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Thermodynamics and Statistical Mechanics

Thermodynamics and Statistical Mechanics. Entropy. Thermodynamic Probability. Distribution. N = 4 U = 3 e k 1 2 3 3 e 1 2 e 1 1 e 1 3 0 3 2 1 w 4 12 4. Combining Systems. Consider two systems. System A: Number of arrangements: w A

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Thermodynamics and Statistical Mechanics

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  1. Thermodynamics and Statistical Mechanics Entropy Thermo & Stat Mech - Spring 2006 Class 17

  2. Thermodynamic Probability Thermo & Stat Mech - Spring 2006 Class 17

  3. Distribution • N = 4 U = 3e • k 1 2 3 • 3e 1 • 2e 1 • 1e 1 3 • 0 3 2 1 • w 412 4 Thermo & Stat Mech - Spring 2006 Class 17

  4. Combining Systems • Consider two systems. • System A: Number of arrangements: wA • System B: Number of arrangements: wB • Combined systems: wA × wB Thermo & Stat Mech - Spring 2006 Class 17

  5. 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 Thermo & Stat Mech - Spring 2006 Class 17

  6. Wave Equation Thermo & Stat Mech - Spring 2006 Class 17

  7. Boundary Conditions Thermo & Stat Mech - Spring 2006 Class 17

  8. Energy of Particles Thermo & Stat Mech - Spring 2006 Class 17

  9. Density of States • The allowed values of k can be plotted in k space, and form a three dimensional cubic lattice. From this picture, we can see that each allowed state occupies a volume of k space equal to, Thermo & Stat Mech - Spring 2006 Class 17

  10. 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, Thermo & Stat Mech - Spring 2006 Class 17

  11. 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, Thermo & Stat Mech - Spring 2006 Class 17

  12. 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. Thermo & Stat Mech - Spring 2006 Class 17

  13. Density of States gis the number of states with the same k, or the number of particles that one k can hold. Thermo & Stat Mech - Spring 2006 Class 17

  14. Density of States • In terms of energy of a particle: Thermo & Stat Mech - Spring 2006 Class 17

  15. Free Electrons Thermo & Stat Mech - Spring 2006 Class 17

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