Maxwell RelationsEdit
Maxwell relations are a compact and practical set of identities in equilibrium thermodynamics that tie together how fundamental state variables like temperature, pressure, volume, and entropy change with one another. They emerge from the fundamental thermodynamic equations and the mathematical fact that mixed second derivatives commute under reasonable smoothness assumptions. Named after James Clerk Maxwell, these relations form a bridge between experimentally accessible data and quantities that are harder to measure directly. They apply to simple, homogeneous, single-component systems described by a well-defined equation of state, and they motivate the use of different thermodynamic potentials, such as internal energy, Helmholtz free energy, Gibbs free energy, and enthalpy.
Foundations and derivation
In classical thermodynamics, the fundamental relations express how a system responds to changes in its natural variables. For a simple compressible system with parasitic effects neglected, the basic differential forms can be written as:
- dU = T dS − P dV
- dF = −S dT − P dV
- dH = T dS + P dV
- dG = −S dT + V dP
Here U is the internal energy, F is the Helmholtz free energy, H is the enthalpy, and G is the Gibbs free energy. Each equation shows a different pair of natural variables (for example, U = U(S,V); F = F(T,V); H = H(S,P); G = G(T,P)). The crucial mathematical point is the Schwarz theorem (the equality of mixed partial derivatives) applied to these potentials. Taking mixed derivatives of the same potential with respect to two variables and equating the results yields a set of identities that relate how one variable changes with another at fixed values of the third variable. When written in terms of P, V, T, and S, these identities take the canonical form known as the Maxwell relations. See also the concept of a thermodynamic potential and the general idea of a Legendre transformation used to switch between natural variables.
The four classical Maxwell relations
From the fundamental relations above, the following Maxwell identities arise:
- (∂T/∂V)_S = −(∂P/∂S)_V
- (∂T/∂P)_S = (∂V/∂S)_P
- (∂S/∂V)_T = (∂P/∂T)_V
- (∂S/∂P)_T = −(∂V/∂T)_P
These equations are typically stated with the variables in their most natural pairs (S,T), (P,V), etc. and can be read as “the way one property changes with a given variable at fixed another property is tied to how a related pair changes in the opposite way.” For example, the first relation says that how T changes with V at fixed S is tied to how P changes with S at fixed V.
In addition to these four, equivalent forms can be written directly from the differential forms of the potentials: - From dF = −S dT − P dV, one obtains (∂P/∂T)_V = (∂S/∂V)_T. - From dG = −S dT + V dP, one obtains (∂S/∂P)_T = −(∂V/∂T)_P. These links can also be understood via a sequence of Legendre transformations between the potentials thermodynamic potential.
Physical interpretation and connections to measurements
Maxwell relations provide a lattice of connections between derivatives that are often easier to measure and those that are more difficult. They enable the extraction of one derivative from another that is experimentally accessible, given reliable data for the relevant equation of state equations of state.
- By combining Maxwell relations with definitions of common thermodynamic coefficients, one can relate measurable quantities like the thermal expansion coefficient α, the isothermal compressibility κ_T, and heat capacities C_P or C_V. For instance, the cross-derivative relations mirror the way entropy responds to changes in P or T, and how volume responds to changes in T or P.
- The relations are particularly valuable when direct measurements of certain derivatives are challenging. For example, (∂S/∂P)_T can be tied to −(∂V/∂T)_P, allowing access to entropy changes through measurements of volume and temperature.
These ideas sit at the intersection of theory and experiment and are routinely used in fields ranging from chemical thermodynamics to materials science and mechanical engineering. See discussions of Gibbs free energy, Helmholtz free energy, and Legendre transformation for further connections.
Applications, generalizations, and limitations
Maxwell relations underpin many practical workflows: - In engineering thermodynamics, they underpin property correlations used in design of engines and refrigerators, where data for P-V-T relationships are combined to derive otherwise inaccessible information. - In chemical thermodynamics, Maxwell relations support the linkage of phase equilibria, solubility data, and calorimetric measurements, helping to understand how systems respond to temperature and pressure changes. - In materials science, they help relate diffusion, elasticity, and thermo-mechanical behavior by connecting derivatives of state functions.
Generalizations extend to more complex systems. For multi-component mixtures, polar or magnetic substances, and systems with additional fields, analogous Maxwell-type relations follow from the appropriate thermodynamic potentials and their natural variables. In non-equilibrium contexts or near critical points, the standard relations assume differentiability and equilibrium; near phase transitions these derivatives can diverge or become ill-behaved, and the straightforward Maxwell identities require careful interpretation or modification.