Name: free expansion: pressure, volume, and temperature
Uploaded: 2023-04-30T12:32:36.000Z
Duration: 1 min 27 s
Channel: JoVE

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pressure-and-volume-in-an-adiabatic-process

Pressure and Volume in an Adiabatic Process

Stellen Sie sich ein Volumen von Luftmolekülen vor, das in der Atmosphäre aufsteigt. Da der Druck mit der Höhe abnimmt, versuchen die Luftmoleküle, den gleichen Druck aufrechtzuerhalten, indem sie ihr Volumen in einem adiabatischen Prozess vergrößern.
Die innere Energieänderung wird in Bezug auf die molare Wärmekapazität geschrieben und neu angeordnet.
Die Verwendung des idealen Gasgesetzes, seine Differenzierung und die Neuanordnung der Begriffe geben einen Ausdruck für die Temperaturänderung.
Die Gleichsetzung der Temperaturänderungsgleichungen, die sich aus dem ersten Hauptsatz der Thermodynamik und dem idealen Gasgesetz ergeben, und die Neuanordnung ergeben eine Gleichung in Bezug auf die molare Wärmekapazität.
Dividieren Sie durch pV und verwenden Sie das Verhältnis zwischen den molaren Wärmekapazitäten. Die Gleichung kann anhand ihres Verhältnisses weiter vereinfacht werden.
Nach der Integration und unter Verwendung eines allgemeinen mathematischen Ausdrucks für Logarithmen erhält man die Bedingung für einen idealen Gas, einen adiabatischen Prozess zu durchlaufen, in Bezug auf Druck und Volumen.
In ähnlicher Weise können die Bedingungen für adiabatische Prozesse in Bezug auf Temperatur-Volumen und Druck-Temperatur geschrieben werden.
Wenn sich also Luft ausdehnt, sinkt die Temperatur, wie sie durch die Temperatur-Volumen-Bedingung gegeben ist, und kondensiert zu Wolken.

<ol>
	<li>Young, H.D. und Freedman, R.A. (2012). Universitätsphysik mit moderner Physik. San Francisco, Kalifornien: Pearson. S. 640, 641</li>
	<li>Walker, Jearl (2007). Halliday und Resnick, Grundlagen der Physik. Wiley. Seiten 573-574</li>
</ol>

druck und volumen in einem adiabatischen prozess

Die freie Ausdehnung eines Gases ist ein adiabatischer Prozess. Es gibt jedoch einige Unterschiede zwischen freier Ausdehnung und adiabatischer Ausdehnung. Bei der freien Ausdehnung wird keine Arbeit verrichtet und es gibt keine Veränderung der inneren Energie. Bei einer adiabatischen Ausdehnung wird jedoch Arbeit verrichtet und es gibt eine Änderung der inneren Energie. Während eines adiabatischen Prozesses wird die Beziehung zwischen Druck und Volumen aus der Bedingung für den adiabatischen Prozess erhalten, das heißt

<img alt="Equation1" src="/files/ftp_upload/13688/13688_PT Equation_1.svg" style="text-align: center; height: 13px;" />

Allerdings gilt diese Bedingung nicht für einen freien Expansionsprozess, auch wenn die Ausdehnung adiabatisch ist. Außerdem befindet sich das Gas bei einem freien Expansionsprozess nur in einem Gleichgewichtszustand am Anfangs- und Endpunkt. Daher können auf einem pV-Diagramm nur diese Anfangs- und Endpunkte dargestellt werden, nicht die Ausdehnung selbst. Zusätzlich dazu sind die Anfangs- und Endtemperaturen eines adiabatischen Prozesses durch den Ausdruck

<img alt="Equation2" src="/files/ftp_upload/13688/13688_PT Equation_2.svg" style="text-align: center; height: 18px;" />

verbunden. Für die freie Ausdehnung gibt es jedoch keine Veränderung der inneren Energie, daher sind die Anfangs- und Endtemperaturen gleich. Anstatt der obigen Gleichung haben wir daher

<img alt="Equation3" src="/files/ftp_upload/13688/13688_PT Equation_3.svg" style="text-align: center; height: 13px;" />

Aus dem Ausdruck für die Temperatur ergibt sich, dass die Anfangs- und Endzustände einer freien Ausdehnung auf der gleichen Isothermen in einem pV-Diagramm liegen. Wenn das Gas als ideal angenommen wird, gibt es bei keiner Änderung der Temperatur auch keine Änderung des Produkts aus Druck und Volumen. Daher können Druck und Volumen für die freie Ausdehnung wie folgt ausgedrückt werden:

<img alt="Equation4" src="/files/ftp_upload/13688/13688_PT Equation_4.svg" style="text-align: center; height: 13px;" />

Druck und Volumen in einem adiabatischen Prozess

Die freie Expansion eines Gases ist ein adiabatischer Prozess. Es gibt jedoch nur wenige Unterschiede zwischen freier Expansion und adiabatischer Expansion. Während der freien Expansion ...

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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 adiabatic. The gas is in equilibrium only at the start and the end of free expansion. Because of that, only the initial and final points can be shown on a pV diagram, not the path of the expansion.

The temperature behavior is also different. For adiabatic expansion, the initial and final temperatures are linked by an equation. For free expansion, the internal energy stays the same, so the initial and final temperatures are equal. This means the two states lie on the same isotherm, or constant-temperature line, on a pV diagram.

If the gas is ideal, a constant temperature means the product of pressure and volume does not change. So for free expansion, pressure and volume can be written with the same value of pV at the initial and final states. This makes free expansion a useful case for comparing adiabatic behavior with a process that changes volume without changing internal energy.

Free Expansion: Pressure, Volume, and Temperature

free-expansion-pressure-volume-temperature

thermodynamic 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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Heat capacity describes how much heat a substance absorbs for a given change in temperature. It is also called thermal capacity, and its SI unit is J/K. Specific heat capacity is the heat needed to raise the temperature of 1 kg of a substance by 1 K, and its SI unit is J/kg·K. The video also explains molar heat capacity. This is the amount of heat added or removed to change the temperature of 1 mole of a substance by 1 K. It can be measured at constant volume or constant pressure, and it is...

heat capacities of an ideal gas ii

Ideal Gas Heat Capacity Differences

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Adiabatic processes change the temperature of an ideal gas without heat transfer. In an adiabatic compression, no heat is added, but work is done on the gas. That work raises the gas temperature. In an adiabatic expansion, the gas does work and its temperature drops. A common example is the compression of a gas-air mixture inside a car engine cylinder. The compression happens so quickly that there is no time for heat to move between the mixture and its surroundings. Even so, the work done on...

pressure and volume in an adiabatic process

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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Thermodynamic potentials are energy-based state functions used to analyze a thermodynamic system. They help describe how a system changes under different conditions. The four main thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These potentials can be written using two of the variables pressure, volume, temperature, and entropy. In each case, the rate of change of the potential with respect to the other two variables gives useful...

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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...

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Explore how free expansion differs from adiabatic expansion, including no work done, unchanged internal energy, equal temperatures, and pV diagram behavior.

Free Expansion: Pressure, Volume, and Temperature | JoVE K12

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Free Expansion: Pressure, Volume, and Temperature

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