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

質量に取り付けられた摩擦のないピストンが取り付けられたpolの圧力で体積VOの理想気体を持つシリンダーを考えてみましょう。圧力が平衡状態にある間、ピストンは静止したままです。
質量が半分になると、ピストンの外圧は比例して減少します。ガスの内圧が高いと、ピストンが前方に押し出され、新たな平衡状態に達します。
ガス量が増加します。システムは、作業を行うことにより新しい熱力学的状態に到達します。
質量を2段階で減らすことで、同じ最終状態が達成されると仮定します。最初に、圧力は0.75poに減少し、行われた作業は2番目の曲線の下の面積で表されます。
第2段階で質量をさらに減らすと、ガスによって行われる仕事は第3曲線の下の面積と等しくなります。したがって、2番目のプロセスでガスによって行われる総仕事は、2つの曲線の下の面積の合計です。
したがって、ガスによって行われる仕事はプロセスごとに異なるため、経路依存の機能です。

<ol>
 <li>ヤング、HD、フリードマン、RA大学物理学と現代物理学。カリフォルニア州サンフランシスコ:ピアソン。628(2012年)。</li>
 <li>OpenStaxです。大学物理学 Vol.2.110ページ[Web版]。<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>. から取得。</li>
</ol>

熱力学状態間のパス

図 1 のパス AC と ABC で表される、理想気体に関係する 2 つの熱力学プロセスを考えてみましょう。

<img alt="Equation1" src="/files/ftp_upload/13676/13676_Figure_PT_1.png" style="text-align: center; height: 250px;" />

パス A から C までの最初のプロセスでは、ガスは一定温度 T に保たれます。 ボリューム V1 から V2 まで拡張されます。

理想気体が行う仕事は次のように表されます。

<img alt="Equation1" src="/files/ftp_upload/13676/13676_Equation_PT_1.svg" style="text-align: center; height: 45px;" />

理想気体方程式から圧力を nRT/V として代入し、項を積分すると、一定温度での理想気体によって行われる仕事は次のように得られます。

<img alt="Equation2" src="/files/ftp_upload/13676/13676_Equation_PT_2.svg" style="text-align: center; height: 30px;" />

2 番目のプロセスでは、パス A から B に対して、最初に理想気体が体積 V1p1 で sub> を V2 にします。 次に、ガスは経路 B から C に沿って一定の体積 V2 で冷却され、圧力が まで低下します。 p2.

パス A から B の場合、作業は一定の圧力の下で行われるため

<img alt="Equation3" src="/files/ftp_upload/13676/13676_Equation_PT_3.svg" style="text-align: center; height: 13px;" />

パス B から C の場合、体積は一定のままであるため、ガスによる仕事も、周囲のガスによる仕事も行われません。 したがって、このプロセスでガスによって行われる仕事の合計は、パス A から B で行われる仕事と同じです。

どちらのプロセスでも、ガスは体積 V1 から体積 V2 まで膨張します。 圧力は p1 から p2 に変化します。 ただし、両方のプロセスで実行される作業は異なります。 これは、系によって行われる作業がパスに依存していることを証明しています。

図1のパスACとABCで表される理想気体を含む2つの熱力学的プロセスについて考えてみます。

パス A から ...

Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

The First Law of Thermodynamics

熱力学第一法則

KEY TERMS AND CONCEPTS

主要用語と概念

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

thermodynamic systems

Open, Closed, and Isolated Systems

A thermodynamic system is the part of the world that scientists study for its heat and energy behavior. It sits inside a surrounding environment, or surroundings. The system and the surroundings can exchange heat and do work across the boundary that separates them. The immediate surroundings affect the system most strongly because they interact with it directly. A kettle of boiling tea shows how this works in everyday life. The tea and the kettle together form the system. If the lid is open,...

work done during volume change

Gas Work in a Piston Cylinder

Gas work in a piston cylinder describes how pressure and volume change are related in thermodynamics. In mechanics, work happens when a force moves an object. In thermodynamics, work done on a system can be estimated when the system changes volume during a process. A gas can be trapped in a cylinder with a movable piston at one end. When the gas expands from volume V1 to volume V2, it pushes on the piston and moves it by a small distance dr. If the pressure on the piston face is p and the...

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...

heat and free expansion

Two Gas Paths to the Same Volume

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internal energy

How Internal Energy Changes in Gas Systems

Internal energy is the energy stored inside a thermodynamic system. It is the sum of the kinetic and potential energies of the molecules or other particles in the system. Molecular kinetic energy includes translational motion, rotation, and vibration. Molecular potential energy comes from interactions between molecules. A system’s location and overall motion do not affect its internal energy. For an ideal monatomic gas, each particle is a single atom. That means there is no rotational or...

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Heat, Work, and Internal Energy

Heat, work, and internal energy are linked by the first law of thermodynamics. This law is a generalized form of energy conservation. It explains how energy changes when heat is added to or removed from a system, and when the system does work or has work done on it. The change in internal energy equals the heat transferred into the system minus the work done by the system. Added heat increases internal energy. For that reason, heat is taken as positive when it enters the system and negative...

first law of thermodynamics: problem-solving

first-law-of-thermodynamics-problem-solving

First Law of Thermodynamics: Solving Problems

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cyclic processes and isolated systems

Net Work in a Cyclic Thermodynamic Process

A cyclic thermodynamic process returns a system to its initial state. In a cycle, the change in internal energy is zero. An isolated system is different because it has zero heat exchange and zero work, so its internal energy stays constant. For a non-isolated system, the internal energy change is zero only when the process is cyclic. The first law of thermodynamics relates internal energy, heat transfer, and work. It states that the change in internal energy is the difference between the net...

isothermal processes

Isothermal Expansion and Heat Transfer

An isothermal process is a thermodynamic process that happens at constant temperature. During the process, heat slowly flows into the system or out of the system so thermal equilibrium is maintained. Phase changes are common examples of isothermal processes. Water can evaporate into steam at a constant temperature, and water can also freeze into ice at a constant temperature. An ideal gas can also undergo isothermal expansion or compression. Here, an ideal gas in an isolated cylinder expands...

isochoric and isobaric processes

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...

heat capacities of an ideal gas i

Ideal Gas Heat Capacity at Constant Pressure

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

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

Ideal Gas Heat Capacity and Degrees of Freedom

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adiabatic processes for an ideal gas

Gas Temperature Changes in Adiabatic Processes

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

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...

thermodynamic potentials

Energy Forms in Thermodynamics

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...

maxwell's thermodynamic relations

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.

joule-thomson effect

Gas Cooling and Heating in Throttling

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

physics

JoVE Core 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

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HS-PS3-5

Path Dependence in Gas Work

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