Thermodynamic Probability - Thermodynamics and Statistical Mechanics - Lecture Slides, Slides of Thermodynamics

Download Thermodynamic Probability - Thermodynamics and Statistical Mechanics - Lecture Slides and more Slides Thermodynamics in PDF only on Docsity! Thermodynamics and Statistical Mechanics Entropy Docsity.com Thermodynamic Probability UN NN N N NNN Nw n j jj n j j i = = ∏ = ⋅⋅⋅⋅ = ∑ ∑ = = 1 1 321 ! ! !!! ! ε Docsity.com Entropy • S = k ln w • SA = k ln wA SB = k ln wB • SA+B = k ln(wA × wB) = k ln wA + k ln wB • SA+B = SA + SB Docsity.com Wave Equation ∇ + =2 2 0ψ ψk k = 2π λ ψ ψ ψ ψ( , , ) ( ) ( ) ( )x y z x y zx y z= 022 2 =+ yy y k dy d ψ ψ k k k kx y z 2 2 2 2= + + d dx kx x x 2 2 2 0 ψ ψ+ = 022 2 =+ zz z k dz d ψψ Docsity.com Boundary Conditions ( )222 2 2 22 2 22 2 22 2 zyx zyx nnn LL n L n L nk ++     =++= ππππ k k k kx y z 2 2 2 2= + + Docsity.com Density of States •All the values of k that have the same magnitude fall on the surface of one octant of a sphere in k space, since nx, ny, and nz are positive. The volume of that octant is given by, V k kk = = 1 8 4 3 1 6 3 3π π Docsity.com Density of States •Then, the volume of a shell that extends from k to k + dk can be obtained by differentiating the expression for Vk, dV k dk k dkk = = 1 6 3 2 2 2π π Docsity.com Density of States •If we divide this expression by the volume occupied by one state, we will have an expression for the number of states between •k and k + dk. dN dV V k dk L L k dk V k dkk s = =     = = π π π π 2 2 2 2 3 3 2 2 2 2 Docsity.com