Aktuelle Version vom 21. Juni 2024, 10:57 Uhr

CPRT.I.C

08 Vom Differential über das Integral zum Zustand der inneren Energie

A in (S,V,N)-Koordinaten: dU(S,V) -> U(S,V,N)

$(3.11.1.1.1)$	$d U$	$=$	${(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) [+ n R T_{0} \frac{d S}{n R} - p_{0} V_{0} \frac{d V}{V}]$	$d U (S, V), d N = 0 \leftarrow (3.9.1.1.2)$
	$U$	$=$	$\int_{S, d V = 0} d U$	$U (S, V, N)$
		$=$	$\int_{S, d V = 0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) n R T_{0} \frac{d S}{n R}$
		$=$	$+ \frac{3}{2} n R T_{0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) + B (V, N)$
	${(\frac{\partial U}{\partial V})}_{S, N}$	$=$	$- \frac{n R}{V} T_{0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) + {(\frac{\partial B}{\partial V})}_{N}$	$\partial_{V} U (S, V, N), B (V, N)$
		$=$	$- p + {(\frac{\partial B}{\partial V})}_{N}$	$p (S, V, N), B (V, N)$
$(3.11.1.1.2)$	${(\frac{\partial U}{\partial V})}_{S, N}$	$=$	$- p$	$\partial_{V} U (S, V, N), p (S, V, N)$
	$B$	$\neq$	$B (V, N)$	$B = const.$
	$U$	$=$	$\int_{V, d S = 0} d U$	$U (S, V, N)$
		$=$	$- \int_{S, d V} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) p_{0} V_{0} \frac{d V}{V}$
		$=$	$- \int_{S, d V} p_{0} V_{0} \frac{1}{V} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) d V$
		$=$	$+ \frac{3}{2} n R T_{0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) + A (S, N)$
	${(\frac{\partial U}{\partial S})}_{V, N}$	$=$	$+ T_{0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) + {(\frac{\partial A}{\partial S})}_{N}$	$\partial_{S} U (S, V, N), A (S, N)$
$(3.11.1.1.3)$		$=$	$+ T + {(\frac{\partial A}{\partial S})}_{N}$	$T (S, V, N), A (S, N)$
	${(\frac{\partial U}{\partial S})}_{V, N}$	$=$	$+ T$	$\partial_{S} U (S, V, N), T (S, V, N)$
	$A$	$\neq$	$A (S, N)$	$A = const.$
	$U$	$=$	$+ \frac{3}{2} n R T_{0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R}) + A + B$	$U (S, V, N)$
$(3.11.1.1.4)$	$U (S_{0}, V_{0}, N)$	$=$	$0$
	$A + B$	$=$	$0$
$(3.11.1.1.5)$	$U$	$=$	$+ \frac{3}{2} n R T_{0} {(\frac{V_{0}}{V})}^{2 / 3} \exp (\frac{2}{3} \frac{S - S_{0}}{n R})$	$U (S, V, N) \leftarrow (3.10.1.1.1)$

B in (S,p,N)-Koordinaten: dU(S,p) -> U(S,p,N)

