5. Einfache Anwendungen

• Zustandsgleichung des idealen Gases:

  
  

  $$\Omega {1 \over h_0^f} \int_{E}^{E+\delta E} \int \prod_{i}^{} \xi$$ = (58)

  $$\overline{p} \overline{X}_{V}^{} {\partial \ln(\Omega) \over \partial V} {N \over V}$$ = (59)

  $$\folgt \nu$$ = (60)

  
  

  F�r die Temperatur gilt
  

  $${1 \over T} \left.\vphantom{ k_B {\partial \ln(\Omega) \over \partial E}}\right. {\partial \ln(\Omega) \over \partial E} \left.\vphantom{ k_B {\partial \ln(\Omega) \over \partial E}}\right\vert _{\overline{E}}^{} \left.\vphantom{ k_B {\partial \ln(\xi(E)) \over \partial E}}\right. {\partial \ln(\xi(E)) \over \partial E} \left.\vphantom{ k_B {\partial \ln(\xi(E)) \over \partial E}}\right\vert _{\overline{E}}^{}$$ = (61)

  $$\folgt \overline{E} \overline{E}$$ = (62)

  
  
  d. h. die innere Energie U des idealen Gases h„ngt von der absoluten Temperatur T ab.

• W„rmekapazit„t

  $$\left(\vphantom{ {\delta Q \over dT}}\right. {\delta Q \over dT} \left.\vphantom{ {\delta Q \over dT}}\right)_{g}^{}$$ (63)

  analog auch bezogen auf die Stoffmenge oder ein Mol. F�r reversible Prozesse mit $\delta$Q = TdS

  $$\left(\vphantom{ {\partial S \over \partial T}}\right. {\partial S \over \partial T} \left.\vphantom{ {\partial S \over \partial T}}\right)_{g}^{}$$ (64)

  weiterhin gilt
  

  $${1 \over T} \left.\vphantom{ {\partial S \over \partial E}}\right. {\partial S \over \partial E} \left.\vphantom{ {\partial S \over \partial E} }\right\vert _{E=U}^{} {\partial k_B \ln(\Omega(U,V)) \over \partial U}$$ = (65)

  $$\left(\vphantom{ {\partial U \over \partial T}}\right. {\partial U \over \partial T} \left.\vphantom{ {\partial U \over \partial T}}\right)_{V}^{} \left(\vphantom{ {\partial U \over \partial V}}\right. {\partial U \over \partial V} \left.\vphantom{ {\partial U \over \partial V}}\right)_{T}^{}$$ dU = (66)

  $$\folgt \delta \left(\vphantom{ {\partial U \over \partial T}}\right. {\partial U \over \partial T} \left.\vphantom{ {\partial U \over \partial T}}\right)_{V}^{} \left[\vphantom{ \left({\partial U \over \partial V}\right)_T + p }\right. \left(\vphantom{ {\partial U \over \partial V}}\right. {\partial U \over \partial V} \left.\vphantom{ {\partial U \over \partial V}}\right)_{T}^{} \left.\vphantom{ \left({\partial U \over \partial V}\right)_T + p }\right]$$ = (67)

  
  
  
  
  

• Kompressionsmodul (isothermer)

  $$\left(\vphantom{ {\partial p \over \partial V}}\right. {\partial p \over \partial V} \left.\vphantom{ {\partial p \over \partial V}}\right)_{T}^{}$$ (68)

  Kompressibilit„t (isotherme): $\kappa$ = ${1 \over K}$

• Ausdehnungkoeffizient (isobar)

  $$\alpha {1 \over V} \left(\vphantom{ {\partial V \over \partial T}}\right. {\partial V \over \partial T} \left.\vphantom{ {\partial V \over \partial T}}\right)_{p}^{}$$ (69)

• C_p - C_V:

  $$\left(\vphantom{ {\partial p \over \partial T}}\right. {\partial p \over \partial T} \left.\vphantom{ {\partial p \over \partial T}}\right)_{V}^{} \left(\vphantom{ {\partial V \over \partial T}}\right. {\partial V \over \partial T} \left.\vphantom{ {\partial V \over \partial T}}\right)_{p}^{} \alpha^{2}_{}$$ (70)

  und damit C_p > C_V. Z. B: f�r ein ideales Gas: c_p = c_V + R (Molw„rmen)

• Adiabatengleichung

  $$\gamma$$ (71)

  bei quasistatischer Expansion eines thermisch isolierten idealen Gases

Beispiel: einatomiges ideales Gas

$$\Omega \halbe$$ = (72)

$${1 \over T} \left.\vphantom{ k_B {\partial \ln(\Omega) \over \partial E}}\right. {\partial \ln(\Omega) \over \partial E} \left.\vphantom{k_B {\partial \ln(\Omega) \over \partial E}}\right\vert _{E=U}^{} {3 N \over 2} {k_B \over U}$$ = (73)

$${3 \over 2} {3 \over 2} \nu$$ U = (74)

$$\Omega {3 N \over 2} \left(\vphantom{ {3 \over 2} \nu R T}\right. {3 \over 2} \nu \left.\vphantom{ {3 \over 2} \nu R T}\right)$$ S = (75)

$${3 \over 2}$$ c_V = (76)

$${5 \over 2}$$ c_p = (77)

$$\gamma {c_p \over c_V} {5 \over 3} \approx$$ = (78)