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

K12 Standards

new-jersey-student-learning-standards-for-science

New Jersey Student Learning Standards for Science

High School

Physical Science

Energy

HS-PS3-4

Path Between Thermodynamics States

Consider a cylinder with an ideal gas of volume Vo at a pressure of po fitted with a frictionless piston attached to a mass. The piston remains stationary while the pressure is in equilibrium.
If the mass reduces by half, the external pressure on the piston reduces proportionally. The higher internal pressure of the gas pushes the piston forward until a new equilibrium is reached.
The gas volume increases. The system reaches a new thermodynamic state by performing work.
Suppose the same final state is achieved by reducing the mass in two stages. Initially, the pressure decreases to 0.75po, and the work done is represented by the area under the second curve.
Reducing the mass further in the second stage, the work done by the gas equals the area under the third curve. Therefore, the total work done by the gas in the second process is the sum of the areas under the two curves.
Thus, the work done by the gas is different for different processes, and, hence, it is a path-dependent function.

pad tussen thermodynamische toestanden

Beschouw de twee thermodynamische processen waarbij een ideaal gas betrokken is, die in figuur 1 worden weergegeven door paden AC en ABC: <img alt="Equation1" src="/files/ftp_upload/13676/13676_Figure_PT_1.png" style="text-align: center; height: 250px;"> In het eerste proces voor pad A tot C wordt het gas op een constante temperatuur T gehouden. Het ondergaat een uitbreiding van volume V1 naar V2.

De arbeid die door een ideaal gas wordt verricht, wordt uitgedrukt als

<img alt="Equation1" src="/files/ftp_upload/13676/13676_Equation_PT_1.svg" style="text-align: center; height: 45px;">

Door de druk te vervangen door nRT/V uit de ideale gasvergelijking en de termen te integreren, wordt het werk dat wordt verricht door een ideaal gas bij constante temperatuur verkregen als

<img alt="Equation2" src="/files/ftp_upload/13676/13676_Equation_PT_2.svg" style="text-align: center; height: 30px;">

In het tweede proces, voor pad A naar B, wordt het ideale gas eerst geëxpandeerd van volume V1 naar V2 bij constante druk p1 door warmte toe te passen. Het gas wordt vervolgens gekoeld met een constant volume V2 langs pad B naar C, zodat de druk daalt tot p2.

Voor pad A naar B wordt er daarom onder constante druk gewerkt

<img alt="Equation3" src="/files/ftp_upload/13676/13676_Equation_PT_3.svg" style="text-align: center; height: 13px;">

Voor pad B tot C wordt, aangezien het volume constant blijft, geen arbeid verricht door het gas of op het gas door de omgeving. Daarom is de totale arbeid die het gas in dit proces verricht hetzelfde als de arbeid die wordt verricht voor pad A naar B.

In beide processen zet het gas uit van volume V1 naar volume V2, zodat het druk verandert van p1 naar p2. Het werk dat in beide processen wordt verricht, is echter verschillend. Dit bewijst dat het werk dat door een systeem wordt gedaan, padafhankelijk is.

Pad tussen thermodynamische toestanden

Consider the two thermodynamic processes involving an ideal gas that are represented by paths AC and ABC in Figure 1:

In the first process for path A to ...

Education

JoVE Core

Physics

Natuurkunde

Unit 1

Onderdeel 1

The First Law of Thermodynamics

De eerste wet van de thermodynamica

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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

thermodynamic systems

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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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heat capacities of an ideal gas ii

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Ideal gas heat capacities depend on whether the gas is heated at constant volume or constant pressure. At constant volume, the heat absorbed only raises the internal energy of the system. No work is done in this case. At constant pressure, the situation is different. The heat absorbed increases the internal energy and also does work. Molar heat capacity is the heat needed to raise the temperature of 1 mole of gas by 1 unit. Because some of the added heat goes into work, more heat is needed at...

heat capacities of an ideal gas iii

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adiabatic processes for an ideal gas

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pressure and volume in an adiabatic process

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

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...

work done in an adiabatic process

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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joule-thomson effect

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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

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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