Thermal Physics Exam 3 Flashcards

Flashcards generated from Thermal Physics Exam 3 lecture notes covering partition functions, quantum statistics, heat capacity, and the Maxwell distribution.

How is entropy related to Helmholtz free energy at fixed volume and number of particles?

S = - (∂F/∂T) at constant V and N

Why is the 1/N! term needed in the total partition function for N identical and indistinguishable states?

The 1/N! term corrects for overcounting microstates when particles are identical and indistinguishable.

Under what conditions do Boltzmann, Fermi-Dirac, and Bose-Einstein distributions apply?

Boltzmann distribution applies to low density, distinguishable particles. Fermi-Dirac applies to high density fermions (e.g., electrons). Bose-Einstein applies to high density bosons (e.g., photons).

What is the conceptual difference between Z and Ξ in terms of the system, the reservoir, and what s represents in the sum?

The partition function sums over the microstates of a system, weighted by their Boltzmann factors. The grand partition function considers a system in contact with a reservoir, explicitly accounting for the number of particles that can be exchanged.

When is the Fundamental Assumption of Statistical Mechanics valid, and does it apply equally across all temperatures in the provided plot?

The fundamental assumption is that all accessible microstates are equally probable only when the system is in thermal equilibrium. The plot suggests this is more valid at higher temperatures as the system explores more states.

What is the approximate spacing between energy levels (Δ) in the system, expressed in terms of Δ/kB?

The approximate spacing between energy levels (Δ) is related to the temperature at which the heat capacity peaks. Eyeballing the graph, the peak seems to be around ~10K. Thus Δ/kB ~ 10K.

How many degrees of freedom does this system have at high temperatures?

The system has one degree of freedom at high temperatures.

Which law(s) of thermodynamics are used and/or illustrated AND is your sketch correct for a real system?

Law(s) of thermodynamics: The Third Law of Thermodynamics states that the entropy of a system approaches a minimum (usually zero) as the temperature approaches absolute zero. The sketch is correct for a simplified system. However, in a real system, there might be residual entropy due to, for example, degeneracy in the ground state.

Calculate the equation for the pressure from the partition function Ztot = (e^(NV)/(vq))* (kT/(2*omega))^(N)

P = NkT(∂ ln(V) / ∂V)|_{T,N} = NkT/V

What is the origin of the e^(-mv^2/2kT) term in the Maxwell Distribution of velocities of a gas?

Boltzmann factor that appears in the probability of a state with energy E: P(E) ~ e^(-E/KT)

What is the origin of the 4πv^2 term in the Maxwell Distribution of velocities of a gas?

Jacobian needed to transform from cartesian velocity components to spherical velocity components in finding the number of states between v and v+dv.