NationMaster - Encyclopedia: Maxwell relations

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Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation in a second derivative is irrelevant. If Φ is a thermodynamic potential and $x i$ and $x j$ are two different natural variables for that potential, then the Maxwell relation for that potential and those variables is: In thermodynamics, there are a large number of equations relating the various thermodynamic quantities. ... The laws of thermodynamics, in principle, describe the specifics for the transport of heat and work in thermodynamic processes. ... Thermodynamic potentials Maxwell relations Bridgmans equations Exact differential (edit) In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as pressure/volume or temperature/entropy. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. ... In Thermodynamics, Bridgmans Thermodynamic equations is actually a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. ... In mathematics, a differential dQ is said to be exact, as contrasted with an inexact differential, if the function Q exists. ... For more elaboration on these equations see: thermodynamic equations. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

$frac{partial }{partial x_j}left(frac{partial Phi}{partial x_i}right)= frac{partial }{partial x_i}left(frac{partial Phi}{partial x_j}right)$

where the partial derivatives are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are n(n-1)/2 possible Maxwell relations where n is the number of natural variables for that potential.

1 The four most common Maxwell relations
2 Derivation of the Maxwell relations
3 General Maxwell relationships
4 See also

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 or entropy S ) and their mechanical natural variable (pressure P or volume V ):

$left(frac{partial T}{partial V}right)_S = -left(frac{partial P}{partial S}right)_Vqquad= frac{partial^2 U }{partial S partial V}$

$left(frac{partial T}{partial P}right)_S = +left(frac{partial V}{partial S}right)_Pqquad= frac{partial^2 H }{partial S partial P}$

$left(frac{partial S}{partial V}right)_T = +left(frac{partial P}{partial T}right)_Vqquad= frac{partial^2 A }{partial T partial V}$

$left(frac{partial S}{partial P}right)_T = -left(frac{partial V}{partial T}right)_Pqquad= frac{partial^2 G }{partial T partial P}$

where the potentials as functions of their natural thermal and mechanical variables are:

$U(S,V),$ - The internal energy

$H(S,P),$ - The Enthalpy

$A(T,V),$ - The Helmholtz free energy

$G(T,P),$ - The Gibbs free energy

In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ...

Derivation of the Maxwell relations

Derivation of the Maxwell equations can be deduced from the differential forms of the thermodynamic potentials:

$dU = TdS-PdV ,$

$dH = TdS+VdP ,$

$dA =-SdT-PdV ,$

$dG =-SdT+VdP ,$

These equations resemble total differentials of the form

$dz = left(frac{partial z}{partial x}right)_y!dx + left(frac{partial z}{partial y}right)_x!dy$

And indeed, it can be shown that for any equation of the form

$dz = Mdx + Ndy ,$

that

$M = left(frac{partial z}{partial x}right)_y, quad N = left(frac{partial z}{partial y}right)_x$

Consider, as an example, the equation $dH=TdS+VdP,$ . We can now immediately see that

$T = left(frac{partial H}{partial S}right)_P, quad V = left(frac{partial H}{partial P}right)_S$

Since we also know that for functions with continuous second derivatives, the mixed partial derivatives are identical, that is, that

$frac{partial}{partial y}left(frac{partial z}{partial x}right)_y = frac{partial}{partial x}left(frac{partial z}{partial y}right)_x = frac{partial^2 z}{partial y partial x} = frac{partial^2 z}{partial x partial y}$

we therefore can see that

and therefore that

$left(frac{partial T}{partial P}right)_S = left(frac{partial V}{partial S}right)_P$

Each of the four Maxwell relationships given above follows similarly from one of the Gibbs equations.

General Maxwell relationships

The above are by no means the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the number of particles is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles N is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be: The particle number, N, is the number of so called elementary particles (or elementary constituents) of a thermodynamical system. ...

$left(frac{partial mu}{partial P}right)_{SN} = left(frac{partial V}{partial N}right)_{SP}qquad= frac{partial^2 H }{partial P partial N}$

where μ is the chemical potential. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations. In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

Each equation can be re-expressed using the relationship

$left(frac{partial y}{partial x}right)_z = 1left/left(frac{partial x}{partial y}right)_zright.$

which are sometimes also known as Maxwell relations.

Contents

The four most common Maxwell relations GA_googleFillSlot("encyclopedia_square");

Derivation of the Maxwell relations

General Maxwell relationships

See also

The four most common Maxwell relations