Name: using maxwell's relations in thermodynamics
Uploaded: 2025-05-22T09:05:59.000Z
Duration: 1 min 23 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.10

SC.912.P.10.2

Maxwell's Thermodynamic Relations

Betrachten Sie die differentielle Form thermodynamischer Potentiale; Die Maxwell-Gleichungen können aus thermodynamischen Potentialen abgeleitet werden.
Die innere Energie ist eine Funktion der Entropie und des Volumens. Da es sich um ein exaktes Differential handelt, hängt seine Ableitung zweiter Ordnung in Bezug auf Volumen und Entropie nicht von der Differenzierungsordnung ab.
Setzt man die Ausdrücke von Temperatur und Druck ein, erhält man die erste Maxwell-Beziehung.
Die Enthalpie hängt von der Entropie und dem Druck ab. Nimmt man die Ableitung zweiter Ordnung und setzt man die Ausdrücke für Temperatur und Volumen ein, erhält man die zweite Maxwell-Beziehung.
Die freie Energie von Helmholtz hängt vom Volumen und der Temperatur ab. Drückt man sie als Ableitung zweiter Ordnung aus und ersetzt man sie durch die Ausdrücke von Druck und Entropie, erhält man die dritte Maxwell-Beziehung.
Die freie Gibbs-Energie ist eine Funktion von Druck und Temperatur. Differenziert man sie zweimal, so erhält man die Beziehungen von Volumen und Entropie, so erhält man die vierte Maxwell-Beziehung.
Die Maxwell-Gleichungen werden verwendet, um komplexe thermodynamische Probleme zu lösen, da sie eine partielle Differenz zwischen Größen, die experimentell schwer zu messen sind, mit einer partiellen Differenz zwischen leicht messbaren Größen in Beziehung setzen.

<ol>
	<li>Blundell, S.J. und Blundell, K.M. (2010). Konzepte der thermischen Physik. Zweite Auflage. New York, USA: Oxford University Press. S. 179.</li>
	<li>Helrich, C.S. (2009). Moderne Thermodynamik mit statistischer Mechanik. Zweite Auflage. Berlin: Springer. S. 69.</li>
</ol>

maxwells thermodynamische beziehungen

Maxwells thermodynamische Beziehungen sind bei der Lösung von Problemen in der Thermodynamik sehr nützlich. Jede der Maxwell-Beziehungen bezieht sich auf eine partielle Ableitung zwischen Größen, die experimentell schwer zu messen sein können, und einer partiellen Ableitung zwischen Größen, die leicht gemessen werden können. Diese Beziehungen sind eine Reihe von Gleichungen, die sich aus der Symmetrie der zweiten Ableitungen und den thermodynamischen Potenzialen ableiten lassen.

Alle thermodynamischen Potenziale sind exakte Differentiale. Daher hängt ihre Ableitung zweiter Ordnung nicht von der Reihenfolge der Differentiation ab. Im Fall von Maxwells Beziehungen wird das thermodynamische Potential in Bezug auf die partiellen Ableitungen dieser Funktion ausgedrückt. Durch Substitution der Werte der partiellen Ableitungen ergeben sich Maxwells Gleichungen. Die vier Maxwellschen Beziehungen lauten:

<img alt="Equation1" src="/files/ftp_upload/14173/14173_Equation_1.png" style="text-align: center; height: 32px;" />

<img alt="Equation2" src="/files/ftp_upload/14173/14173_Equation_2.png" style="text-align: center; height: 35px;" />

<img alt="Equation3" src="/files/ftp_upload/14173/14173_Equation_3.png" style="text-align: center; height: 34px;" />

<img alt="Equation4" src="/files/ftp_upload/14173/14173_Equation_4.png" style="text-align: center; height: 35px;" />

Diese Gleichungen setzen Entropieänderungen, die schwer zu messen sind, mit Änderungen anderer thermodynamischer Variablen wie Temperatur, Volumen und Druck in Verbindung, die leichter zu messen sind. Zum Beispiel gibt der Term auf der linken Seite der letzten Gleichung die Änderung des Drucks mit der Temperatur bei konstantem Volumen an. Diese Größe kann leicht im Labor gemessen werden. Der Term auf der rechten Seite der Gleichung ist jedoch komplizierter, da die Änderung der Entropie mit dem Volumen bei konstanter Temperatur schwer zu messen ist.

