Name: path dependence in gas work
Uploaded: 2023-04-30T12:31:19.000Z
Duration: 1 min 21 s
Channel: JoVE

K12 Standards

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New Jersey Student Learning Standards for Science

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Physical Science

Energy

HS-PS3-4

Path Between Thermodynamics States

Stellen Sie sich einen Zylinder mit einem idealen Gas des Volumens Vo bei einem Druck von po vor, der mit einem reibungsfreien Kolben ausgestattet ist, der an einer Masse befestigt ist. Der Kolben bleibt stationär, während sich der Druck im Gleichgewicht befindet.
Verringert sich die Masse um die Hälfte, verringert sich der äußere Druck auf den Kolben proportional. Durch den höheren Innendruck des Gases wird der Kolben so lange nach vorne gedrückt, bis ein neues Gleichgewicht erreicht ist.
Das Gasvolumen nimmt zu. Das System erreicht durch die Ausführung von Arbeit einen neuen thermodynamischen Zustand.
Angenommen, der gleiche Endzustand wird erreicht, indem die Masse in zwei Stufen reduziert wird. Zunächst sinkt der Druck auf 0,75 pö, und die geleistete Arbeit wird durch die Fläche unter der zweiten Kurve dargestellt.
Reduziert man die Masse in der zweiten Stufe weiter, so entspricht die vom Gas geleistete Arbeit der Fläche unter der dritten Kurve. Daher ist die Gesamtarbeit, die das Gas im zweiten Prozess leistet, die Summe der Flächen unter den beiden Kurven.
Die Arbeit, die das Gas leistet, ist also für verschiedene Prozesse unterschiedlich und daher eine pfadabhängige Funktion.

<ol>
 <li>Young, H.D., Freedman, RA Universitätsphysik mit moderner Physik. San Francisco, Kalifornien: Pearson. 628 (2012).</li>
 <li>OpenStax. Universitätsphysik Vol.2. Seite 110 [Webversion]. Abgerufen von <a href="https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy.">https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy.</a>.</li>
</ol>

weg zwischen thermodynamischen zuständen

Betrachten Sie die beiden thermodynamischen Prozesse, die ein ideales Gas repräsentieren und durch die Wege AC und ABC in Abbildung 1 dargestellt sind:

<img alt="Equation1" src="/files/ftp_upload/13676/13676_Figure_PT_1.png" style="text-align: center; height: 250px;" />

Im ersten Prozess, für den Weg von A nach C, wird das Gas bei konstanter Temperatur T gehalten. Es unterliegt einer Expansion vom Volumen V1 zum Volumen V2.

Die Arbeit, die von einem idealen Gas geleistet wird, wird wie folgt ausgedrückt:

<img alt="Equation1" src="/files/ftp_upload/13676/13676_Equation_PT_1.svg" style="text-align: center; height: 45px;" />

Substituiert man den Druck als nRT/V aus der idealen Gasgleichung und integriert die Terme, so erhält man die Arbeit, die von einem idealen Gas bei konstanter Temperatur geleistet wird:

<img alt="Equation2" src="/files/ftp_upload/13676/13676_Equation_PT_2.svg" style="text-align: center; height: 30px;" />

Im zweiten Prozess, für den Weg von A nach B, wird das ideale Gas zunächst von Volumen V1 auf Volumen V2 bei konstantem Druck p1 durch zugeführte Wärme expandiert. Das Gas wird dann bei konstantem Volumen V2 entlang des Weges B auf C abgekühlt, so dass der Druck auf p2 abfällt.

Für den Weg von A nach B wird Arbeit bei konstantem Druck geleistet, daher gilt:

<img alt="Equation3" src="/files/ftp_upload/13676/13676_Equation_PT_3.svg" style="text-align: center; height: 13px;" />

Für den Weg von B nach C wird, da das Volumen konstant bleibt, keine Arbeit durch das Gas oder die Umgebung auf das Gas ausgeübt. Daher ist die vom Gas bei diesem Prozess geleistete Gesamtarbeit die gleiche wie die für den Weg A nach B geleistete Arbeit.

In beiden Prozessen dehnt sich das Gas vom Volumen V1 auf das Volumen V2 aus, so dass sich sein Druck von p1 zu p2 ändert. Die dabei geleistete Arbeit ist jedoch unterschiedlich. Dies beweist, dass die geleistete Arbeit eines Systems von dem gewählten Weg abhängig ist.

Weg zwischen thermodynamischen Zuständen

Betrachten Sie die beiden thermodynamischen Prozesse mit einem idealen Gas, die in Abbildung 1 durch die Pfade AC und ABC dargestellt sind:

Im ersten Prozess für den Pfad A bis ...

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The First Law of Thermodynamics

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KEY TERMS AND CONCEPTS

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path dependence in gas work

Work done by an ideal gas can depend on the path it takes between two thermodynamic states. The transcript compares two routes from state A to state C for the same gas. In both cases, the gas expands from volume V1 to volume V2, but the work is not the same.

Along path AC, the gas stays at constant temperature T. It expands from V1 to V2. The work done by the gas is found from the pressure-volume relation for an ideal gas. After substituting pressure as nRT/V and integrating, the work for this isothermal process is obtained.

Along path ABC, the gas follows two steps. First, it expands from V1 to V2 at constant pressure p1 by applying heat. Then it is cooled at constant volume V2, and its pressure drops to p2. During the constant-pressure step AB, work is done by the gas. During the constant-volume step BC, no work is done because the volume does not change.

The total work for path ABC is therefore the same as the work done during AB. Even though both paths connect the same start and end states and the pressure changes from p1 to p2, the work values are different. This shows that work done by a system is path-dependent.

Path Dependence in Gas Work

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path between thermodynamics states

Work done by an ideal gas can depend on the path it takes between two thermodynamic states. The transcript compares two routes from state A to state C for the same gas. In both cases, the gas expands from volume V1 to volume V2, but the work is not the same. Along path AC, the gas stays at constant temperature T. It expands from V1 to V2. The work done by the gas is found from the pressure-volume relation for an ideal gas. After substituting pressure as nRT/V and integrating, the work for this...

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key terms and concepts

Compare two ideal-gas paths between the same states and see why an isothermal route and a constant-pressure-plus-volume route give different work values.

Path Dependence in Gas Work | JoVE K12

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HS-PS3-5

Path Dependence in Gas Work

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