$(3.11.1.2.1)$	$d U$	$=$	$\frac{3}{5} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) [+ n R T_{0} \frac{d S}{n R} + p_{0} V_{0} \frac{d p}{p}]$	$d U (S, p), d N = 0 \leftarrow (3.9.2.2.7)$
	$U$	$=$	$\int_{S, d p = 0} d U$	$U (S, p, N)$
		$=$	$\int_{S, d p = 0} \frac{3}{5} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) n R T_{0} \frac{d S}{n R}$
		$=$	$+ \frac{5}{2} \cdot \frac{3}{5} n R T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + B (p, N)$
		$=$	$+ \frac{3}{2} n R T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + B (p, N)$
	${(\frac{\partial U}{\partial p})}_{S, N}$	$=$	$+ \frac{2}{5} \cdot \frac{3}{2} \frac{n R}{p} T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + {(\frac{\partial B}{\partial p})}_{N}$	$\partial_{p} U (S, p, N), B (p, N)$
		$=$	$+ \frac{3}{5} \frac{n R}{p} T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + {(\frac{\partial B}{\partial p})}_{N}$	$\partial_{p} U (S, p, N), B (p, N)$
		$=$	$+ \frac{3}{5} V + {(\frac{\partial B}{\partial p})}_{N}$	$\partial_{p} U (S, p, N), V (S, p, N), B (p, N)$
	${(\frac{\partial U}{\partial p})}_{S, N}$	$=$	${(\frac{\partial (H - p V)}{\partial p})}_{S, N} = V - V {(\frac{\partial p}{\partial p})}_{S, N} - p {(\frac{\partial V}{\partial p})}_{S, N}$	$(S, p, N) : \partial_{p} U, \partial_{p} H, \partial_{p} V$
	$- p {(\frac{\partial V}{\partial p})}_{S, N}$	$=$	$+ \frac{3}{5} V$	$(S, p, N) : \partial_{p} V, V \leftarrow (3.9.2.2.3)$
$(3.11.1.2.2)$	${(\frac{\partial U}{\partial p})}_{S, N}$	$=$	$+ \frac{3}{5} V$	$(S, p, N) : \partial_{p} U, V$
	$B$	$\neq$	$B (p, N)$	$B = const.$
	$U$	$=$	$\int_{p, d S = 0} d U$	$U (S, V, N)$
		$=$	$\int_{p, d S = 0} \frac{3}{5} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) p_{0} V_{0} \frac{d p}{p}$
		$=$	$\int_{p, d S = 0} \frac{3}{5} \frac{1}{p} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) p_{0} V_{0} d p$
		$=$	$+ \frac{5}{2} \cdot \frac{3}{5} p_{0} V_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + A (S, N)$
		$=$	$+ \frac{3}{2} p_{0} V_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + A (S, N)$
	${(\frac{\partial U}{\partial S})}_{p, N}$	$=$	$+ \frac{2}{5} \frac{1}{n R} \cdot \frac{3}{2} p_{0} V_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + {(\frac{\partial A}{\partial S})}_{N}$	$\partial_{S} U (S, p, N), A (S, N)$
		$=$	$+ \frac{3}{5} T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + {(\frac{\partial A}{\partial S})}_{N}$	$\partial_{S} U (S, p, N), A (S, N)$
		$=$	$+ \frac{3}{5} T + {(\frac{\partial A}{\partial S})}_{N}$	$\partial_{S} U (S, p, N), T (S, p, N), A (S, N)$
	${(\frac{\partial U}{\partial S})}_{p, N}$	$=$	${(\frac{\partial (H - p V)}{\partial S})}_{p, N} = T - p {(\frac{\partial V}{\partial S})}_{p, N}$	$(S, p, N) : \partial_{S} U, \partial_{S} H, \partial_{S} V, T$
	$- S {(\frac{\partial V}{\partial S})}_{p, N}$	$=$	$- \frac{2}{5} \frac{S}{n R} V$	$(S, p, N) : \partial_{S} V, S, V \leftarrow (3.9.2.3.3)$
$(3.11.1.2.3)$	$- p {(\frac{\partial V}{\partial S})}_{p, N}$	$=$	$- \frac{2}{5} T$	$\partial_{S} V (S, p, N), T (S, p, N)$
$(3.11.1.2.4)$	${(\frac{\partial U}{\partial S})}_{p, N}$	$=$	$+ \frac{3}{5} T$	$\partial_{S} U (S, p, N), T (S, p, N)$
	$A$	$\neq$	$A (S, N)$	$A = const.$
	$U$	$=$	$\frac{3}{2} n R T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R}) + A + B$	$U (S, p, N)$
$(3.11.1.2.5)$	$U (S_{0}, p_{0}, N)$	$=$	$0$
	$A + B$	$=$	$0$
$(3.11.1.2.6)$	$U$	$=$	$\frac{3}{2} n R T_{0} {(\frac{p}{p_{0}})}^{2 / 5} \exp (\frac{2}{5} \frac{S - S_{0}}{n R})$	$U (S, p, N) \leftarrow (3.10.2.2.2)$

C in (T,V,N)-Koordinaten: dU(T,V) -> U(T,V,N)

