Summary Boltzman statistics FermiDirac statistics BoseEinstein statistics MaxwellBoltzmann

Summary • • • Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13 -4: Show that for a system of N particles obeying Maxwell-Boltzmann statistics, the occupation number for the jth energy level is given by

13. 7 The Connection between Classical and Statistical thermodynamics U = Σ NJ EJ where EJ = EJ (x) x is an extensive property such as volume (see eqn 12. 21!) Differentiate the above equation Since

define For two states with the same X For classical thermodynamics

Since d. Q = T·d. S Yd. X and d. W = Pd. V = The first equation illustrates that the heat transfer is energy resulting in a net redistribution of particles among the available energy levels involving NO WORK!

The equation could be interpreted as: An increase in the system’s internal energy could be brought by a decrease in volume with an associated increase in the EJ. the energy levels are shifted to higher values with no redistribution of the particles among the levels. For an open system, the change in the internal energy is where μ is the chemical potential defined on a per particle.

The Helmholtz function (F = U –TS) can be written as For MB statistics

Now the MB distribution can be rewritten as (similar to those derived in FD and BE statistics)

13. 8 Comparison of the three Distributions a = 1 for FD statistics a = -1 for BE statistics a = 0 for MB statistics

• BE curve: The distribution is undefined for x < 0. particles tend to condense in regions where Ej is small, that is, in the lower energy state. • FD curve: At the lower levels with Ej – u negative the quantum states are nearly uniformly populated with one particle per state. • MB curve: lies between BE and FD curves and is only valid for the dilute gas region: many states are unoccupied. • Statistical equilibrium is a balance between the randomizing forces of thermal agitation, tending to produce a uniform population of the energy levels, and the tendency of mechanical systems to sink to the states of lowest energy.

Alternative Statistical Models • Microcanonical ensemble: treats a single material sample of volume V consisting of an assembly of N particles with fixed total energy U. The independent variables are V, N, and U. • The canonical ensemble: considers a collection of Na identical assemblies, each of volume V. A single assembly is assumed to be in contact through a diathermal wall with a heat reservoir of the remaining Na -1 assemblies. The independent variables are V, N, and T, where T is the temperature of the reservoir. • Grand canonical ensemble: consists of open assemblies that can exchange both energies and particles with a reservoir. This is the most general and most abstract model. The independent variables are V, T and u, where u is the chemical potential.

Further comments on degeneracy • A microstate is said to be non-degenerate if no two particles have the same energy. • Alternative way of defining a non-degenerate system is to say that the number of quantum states with E <k. T, >> N, where k is Boltzmann constant, T is the temperature and N is the number of particles. • Equation 12. 24 can be employed to calculate the number of states contained within the octant.

• Example: Calculate the total number of accessible microstates in a system where 100 units of energy have been distributed among three distinguishable particles of zero spin. • Solution:

• 13 -10 (a) using results from chapter 9, show that (b) It follows from the statistical definition of the entropy that Consider a system with a chemical potential u = - 0. 3 e. V. By what factor is the number of possible microstates of the system increased when a single particle is added to it at room temperature? (k = 8. 617 x 10 -5 e. VK-1) • Solution: (a) using equation 13. 53 directly

Chapter 14: The Classical Statistical Treatment of an Ideal Gas

14. 1 Thermodynamic properties from the Partition Function • All thermodynamic properties can be expressed in terms of the logarithm of the partition function and its derivatives. • Thus, one only needs to evaluate the partition function to obtain its thermodynamic properties!

• From chapter 13, we know that for dilute gas system, M-B statistics is applicable, where S= U/T + Nk (ln Z – ln N +1) F= -Nk. T (ln Z – ln N +1) μ= = – k. T (ln. Z – ln. N) • Now one can derive expressions for othermodynamic properties based on the above relationship.

1. Internal Energy: We have …

differentiating the above Z equation with respect to T, we have … = (keeping V constant means ε(V) is constant) Therefore, or

2. Gibbs Function since G = μ N G = - Nk. T (ln Z – ln N) 3. Enthalpy G = H – TS → H = G + TS H =-Nk. T(ln. Z - ln N) + T(U/T + Nkln. Z – Nkln. N + Nk) = -Nk. Tln. Z + Nk. Tln. N + U + Nk. Tln. Z – Nk. Tln. N + Nk. T = U + Nk. T = Nk. T 2 + Nk. T = Nk. T (1 + T · )

4. Pressure