Maxwells Thermodynamische Beziehungen

Die thermodynamischen Beziehungen von Maxwell sind sehr nützlich bei der Lösung von Problemen in der Thermodynamik. Jede von Maxwells Relationen bezieht sich auf einen Teil ...

Education

Lehre

JoVE Core

Physics

Physik

Unit 1

Einheit 1

The First Law of Thermodynamics

Der erste Hauptsatz der Thermodynamik

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

using maxwell's relations in thermodynamics

Maxwell's thermodynamic relations help solve problems in thermodynamics. They connect partial differentials for quantities that are hard to measure in the lab with quantities that are easier to measure. This makes them useful for working with temperature, pressure, volume, and entropy.

These relations come from thermodynamic potentials and the symmetry of second derivatives. Thermodynamic potentials are exact differentials, so the order of second differentiation does not change the result. When the partial derivatives of these functions are substituted, Maxwell's equations are obtained.

There are four Maxwell's relations. Together, they link changes in entropy, which are difficult to measure, to changes in other thermodynamic variables that are easier to observe. For example, one relation gives the change in pressure with temperature at constant volume. That term can be measured in the laboratory. The matching entropy term with volume at constant temperature is harder to measure directly.

Using Maxwell's Relations in Thermodynamics

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Open, Closed, and Isolated Systems

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Constant Volume and Pressure in Thermodynamics

Thermodynamic processes can happen at constant volume or at constant pressure. An isochoric process is a constant-volume process. In this case, the volume does not change, so the work done by the system is zero. The first law of thermodynamics still applies, so any heat added or removed changes the internal energy of the system only. A simple example is heating 1000 g of water from 40 degrees celsius to 70 degrees celsius at constant volume. Because the process is isochoric, no work is done by...

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Ideal Gas Heat Capacity Differences

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Gas Temperature Changes in Adiabatic Processes

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Free Expansion: Pressure, Volume, and Temperature

Free expansion shows how pressure, volume, and temperature behave when a gas expands without doing work. It is an adiabatic process, but it is not the same as adiabatic expansion. In free expansion, no work is done and the internal energy does not change. In adiabatic expansion, work is done and the internal energy changes. For an adiabatic process, the pressure-volume relation comes from the adiabatic condition. That relation does not apply to free expansion, even though the expansion is...

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Adiabatic Compression in an Engine Cylinder

Adiabatic compression in an engine cylinder changes the pressure, temperature, and volume of a gas-air mixture. In this setup, gasoline vapor is injected into the cylinder when the piston is in its expanded position. The mixture starts at 20 °C, 1.00 x 10^5 N/m^2, and 240 cm^3. The mixture is then compressed adiabatically to 40 cm^3. Adiabatic means no heat is exchanged with the surroundings during the process. For this calculation, the compression is treated as quasi-static, even though real...

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maxwell's thermodynamic relations

Maxwell's thermodynamic relations help solve problems in thermodynamics. They connect partial differentials for quantities that are hard to measure in the lab with quantities that are easier to measure. This makes them useful for working with temperature, pressure, volume, and entropy. These relations come from thermodynamic potentials and the symmetry of second derivatives. Thermodynamic potentials are exact differentials, so the order of second differentiation does not change the result.

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Gas cooling and heating in throttling is explained by the Joule-Thomson effect, also called the Joule-Kelvin effect. It describes how a fluid changes temperature when it passes through a valve or porous plug in a thermally insulated system. This type of flow is called a throttling process. In the experiment, a high-pressure gas is forced through a throttle valve or porous plug into a lower-pressure region. The gas expands as it moves into the larger volume. Work is done on the gas as it is...

key terms and concepts

Learn how Maxwell's thermodynamic relations connect hard-to-measure entropy changes with measurable temperature, pressure, and volume relationships.

Using Maxwell's Relations in Thermodynamics | JoVE K12

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Using Maxwell's Relations in Thermodynamics

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