1. Thermodynamics 1.1. A macroscopic view of matter

Transcription

1 1. Thermodynamics 1.1. A macroscopic view of matter Intensive: independent of the amount of substance, e.g. temperature,pressure. Extensive: depends on the amount of substance, e.g. internal energy, enthalpy. Specific Volume: Mechanical work along a reversible path A B: Compressibility: = = 1 density: Heat transfer: = = = Coefficient of thermal expansion: = 1 Ideal gas law (limit of low density and high temperature): = = Avogadro s constant: = Heat capacity for constant volume: = 2 = First law of thermodynamics: 3 2 = Note that both ΔQ and ΔW are pathdependent, while ΔU is path-independent and thus a state variable! = Magnetic systems: = = First law: = + When a magnetic field is applied: = Page 1 of 13

2 External magnetic field: H (intensive) Magnetization: M (extensive) H -P M V Ferromagnetic systems: See L1.8 or Huang p Magnetic equation of state: = = 1.2. Heat and Entropy Heat equations: See L2.1 or Huang p. 15 Enthalpy: = + Heat capacities: = = Internal energy of an ideal gas: = = Adiabatic transformation: ΔQ = 0 always reversible Adiabatic exponent: = = = = Cyclic process: Efficiency of a carnot cycle: = 0 = = = 1 For any cyclic process: Page 2 of 13

3 Entropy: Equals sign applies for reversible transformations only! SB SA =, = + ln + ln For monatomic gases = and thus:, = + ln Sacker-Tetrode: with: = 5 2 ln = h = Using Thermodynamics Energy equation: = = Measurable coefficient: = 1 = 1 = 1 Derivation of the above-mentioned laws and some other useful relations: See Huang p TdS-equations: See L3.2 or Huang p. 35 Reversible expansion: = 0 Irreversible expansion: = ln > 0 Page 3 of 13

4 = = Helmholtz Free Energy: = = = Gibbs Free Energy = + = Maxwell Relations: See L3.6 or Huang p. 42 = = Chemical potential: If the number of particles N changes, the chemical potential μ describes the amount of energy needed to add one particle to a thermally and mechanically isolated system. First law becomes then: = + =, =, =, =, First order phase transitions: First derivatives are continuous Coexisting phases have the same P and T, but can differ in V and S. Δv = Change in specific volume Latent heat: = Clausius-Clapeyron-Equation: = = Page 4 of 13

5 Van-der-Waals-equation for real gases: + = = 8 27 = 27 = 3 When system breaks up in a mixture of phases with different ρ: = See L4.6 and L4.7 Page 5 of 13

6 2. Statistical Physics STP (standard temperature and pressure): = , = 1 Cross-section of molecules: Mean free path: = = = 3 Probability of finding a particle k steps away from the starting point after a random walk with n total steps:, = 1 2 =, = 0 = 4 Probability of finding a particle at a distance x from the starting point after a time t: 1, = 4 = 2 where x 0 is size of a step and t 0 the duration of the random walk Hamiltonian H of a system:, = For ideal gases, the internal energy is equal to the kinetic energy, as there is no potential U. Phase space Γ-space: 6N-dimensional space (3 for the position and 3 for 3 for the momenta of each particle), where every point represents a state of the system. The evolution of a system is therefore represented by a trajectory. μ-space: 6-dimensional space where every particle is represented by a point that defines its position and momentum. Phase space: See Huang p. 69 Page 6 of 13

7 Distribution function: =,, =, For any quantity O, the ensemble average :, =, 2 =,,, =,,, =,, = ln, =,, = 1 = 2! 2, =, = ln + See L5.7 With a N!-times smaller number of microstates:, = ln + = Number of microstates (correct Boltzmann counting): = 1!!! See Huang p. 74 Page 7 of 13

8 Distribution entropy (Boltzmann s H- function): = ln = = = = = ln For ideal gases: = = 1 = 3 2 = 2 = 2 = = = 2 = = = 2 Pressure of ideal gas: = 2 = With: = = 1 3 = 1 3 = 2 3 Specific heat: = 1 = 2 Page 8 of 13

9 Equipartition of energy: Each degree of freedom adds to Distribution of speed: = 1 = 1 = 0 = 3 Mean kinetic energy: 1 2 = 3 2 Most probable speed: = 2 For ideal gases: = = 3 2 = 3 2, = ln + For constant N: For constant N: = ln ln + Probability of finding the system in state i: = Transport phenomena Total flux: See Huang p. 99 = = 2 Hydrodynamic regime: See L7.1 = Page 9 of 13

10 Microcanonical ensemble: All members have the same energy = ln is the total number of states at energy E Absolute temperature: 1 = Canonical ensemble: Distribution function of system 1:, =, =, = 1 Partition function:,, =! h = ln, =, Energy fluctuation: = Negligible for See L8.3 = = 1 = = 1 = / = / = 1 = 1 = 1 Page 10 of 13

11 Ideal gas: =, = 1! h = 2, = ln, = ln 1 = 1 = = ln, = 5 2 ln = ln ln 2 h Page 11 of 13

12 3. Mathematical Appendix Stirling approximation: ln! ln Huang p. 305 Exact differential:, = + Huang p. 306 Partial derivatives: = w is some function of the three variables x, y and z. = 1 Another useful relationship: + = Huang p. 307 Chain rule: = 1 Huang p. 307 Taylor-Approximation in the neighbourhood of x 0 : + 1! + 2! + +! Gaussian Integrals: = = 3 4 / = 1 2 / Huang p. 83 Page 12 of 13

13 4. Constants and units 1 = = = = = h = Page 13 of 13