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

K12 Standards

new-york-state-science-learning-standards

New York State Science Learning Standards

High School

Physical Science

HS. Energy

HS-PS3-3.

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

Pressure and Volume in an Adiabatic Process

大気中に上昇する大量の空気分子を考えてみましょう。高度とともに圧力が減少すると、空気分子は断熱プロセスの下で体積を増やすことにより、同じ圧力を維持しようとします。
内部エネルギー変化は、モル熱容量で記述され、再配置されます。
理想気体の法則を使用し、それを区別し、項を並べ替えると、温度変化の表現が得られます。
熱力学の第一法則と理想気体の法則から得られた温度変化方程式を同等にし、再配置すると、モル熱容量に関する方程式が得られます。
全体をpVで除算し、モル熱容量の関係を使用します。方程式は、それらの比率を使用してさらに簡略化できます。
積分し、対数の一般数式を使用すると、理想的な気体が断熱過程を経る条件は、圧力と体積の観点から得られます。
同様に、断熱プロセスの条件を温度-体積と圧力-温度の観点から記述できます。
したがって、空気が膨張すると、温度-体積条件によって与えられる温度が下がり、凝縮して雲を形成します。

<ol>
	<li>ヤング、H.D.およびフリードマン、RA(2012)。現代物理学と大学物理学。カリフォルニア州サンフランシスコ:ピアソン。640、641ページ</li>
	<li>ウォーカー、ジャール(2007)。HallidayとResnickの物理学の基礎。ワイリー。573-574ページ</li>
</ol>

断熱過程における圧力と体積

気体の自由膨張は断熱プロセスです。 ただし、自由膨張と断熱膨張にはほとんど違いがありません。 自由膨張中は仕事は行われず、内部エネルギーは変化しません。 しかし、断熱膨張の場合、仕事が行われ、内部エネルギーに変化が生じます。 断熱過程では、圧力と体積の関係は断熱過程の条件から得られます。つまり、 

<img alt="Equation1" src="/files/ftp_upload/13688/13688_PT Equation_1.svg" style="text-align: center; height: 13px;" />

ただし、自由膨張プロセスの場合、断熱膨張であってもこの条件は当てはまりません。 また、自由膨張過程では、気体が平衡状態となるのは開始点と終了点のみです。 したがって、pV ダイアグラムでは、展開自体ではなく、最初と最後の点のみをプロットできます。 これに加えて、断熱プロセスの初期温度と最終温度は次の式によって関連付けられます。

<img alt="Equation2" src="/files/ftp_upload/13688/13688_PT Equation_2.svg" style="text-align: center; height: 18px;" />

しかし、自由膨張の場合、内部エネルギーに変化はありません。 したがって、初期温度と最終温度は等しくなります。 したがって、上記の式の代わりに、

<img alt="Equation3" src="/files/ftp_upload/13688/13688_PT Equation_3.svg" style="text-align: center; height: 13px;" />

温度の式から、自由膨張の初期状態と最終状態は、pV 図の同じ等温線上にあります。 さて、気体が理想的であると仮定すると、温度の変化がないので、圧力と体積の積も変化しません。 したがって、自由膨張の場合、圧力と体積は次のように表すことができます。

<img alt="Equation4" src="/files/ftp_upload/13688/13688_PT Equation_4.svg" style="text-align: center; height: 13px;" />

ガスの自由膨張は断熱プロセスです。ただし、自由膨張と断熱膨張の間にはいくつかの違いがあります。フリーエクスパンション中は、...

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

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

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

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...

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...

first law of thermodynamics

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

The first law of thermodynamics links internal energy, heat, and work. It states that the change in internal energy of a system equals the net heat transfer into the system minus the net work done by the system. This is a general form of energy conservation, and it can be used for any thermodynamic process. To solve a first-law problem, start by identifying the thermodynamic system. Then list the initial and final states of the process. Next, identify the known and unknown quantities. It is...

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

Ideal gas heat capacity depends on degrees of freedom, which are the independent ways a gas molecule can move, rotate, and vibrate. If d is the number of degrees of freedom, the molar heat capacity at constant volume can be written in terms of d. The molar heat capacity at constant pressure can also be written using the relationship between the two heat capacities. The ratio of molar heat capacity at constant pressure to molar heat capacity at constant volume is called the ratio of heat...

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

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

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

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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Free Expansion: Pressure, Volume, and Temperature

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