Name: using maxwell's relations in thermodynamics
Uploaded: 2025-05-22T09:05:59.000Z
Duration: 1 min 23 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.10

SC.912.P.10.2

Maxwell's Thermodynamic Relations

考虑热力学势的微分形式；麦克斯韦方程组可以从热力学势推导出来。
内能是熵和体积的函数。由于内能是一个全微分，其对体积和熵的二阶导数与求导顺序无关。
代入温度和压强的表达式，可得到第一个麦克斯韦关系式。
焓取决于熵和压强。取二阶导数并代入温度和体积的表达式，可得到第二个麦克斯韦关系式。
亥姆霍兹自由能取决于体积和温度。将其表示为二阶导数，并代入压力和熵的表达式，可得到第三个麦克斯韦关系式。
吉布斯自由能是压力和温度的函数。对其求二次微分，并代入体积和熵的关系式，可得到第四个麦克斯韦关系式。
麦克斯韦方程组可用于求解复杂的热力学问题，因为它将难以通过实验直接测量的物理量之间的偏微分关系，转化为易于测量的物理量之间的偏微分关系。

<ol>
	<li>布伦德尔，SJ 和布伦德尔，KM（2010 年）。热物理学中的概念。再版。美国纽约:牛津大学出版社，第 179 页。</li>
	<li>赫尔里希，CS（2009 年）。现代热力学与统计力学。再版。柏林:施普林格。第 69 页。</li>
</ol>

麦克斯韦热力学关系

麦克斯韦的热力学关系对于解决热力学问题非常有用。 每个麦克斯韦关系都将难以通过实验测量的量之间的偏微分与可以容易测量的量之间的偏微分联系起来。 这些关系式是由二阶导数和热力学势的对称性推导出来的一组方程。 所有热力学势都是精确微分。 因此，它们的二阶导数不依赖于微分阶数。 在麦克斯韦关系的情况下，热力学势用该函数的偏导数表示。 代入偏导数的值即可得到麦克斯韦方程组。 四个麦克斯韦关系是： <img alt="Equation1" src="/files/ftp_upload/14173/14173_Equation_1.png" style="text-align: center; height: 32px;"> <img alt="Equation2" src="/files/ftp_upload/14173/14173_Equation_2.png" style="text-align: center; height: 35px;"> <img alt="Equation3" src="/files/ftp_upload/14173/14173_Equation_3.png" style="text-align: center; height: 34px;"> <img alt="Equation4" src="/files/ftp_upload/14173/14173_Equation_4.png" style="text-align: center; height: 35px;"> 这些方程将难以测量的熵变化与更容易测量的其他热力学变量（如温度、体积和压力）的变化联系起来。 例如，在最后一个方程中，左侧项给出了恒定体积下压力随温度的变化。 这个量可以在实验室中轻松测量。 然而，方程右边的项比较复杂，因为很难测量恒温下熵随体积的变化。

麦克斯韦热力学关系在解决热力学问题时非常有用。每一个麦克斯韦关系都将一个偏导数……

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

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. When the partial derivatives of these functions are substituted, Maxwell's equations are obtained.

There are four Maxwell's relations. Together, they link changes in entropy, which are difficult to measure, to changes in other thermodynamic variables that are easier to observe. For example, one relation gives the change in pressure with temperature at constant volume. That term can be measured in the laboratory. The matching entropy term with volume at constant temperature is harder to measure directly.

Using Maxwell's Relations in Thermodynamics

using-maxwells-relations-in-thermodynamics

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

Heat added to a thermodynamic system can lead to the same final state through different paths. The path matters because work and heat depend not only on the initial and final states, but also on the steps in between. In this topic, an ideal gas is used to show two different ways to reach the same volume. In the first case, the gas is in a cylinder with a piston and is heated at a constant temperature. As the gas molecules absorb energy, the gas expands slowly in a controlled isothermal process.

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

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

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

Learn how Maxwell's thermodynamic relations connect hard-to-measure entropy changes with measurable temperature, pressure, and volume relationships.

Using Maxwell's Relations in Thermodynamics | JoVE K12

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Using Maxwell's Relations in Thermodynamics

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