Statistical methods can be applied to molecules, photons, wave functions elastic waves in solids Particle
2.1 Basic Concepts
A basic idea embodied in statistical thermodynamics is: Even when a material is in equilibrium on a macroscopic scale, it is dynamic on a microscopic scale.
For example An atom in a solid vibrates at a frequency in the order of 1013 s1 Gas molecules have velocities of order of 102 ms1
When we observe a property of a system, we really see the average of this property in all of the microstates that the material passed through during the observation time
Entropy S
In science, the S is generally interpreted in 1. Macroscopic classical thermodynamics 2. Microscopic statistical thermodynamics 3. Information theory The statistical definition of S is the fundamental one while the other two can be mathematically derived from it, but not vice versa.
» A finite array of microstates satisfies two constraints
N
Where:
i
i
N;
N
i i
i
E U
N : the number of particles; i : energy level; E : total energy of N particles; i : the energy of ith level
Microscopic (statistical) thermodynamics : » Attempts to compute absolute values of thermodynamic quantities based on a statistical averaging of such properties of individual atoms or molecules as the m, V, bond strength, vibration frequencies » Relies heavily on quantum mechanics, and knowledge of molecular motion and structures.
»Be concerned with relative changes of
macroscopic properties, such as P, T, Cp,m »Does not require any knowledge of the atomistic nature of matters. »Can ignore the existence of molecules, molecular complexity, and quantum mechanics
and separate shelves with slashes: the number of dot: N ; slash: g1 Total location: N + g 1
The problem is now to reduce to problem 3: how many ways can we select N distinguishable dot locations and g 1 distinguishable slash locations from N+g1 dot and slash locations?
The first problem that we must cope with Liken it to the problem of placing N copies
of the same book amonate the books with identical dots,
To be derived
Dynamic distribution Distribution consequences Macrostate »Be characterized by a few state variables, such as T, V, and U etc. »Be the state of all particles in the system »Passes very rapidly through many microstates during the observation time
The number of microstates for each macrostate will be equal to the number of ways in which we can choose these Ni from N particles.
Clearly, what is required here is a statistical treatment of the problem.
Statistical Thermodynamics
统计热力学基础
There are two approaches to the study of thermodynamics: Macroscopic thermodynamics Microscopic thermodynamics
Macroscopic thermodynamics :
i 1
Problem 3.
How many ways can we select N distinguishable objects from a set of g distinguishable objects? Solution: This problem can now reduce to special case of problem 2, Such that: there are 2 shelves to put N books and gN books The answer is: g!/[N!(gN)!]
Problem 4.
If there is no limit on the number of objects in any box, how many ways can we put N indistinguishable objects into g distinguishable boxes? Solution:
Principle of equaling a priori probability
(1)试验的所有可能结果是有限的; (2)每一种可能结果出现的可能性(概率)相等
Problem 1. How many ways can we arrange N distinguishable objects? Solution:
The answer is: (N+g1)!/[N !(g1)!]
Problem 5.
How many ways can we put N distinguishable objects into g distinguishable boxes? Solution: Each of N different books can be put on any of g shelves The 1st book: g ways; the 2nd book: g ways and so on The answer is: gN
»degeneracy of level (g) the number of states with the equal energy
Shifting of particles from one state to another on the same level constitutes a new arrangement
Problem 2. How many ways can we put N distinguishable objects into r different boxes, (regardless of order within the boxes), such that there are N1 objects in the first box, N2 in the second, , and Nr in the r-th box? Solution: N ! must be divided by N1!, N2!, , Ni!, , Nr1!, Nr! to account for the meaningless rearrangements. r The answer is: N ! / N r !
What we can do is to take advantage of the large number of atoms in the system to make statistical descriptions of the behavior of the atoms that make up the material based on their microscopic behavior What we really want is Statistical information regarding the motions of all particles in the system