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

考虑一个装有理想气体的圆柱形容器，其初始体积为 Vo，初始压强为 po，容器配备了一个无摩擦活塞，并连接一个质量块。当压强处于平衡状态时，活塞保持静止。
如果质量减半，活塞所受的外部压力将相应减小。气体内部较高的压力会推动活塞向前移动，直至达到新的平衡状态。
气体体积增大。系统通过做功达到新的热力学状态。
假设通过两个阶段减少质量来达到相同的终态。最初，压力降至 0.75po，所做的功由第二条曲线下方的面积表示。
在第二阶段进一步减少质量时，气体所做的功等于第三条曲线下方的面积。因此，气体在第二个过程中所做的总功是两条曲线下方面积之和。
因此，气体所做的功因过程不同而不同，故为路径依赖函数。

<ol>
 <li>Young， H.D.， Freedman， RA 大学物理学与现代物理学。加利福尼亚州旧金山:皮尔逊。628 （2012 年）。</li>
 <li>OpenStax 的。大学物理学第 2 卷。第 110 页 [Web 版本]。取自<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>

热力学状态之间的路径

考虑涉及理想气体的两个热力学过程，由图 1 中的路径 AC 和 ABC 表示： 

 <img alt="Equation1" src="/files/ftp_upload/13676/13676_Figure_PT_1.png" style="text-align: center; height: 250px;"> 

 在路径A到C的第一个过程中，气体保持在恒定温度T。 它从体积V1 扩展到 V2。 

 理想气体所做的功表示为 

 <img alt="Equation1" src="/files/ftp_upload/13676/13676_Equation_PT_1.svg" style="text-align: center; height: 45px;"> 

 从理想气体方程中代入压力为nRT/V并对各项进行积分，可得到理想气体在恒温下所做的功为 

 <img alt="Equation2" src="/files/ftp_upload/13676/13676_Equation_PT_2.svg" style="text-align: center; height: 30px;"> 

 在第二个过程中，对于A到B的路径，理想气体首先从体积V1 膨胀到 V2 通过在恒压p1 的情况下加热。 然后气体以恒定体积V2沿路径B冷却到C，使其压力降至 p2。 

 对于A到B的路径，工作是在恒定压力下完成的，因此 

 <img alt="Equation3" src="/files/ftp_upload/13676/13676_Equation_PT_3.svg" style="text-align: center; height: 13px;"> 

 对于B到C的路径，由于体积保持恒定，因此气体或周围环境对气体没有做功。 因此，气体在此过程中所做的总功与路径A到B所做的功相同。 

 在这两个过程中，气体都从体积 V1 膨胀到体积 V2，因此其体积 压力从p1变化到p2。 然而，这两个过程中完成的工作是不同的。 这证明系统所做的工作是路径依赖的。

考虑图1中由路径AC和ABC表示的涉及理想气体的两个热力学过程：

在路径A到C的第一个过程中……

Education

教育

JoVE Core

Physics

物理学

Unit 1

第一单元

The First Law of Thermodynamics

热力学第一定律

KEY TERMS AND CONCEPTS

关键术语与概念

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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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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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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Path Dependence in Gas Work

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