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

Consider the differential form of thermodynamic potentials; Maxwell&#39;s equations can be derived from thermodynamics potentials.
Internal energy is a function of entropy and volume. Since it is an exact differential, its second-order derivative with respect to volume and entropy does not depend on the order of differentiation.
Substituting the expressions of temperature and pressure gives the first Maxwell&#39;s relation.
Enthalpy depends on entropy and pressure. Taking the second-order derivative and substituting the expressions for temperature and volume gives the second Maxwell&#39;s relation.
Helmholtz free energy depends on volume and temperature. Expressing it as a second-order derivative and substituting the expressions of pressure and entropy gives the third Maxwell&#39;s relation.
Gibbs free energy is a function of pressure and temperature. Differentiating it twice and substituting the relations of volume and entropy gives the fourth Maxwell&#39;s relation.
Maxwell&#39;s equations are used to solve complex thermodynamic problems as they relate a partial differential between quantities that are hard to measure experimentally to a partial differential between easily measurable quantities.

maxwell's thermodynamic relations

Maxwell's thermodynamic relations are very useful in solving problems in thermodynamics. Each of Maxwell's relations relates a partial differential between quantities that can be hard to measure experimentally to a partial differential between quantities that can be easily measured. These relations are a set of equations derivable from the symmetry of the second derivatives and the thermodynamic potentials.
All thermodynamic potentials are exact differentials. Therefore, their second-order derivative does not depend on the order of differentiation. In the case of Maxwell's relations, the thermodynamic potential is expressed in terms of the partial derivatives of that function. Substituting the values of the partial derivatives gives Maxwell's equations. The four Maxwell's relations are:
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These equations relate entropy changes, which are difficult to measure, with changes in other thermodynamic variables, like temperature, volume, and pressure, that are easier to measure. For example, in the last equation, the left-hand side term gives the change in pressure with the temperature at constant volume. This quantity can be easily measured in a laboratory. However, the term on the right-hand side of the equation is more complicated, as it is hard to measure the entropy change with volume at a constant temperature.

Maxwell&#39;s thermodynamic relations are very useful in solving problems in thermodynamics. Each of Maxwell&#39;s relations relates a partial ...

Education

JoVE Core

Physics

Unit 1

The First Law of Thermodynamics

KEY TERMS AND CONCEPTS

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,...

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first law of thermodynamics: problem-solving

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First Law of Thermodynamics: Solving Problems

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cyclic processes and isolated systems

Net Work in a Cyclic Thermodynamic Process

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isothermal processes

Isothermal Expansion and Heat Transfer

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Constant Volume and Pressure in Thermodynamics

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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

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adiabatic processes for an ideal gas

Gas Temperature Changes in Adiabatic Processes

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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...

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 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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