PCRT.I.D.07

PCRT.I.D

07 Zustandsfunktionen des idealen Gases bei (T,V;N=const)

A Freie Energie und molare freie Energie

$$	$-TS$	$=$	$-nRT\ln \left[{\left(\frac{T}{T_0}\right)}^{3/2}\left(\frac{V}{V_0}\right)\right]-S_{0}T$	$S(T,V,N)$
$$	$S_0$	$=$	$\frac{5}{2}nR$	$S(T,V,N)$
$$	$U$	$=$	$\frac{3}{2}nRT$	$U(T,V,N)$
$$	$F$	$=$	$U-TS$	$F(T,V,N),U(T,V,N),S(T,V,N)$
$$	$F$	$=$	$-nRT-nRT\ln \left[{\left(\frac{T}{T_0}\right)}^{3/2}\left(\frac{V}{V_0}\right)\right]$	$F(T,V,N)$
$(01)$	$F$	$=$	$+nRT\ln \left[{\left(\frac{T_0}{T}\right)}^{3/2}{\left(\frac{1}{e}\right)}^{2/2}\right]+nRT\ln \left(\frac{V_0}{V}\right)$	$F(T,V,N),T_0<T,V_0<V,T\uparrow :F\downarrow$
$(02)$	$f$	$=$	$+RT\ln \left[{\left(\frac{T_0}{T}\right)}^{3/2}{\left(\frac{1}{e}\right)}^{2/2}\right]+RT\left(\frac{v_0}{v}\right)$	$f(T,v),T_0<T,v_0<v,T\uparrow :f\downarrow$
$(03)$	$f$	$=$	$+RT\ln \left[{\left(\frac{T_0}{T}\right)}^{5/2}{\left(\frac{1}{e}\right)}^{2/2}\right]+RT\left(\frac{p}{p_0}\right)$	$f(T,p),T_0<T,p<p_0,T\uparrow :f\downarrow$
$(04)$	$f$	$=$	$+pv\ln \left[{\left(\frac{p_0}{p}\right)}^{3/2}{\left(\frac{v_0}{v}\right)}^{5/2}{\left(\frac{1}{e}\right)}^{2/2}\right]$	$f(v,p),v_0<v,p_0<p,v\uparrow ,p\uparrow :f\downarrow$

Übung a

Wir berechnen $(T/F)(\partial F/\partial T)$, $(V/F)(\partial F/\partial V)$, $(TV/U)({\partial }^{2}F/\partial T\partial V)$ für ein ideales Gas im (T,V,N)-Koordinatensystem.

B Innere Energie und molare innere Energie

$(01)$	$U$	$=$	$\frac{3}{2}nRT$	$U(T,V,N)$
$(02)$	$u$	$=$	$\frac{3}{2}RT$	$u(T,v)$

C Freie Enthalpie und chemisches Potential

$$	$+pV$	$=$	$+nRT$	$p(T,V,N)$
$$	$G$	$=$	$F+pV$	$G(T,V,N),F(T,V,N),p(T,V,N)$
$$	$G$	$=$	$-nRT\ln \left[{\left(\frac{T}{T_0}\right)}^{3/2}\left(\frac{V}{V_0}\right)\right]$	$G(T,V,N)$
$(01)$	$G$	$=$	$+nRT\ln {\left(\frac{T_0}{T}\right)}^{3/2}+nRT\left(\frac{V_0}{V}\right)$	$G(T,V,N),T_0<T,V_0<V,T\uparrow :G\downarrow$
$(02)$	$\mu$	$=$	$+RT\ln {\left(\frac{T_0}{T}\right)}^{3/2}+RT\ln \left(\frac{v_0}{v}\right)$	$\mu (T,v)=g(T,v),T_0<T,v_0<v,T\uparrow :\mu \downarrow$
$(03)$	$\mu$	$=$	$+RT\ln {\left(\frac{T_0}{T}\right)}^{5/2}+RT\ln \left(\frac{p}{p_0}\right)$	$\mu (T,p)=g(T,p),T_0<T,p<p_0,T\uparrow :\mu \downarrow$
$(04)$	$\mu$	$=$	$+pv\ln \left[{\left(\frac{p_0}{p}\right)}^{3/2}{\left(\frac{v_0}{v}\right)}^{5/2}\right]$	$\mu (v,p)=g(v,p),v_0<v,p_0<p,v\uparrow ,p\uparrow :\mu \downarrow$

D Enthalpie und molare Enthalpie

$(01)$	$H$	$=$	$\frac{5}{2}nRT$	$H(T,V,N)$
$(02)$	$h$	$=$	$\frac{5}{2}RT$	$h(T,v)$