18 January 2017
L1: Statistical Thermodyamics and the Boltzman Factor
I. Intro to Statistical Thermodynamics
A. Thermodynamics
1. Thermo = temp/heat, dynamics = motion/change
a. In a classical view, there is a continuum of energies
b. In a QM view, there are distinct states (QM is universal)
c. Looks at how the heat changes the occupation or distribution of energy levels
d. Developed before quantum (empirically developed in 1800s)
e. Laws are universal, based on empirical foundations, 0 Th Law + 3 Laws that
describe all of thermo
B. Statistical Mechanics
1. Links the QM and thermodynamics
a. Properties of individual molecules and atoms from Q0 (H =0E) 0
b. Macroscopic thermodynamic properties, ∆G = ∆H – T∆S = RTlnK P
c. Ex: useful when looking at the ideal gas law (PV = nRT)
• P, pressure should just be the summation of all the tiny energies of each
molecule of a gas hitting the walls of a chamber
• Hugely impractical energy calculation, other ways of looking at with
statistical mechanics
2. Goals
a. To describe macroscopic thermodynamic properties in terms of microscopic
atomic and molecular properties
3. Properties of a system can be described at two levels
a. Macroscopic thermodynamic description (p, V, n, C , H, A, Gv…)
b. Microscopic description that specifies the state of each molecule, can use classical
or quantum mechanics, uses more than 10 variables, need to update every 10 -15s
c. Either classical or quantum description is impractical
• Statistical mechanics describes macroscopic mechanics in statistical terms (ie
average, most probable results)
II. The Boltzman Factor + Probability
A. Probability and Energy States
1. Probability of a molecule in state I, with energy 𝜀 = P𝜀 𝑖 i𝑖
2. Independent probabilities
a. Ex: 2 people toss a coin, each time a 50% change of being heads, so for two times
(0.5)(0.5) = 0.25 change of both being heads
b. Independent energies occur when they are independent degrees of freedom
• P ij 𝑖 𝜀 )𝑗 (P𝜀 )iP𝑖 ) j𝑗
3. For energy states, think of as a function of each energy state… must use exponentials
a. 𝑒 𝐶(𝜀𝑖+𝜀 𝑗 = 𝑒 𝐶𝜀𝑖+ 𝑒 𝐶𝜀𝑗
𝐶𝜀
b. Thus, P𝜀 ∝ 𝑒 − 𝑇 , P𝜀 should go down as energy goes up
i 𝑖 i 𝑖
• As 𝜀 𝑖ncreases, there is a smaller probability of finding the molecule in that
state (function drops off faster) • At low T, drops off slower (smaller area under curve), at colder temperatures
you would not expect higher energies to be more probable
c. The probability depends on the ratio of the energy to the temperature
d. We expect high-energy states to be less probable than low-energy states, and that
they become more probable at high T

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