$(3.11.1.3.1)$	$d U$	$=$	$\frac{3}{2} n R d T$	$d U (T, V), d N = 0 \leftarrow (3.9.3.3.3)$
	$U$	$=$	$\int_{T, d V = 0} d U$	$U (T, V, N)$
		$=$	$\int_{S, d V = 0} \frac{3}{2} n R d T$
		$=$	$+ \frac{3}{2} n R T + B (V, N)$
	${(\frac{\partial U}{\partial V})}_{T, N}$	$=$	$+ {(\frac{\partial B}{\partial V})}_{N}$	$\partial_{V} U (T, V, N), B (V, N)$
	${(\frac{\partial U}{\partial V})}_{T, N}$	$=$	${(\frac{\partial (F + T S)}{\partial V})}_{T, N} = - p + S {(\frac{\partial T}{\partial V})}_{T, N} + T {(\frac{\partial S}{\partial V})}_{T, N}$	$(T, V, N) : \partial_{V} U, \partial_{V} H, \partial_{V} S$
	$- V {(\frac{\partial S}{\partial V})}_{T, N}$	$=$	$- n R$	$\partial_{V} S (T, V, N) \leftarrow (3.9.3.2.1)$
	$+ T {(\frac{\partial S}{\partial V})}_{T, N}$	$=$	$+ \frac{n R T}{V} = + p$	$(T, V, N) : \partial_{V} S, p$
$(3.11.1.3.2)$	${(\frac{\partial U}{\partial V})}_{T, N}$	$=$	$0$	$(T, V, N) : \partial_{V} U$
	$B$	$\neq$	$B (V, N)$	$B = const.$
	$U$	$=$	$\int_{V, d T = 0} d U$	$U (T, V, N)$
		$=$	$\int_{V, d T = 0} 0 d V$
		$=$	$+ A (T, N)$
	${(\frac{\partial U}{\partial T})}_{V, N}$	$=$	$+ {(\frac{\partial A}{\partial T})}_{N}$	$\partial_{T} U (T, V, N), A (T, N)$
	${(\frac{\partial U}{\partial T})}_{V, N}$	$=$	${(\frac{\partial (F + T S)}{\partial T})}_{V, N} = - S + S {(\frac{\partial T}{\partial T})}_{V, N} + T {(\frac{\partial S}{\partial T})}_{V, N}$	$(T, V, N) : \partial_{T} U, \partial_{T} F, \partial_{T} S$
	$- T {(\frac{\partial S}{\partial T})}_{V, N}$	$=$	$- \frac{3}{2} n R$	$\partial_{T} S (T, V, N) \leftarrow (3.9.3.3.1)$
$(3.11.1.3.3)$	${(\frac{\partial U}{\partial T})}_{V, N}$	$=$	$+ \frac{3}{2} n R$	$\partial_{T} U (T, V, N)$
	${(\frac{\partial A}{\partial T})}_{N}$	$=$	$+ \frac{3}{2} n R$	$A (T, N)$
	$A (T, N)$	$=$	$+ \frac{3}{2} n R T$
	$U$	$=$	$+ \frac{3}{2} n R T + B$	$U (T, V, N)$
$(3.11.1.3.4)$	$U (T_{0}, V_{0}, N)$	$=$	$0$
	$A + B$	$=$	$+ \frac{3}{2} n R T$
$(3.11.1.3.5)$	$U$	$=$	$\frac{3}{2} n R T$	$U (T, V, N) \leftarrow (3.10.3.2.1)$

CPRT.I.C.08: Unterschied zwischen den Versionen

Aktuelle Version vom 21. Juni 2024, 10:57 Uhr

Inhaltsverzeichnis

08 Vom Differential über das Integral zum Zustand der inneren Energie

A in (S,V,N)-Koordinaten: dU(S,V) -> U(S,V,N)

B in (S,p,N)-Koordinaten: dU(S,p) -> U(S,p,N)

C in (T,V,N)-Koordinaten: dU(T,V) -> U(T,V,N)

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CPRT.I.C.08: Unterschied zwischen den Versionen

Aktuelle Version vom 21. Juni 2024, 10:57 Uhr

08 Vom Differential über das Integral zum Zustand der inneren Energie

A in (S,V,N)-Koordinaten: dU(S,V) -> U(S,V,N)

B in (S,p,N)-Koordinaten: dU(S,p) -> U(S,p,N)

C in (T,V,N)-Koordinaten: dU(T,V) -> U(T,V,N)

Navigationsmenü

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