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Thermische Zustandsgleichung Beispiel Essay

Hier klicken zum Ausklappen Gegeben sei das obige U-Rohr-Manometer. Das U-Rohr ist oben links geschlossen und mit Stickstoff gefüllt. Danach folgt das Quecksilber mit einem ersichtlichen Höhenunterschied und der Behälter, welcher mit einem beliebigen Gas gefüllt ist. Der Stickstoff soll näherungsweise als ideales Gas gelten. Wie groß ist der absolute Druck in dem Behälter?

Bei einem U-Rohr-Manometer berechnet man den absoluten Druck innerhalb des Behälters mit:

$p = p_b + \rho \; h \; g$

Der Bezugsdruck $p_b$ ist hier der Druck, welcher vom Stickstoff ausübt wird. Das bedeutet es muss zunächst der Bezugsdruck bestimmt werden, um dann den absoluten Druck im Behälter berechnen zu können. 

Der Bezugsdruck (also der Druck des Stickstoffs) kann mittels der thermischen Zustandsgleichung bestimmt werden, weil angenommen wird, dass der Stickstoff näherungsweise als ideales Gas gilt:

$p_b V = m \; R_i \; T$


Das Volumen von dem Stickstoff kann berechnet werden durch die Höhe der Säule, in welcher sich der Stickstoff befindet, multipliziert mit der Fläche. Da es sich um einen Durchmesser der Säule von $d = 4mm$ handelt, kann man die Fläche folgendermaßen berechnen:

$A = \pi \cdot r^2$

Es handelt sich um einen Säule, welche einen kreisförmigen Querschnitt besitzt.

$A = \pi \cdot 2^2 mm^2 = 12,566 mm^2$.

Das Volumen berechnet sich nun mit der Höhe der Säule in welcher der Stickstoff enthalten ist:

$V = 500 mm \cdot 12,566 mm^2 = 6.283 mm^3 = 6.283 \cdot 10^{-9} m^3$.

Die Masse ist gegeben mit $m = 0,02g = 2 \cdot 10^{-5} kg$.

Die spezifische Gaskonstante kann aus Tabellen abgelesen werden und beträgt für Stickstoff:

$R_i = 296,8 \frac{J}{kg K}$.

Die Temperatur ist gegeben mit $t = 0°C$:

$T = 273,15 K$

Hier klicken zum Ausklappen WICHTIG: Die Einheiten müssen immer korrekt umgerechnet werden damit das richtige Ergebnis resultiert!!!

Es kann nun die thermische Zustandsgleichung nach $p_b$ aufgelöst und die Werte eingesetzt werden:

$p_b = \frac{m \; R_i \; T}{V}$

$p_b = \frac{2 \cdot 10^{-5} kg \cdot 296,8 \frac{J}{kg K} \cdot 273,15 K}{6.283 \cdot 10^{-9} m^3}$

Hier klicken zum Ausklappen $p_b = 258.064,36 Pa$                 Bezugsdruck (Stickstoff)

Es wurde nun der Bezugsdruck bestimmt. Aus der Grafik kann man anhand des Höhenunterschieds des Quecksilbers erkennen, dass der Bezugsdruck größer ist als der Druck in dem Behälter. Der absolute Druck in dem Behälter lässt sich nun mit der Gleichung für das U-Rohr-Manometer bestimmen:

$p = p_b - \rho \; h \; g$

Das Minuszeichen deswegen, weil der Bezugsdruck größer ist als der Druck im Behälter. Die Druckdifferenz $p_d = \rho h g$ ist demnach negativ. Die Dichte für das Quecksilber beträgt $\rho = 13.550 kg/m^3$.

$p = 258.064,36 Pa - 13.550 kg/m^3 \cdot 0,1 m  \cdot 9,81 m/s^2$

Hier klicken zum Ausklappen $p = 244.771,81 Pa$.                   Absolutdruck im Behälter

Great essay about publishing and community: 'the web is a customer service medium' by paul ford: via @scottros

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