy is?

37. In the context of classical thermodynamics, entropy has been introduced as a measure of the disorder of a system. It is therefore very reasonable to expect the entropy to be related to W, the number of ways of distributing the molecules in a system among their energy levels. Boltzmann showed that the entropy is in fact related to the most-probable distribution W max (if a system contains more than a few hundred particles, that is the only distribution that need be considered): Statistical mechanical entropy

38. This relationship provides a molecular interpretation of entropy, in that it directly relates it to the way the molecules in a system are distributed among its different energy levels, which in turn depends on the energy levels themselves as well as the temperature of a system. Statistical mechanical entropy

39. Does this definition make sense given what we know from classical thermodynamics? As T approaches 0K, all molecules will be in the lowest energy state and the Boltzmann distribution will approach unity (hence S=0) As T becomes progressively higher, the entropy increases Statistical mechanical entropy

40. From the point of view of statistical mechanics, the third law of thermodynamics is a simple consequence of the occupation of the lowest available energy levels at temperature approaching the absolute zero. The second principle of thermodynamics states that the equilibrium state is the state of maximum entropy for an isolated system. From the statistical mechanical perspective, the equilibrium state of a system represents the most probable distribution and has maximum randomness. Statistical mechanical entropy

41. Looking at it from the opposite perspective, you can imagine that the Boltzmann distribution and the definition of entropy given above are statistical mechanical formulations of the second and third principles of thermodynamics (while the first principle, conservation of energy, was explicitly incorporated in the derivation of the Boltzmann distribution).