1. Multiplicity, Entropy, Second & Third Law Of Thermodynamics Flashcards
Multiplicity of Einstein Solid
Ω(N, q) = (q + N - 1)! / q! (N- 1)!
Multiplicity of Two-State Paramagnet System
Ω(N up) = N! / N up! N down!
Fundamental Assumption of Statistical Mechanics
In an isolated system in thermal equilibrium, all accessible microstates are equally probable.
Second Law of Thermodynamics
For spontaneous processes, the entropy of the universe increases and approaching greater disorder and randomness.
A system will naturally evolve towards the macro state with the highest multiplicity (most possible arrangements of its micro states)
Multiplicity of Large Einstein Solid
lnΩ = ln((q + N)! / q!N1)
If q»_space; N
Ω(N, q) = e^Nln(q/N)e^N = (eq/N)^N
If q «_space;N
Ω(N, q) = (Ne/q)^q
Multiplicity of Mono-atomic Ideal Gas
In 3D:
Ω = V Vp / h^3
Ω N = 1/N! (V^N/ h^3N)
1D:
L Lp/ Δx Δpx = L Lp/h
Entropy
(Relationship between Entropy and Multiplicity)
S = klnΩ
Entropy of Large Einstein Solid
S = kln(eq/N)^N = Nk[ln(q/N) + 1]
Entropy of an Ideal Gas
S = Nk[ ln( V/N (4pimU/3Nh^2)^3/2) + 5/2] (Sackur-Tetrode Equation)
Change in Entropy of Ideal Gas
Change in S = Nkln(Vf/Vi) (General)
Change in S = Q/T (any Quasistatic process)
Entropy of Mixing
Change in Sa = Nkln(Vf/Vi) = Nkln2
Change in S total = Change in Sa + Change in Sb = 2Nkln2
Relationship between Temperature, Entropy, and Potential Energy
1/T = (dS/dU)N, V —> T = (dU/dS) N, V
Energy will always tend to close into Theo object with stepper S vs. U graph and out of object with shallows S vs. U graph. (Steep slope is low temp, shallow is high temp)
Thermodynamic Identity & How it’s derived
dU = TdS - PdV
S = f(U, V)
dS = dS/dU dU + dS/dV dV
dS = 1/T dU + P/T dV
dU = TdS - PdV
Change in Entropy
dS = Q/T (constant volume, no work)
dS = CvdT/T (T is changing)
- integrate for ΔS
Third Law of Thermodynamics
The entropy of a system approaches a constant value as the temperature approaches absolute zero. (System’s unique, lowest energy state)
