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Maxwell relations

Maxwell relations is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Maxwell relations rather than just read about it. In short: The Maxwell relations in thermodynamics can be derived from the symmetry of second derivatives and the definitions of the thermodynamic potentials, or from Jacobian determinants. The most common Maxwell relations involve the potential functions U {\displaystyle U} (the total internal energy), H {\displaystyle H} (enthalpy), A {\displaystyle A} (Helmholtz free energy), and G {\displaystyle G} (Gibbs free energy), and…

Maxwell relations — main illustration
Maxwell relations — illustration

Key takeaways

  • Maxwell relations belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Maxwell relations to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Maxwell relations from memory before moving on to harder problems.

Reference excerpt

The Maxwell relations in thermodynamics can be derived from the symmetry of second derivatives and the definitions of the thermodynamic potentials, or from Jacobian determinants. The most common Maxwell relations involve the potential functions U {\displaystyle U} (the total internal energy), H {\displaystyle H} (enthalpy), A {\displaystyle A} (Helmholtz free energy), and G {\displaystyle G} (Gibbs free energy), and the functions of state P {\displaystyle P} (pressure), T {\displaystyle T} (absolute temperature), V {\displaystyle V} (volume), and S {\displaystyle S} (entropy). They are named for the physicist James Clerk Maxwell, who first presented them in his text Theory of Heat (1872). Maxwell's relations are useful in problem-solving quantities that are difficult to measure to those that are easier to work with. Numerous mnemonic devices exist for remembering them, notably, the thermodynamic square.

Equations The structure of Maxwell relations is a statement of equality among the second derivatives for continuous functions. It follows directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant (Schwarz theorem). In the case of Maxwell relations the function considered is a thermodynamic potential and x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} are two different natural variables for that potential,

where the partial derivatives are taken with all other natural variables held constant. For every thermodynamic potential there are 1 2 n ( n − 1 ) {\textstyle {\frac {1}{2}}n(n-1)} possible Maxwell relations where n {\displaystyle n} is the number of natural variables for that potential.

The four most common Maxwell relations

The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature T {\displaystyle T} , or entropy S {\displaystyle S} ) and their mechanical natural variable (pressure P {\displaystyle P} , or volume V {\displaystyle V} ):

where the potentials as functions of their natural thermal and mechanical variables are the internal energy U ( S , V ) {\displaystyle U(S,V)} , enthalpy H ( S , P ) {\displaystyle H(S,P)} , Helmholtz free energy F ( T , V ) {\displaystyle F(T,V)} , and Gibbs free energy G ( T , P ) {\displaystyle G(T,P)} . The thermodynamic square can be used as a mnemonic to recall and derive these relations. The usefulness of these relations lies in their quantifying entropy changes, which are not directly measurable, in terms of measurable quantities like temperature, volume, and pressure. Each equation can be re-expressed using the reciprocal relation ( ∂ y ∂ x ) z = 1 / ( ∂ x ∂ y ) z . {\displaystyle \left({\frac {\partial y}{\partial x}}\right)_{z}=1{\biggl /}\left({\frac {\partial x}{\partial y}}\right)_{z}.}

Derivations

First derivation For a given set of four real variables ( x , y , z , w ) {\displaystyle (x,y,z,w)} , restricted to move on a 2-dimensional C 2 {\displaystyle C^{2}} surface in R 4 {\displaystyle \mathbb {R} ^{4}} . Knowing two of them enables the remaining two to be determined. In particular, one may take any two variables as the independent variables, and let the other two be the dependent variables, then take all these partial derivatives. This derivation exploits the reciprocal relation

… excerpt ends here. Continue reading the full article.

Illustrations

Maxwell relations illustration
Maxwell relations: Flow chart showing the paths between the Maxwell relations. P {\displaystyle P} is pressure, T {\displaystyle T} temperature, V {\displaystyle V} volume, S {\displaystyle S} entropy, α {\displaystyle \alpha } coefficient of thermal expansion, κ {\displaystyle \kappa } compressibility, C V {\displaystyle C_{V}} heat capacity at constant volume, C P {\displaystyle C_{P}} heat capacity at constant pressure.
Flow chart showing the paths between the Maxwell relations. P {\displaystyle P} is pressure, T {\displaystyle T} temperature, V {\displaystyle V} volume, S {\displaystyle S} entropy, α {\displaystyle \alpha } coefficient of thermal expansion, κ {\displaystyle \kappa } compressibility, C V {\displaystyle C_{V}} heat capacity at constant volume, C P {\displaystyle C_{P}} heat capacity at constant pressure.

Worked examples

Example 1 — a first encounter with Maxwell relations

Start with the simplest possible case. Write down what Maxwell relations claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Maxwell relations before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Maxwell relations ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Maxwell relations

In research
Maxwell relations appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Maxwell relations in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Maxwell relations is common in secondary-school and first-year university syllabi. It links to neighbouring topics James Clerk Maxwell, Thermodynamic equations, so understanding it makes those chapters shorter.
In everyday life
Look for Maxwell relations outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Maxwell relations in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Maxwell relations means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Maxwell relations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Maxwell relations in simple terms?

The Maxwell relations in thermodynamics can be derived from the symmetry of second derivatives and the definitions of the thermodynamic potentials, or from Jacobian determinants. The most common Maxwell relations involve the potential functions U {\displaystyle U} (the total internal energy), H {\d…

Why does Maxwell relations matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Maxwell relations?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Maxwell relations.

Tags

  • James Clerk Maxwell
  • Thermodynamic equations

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