phys documents (dragged) 35

phys documents (dragged) 35 - S = 5 2 kN kN ln ² V(2 π...

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38 Physics Formulary by ir. J.C.A. Wevers 8.12 Statistical basis for thermodynamics The number of possibilities P to distribute N particles on n possible energy levels, each with a g -fold degen- eracy is called the thermodynamic probability and is given by: P = N ! ± i g n i i n i ! The most probable distribution, that with the maximum value for P , is the equilibrium state . When Stirling’s equation, ln( n !) n ln( n ) - n is used, one Fnds for a discrete system the Maxwell-Boltzmann distribution. The occupation numbers in equilibrium are then given by: n i = N Z g i exp ² - W i kT ³ The state sum Z is a normalization constant, given by: Z = i g i exp( - W i /kT ) . ±or an ideal gas holds: Z = V (2 π mkT ) 3 / 2 h 3 The entropy can then be deFned as: S = k ln( P ) . ±or a system in thermodynamic equilibrium this becomes: S = U T + kN ln ² Z N ³ + kN U T + k ln ² Z N N ! ³ ±or an ideal gas, with U = 3 2 kT then holds:
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Unformatted text preview: S = 5 2 kN + kN ln ² V (2 π mkT ) 3 / 2 Nh 3 ³ 8.13 Application to other systems Thermodynamics can be applied to other systems than gases and liquids. To do this the term d W = pdV has to be replaced with the correct work term, like d W rev =-Fdl for the stretching of a wire, d W rev =-γ dA for the expansion of a soap bubble or d W rev =-BdM for a magnetic system. A rotating, non-charged black hole has a temparature of T = ¯ hc/ 8 π km . It has an entropy S = Akc 3 / 4¯ h κ with A the area of its event horizon. ±or a Schwarzschild black hole A is given by A = 16 π m 2 . Hawkings area theorem states that dA/dt ≥ . Hence, the lifetime of a black hole ∼ m 3 ....
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