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In mathematics, a differential dQ is said to be exact, as contrasted with an inexact differential, if the function Q exists. It is always possible to calculate the differential dQ of a given function Q(x, y, z). However, if dQ is arbitrarily given, the function Q generally does not exist. 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. ... Maxwells relations are a set of equations in Thermodynamics which are derivable from the definitions of the four thermodynamic potentials. ... In Thermodynamics, Bridgmans Thermodynamic equations is actually a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. ... For more elaboration on these equations see: thermodynamic equations. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In physics, an inexact differential, as contrasted with an exact differential, of a function f is denoted: ; as is true of point functions. ... Partial plot of a function f. ...

1 Overview
2 Some useful equations derived from exact differentials in two dimensions
3 See also
4 References
5 External links

Overview

In one dimension, a differential

$dQ = A(x)dx,$

is always exact. In two dimensions, in order that a differential

$dQ = A(x, y)dx + B(x, y)dy,$

be an exact differential in a simply-connected region R of the xy-plane, it is necessary and sufficient that between A and B there exists the relation: A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

$left( frac{partial A}{partial y} right)_{x} = left( frac{partial B}{partial x} right)_{y}$

In three dimensions, a differential

$dQ = A(x, y, z)dx + B(x, y, z)dy + C(x, y, z)dz,$

is an exact differential in a simply-connected region R of the xyz-coordinate system if between the functions A, B and C there exists the relations:

$left( frac{partial A}{partial y} right)_{x,z} !!!= left( frac{partial B}{partial x} right)_{y,z}$ ; $left( frac{partial A}{partial z} right)_{x,y} !!!= left( frac{partial C}{partial x} right)_{y,z}$ ; $left( frac{partial B}{partial z} right)_{x,y} !!!= left( frac{partial C}{partial y} right)_{x,z}$

These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential dQ, that is a function of four variables to be an exact differential, there are six conditions to satisfy.

In summary, when a differential dQ is exact:

the function Q exists;
$int_i^f dQ=Q(f)-Q(i)$ , independent of the path followed.

In thermodynamics, when dQ is exact, the function Q is a state function of the system. The thermodynamic functions U, S, H, A and G are state functions. Generally, neither work nor heat is a state function. An exact differential is sometimes also called a 'total differential', or a 'full differential', or, in the study of differential geometry, it is termed an exact form. In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given...

Some useful equations derived from exact differentials in two dimensions

(See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations) In Thermodynamics, Bridgmans Thermodynamic equations is actually a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. ... In thermodynamics, there are a large number of equations relating the various thermodynamic quantities. ...

Suppose we have five state functions $z, x, y, u$ , and $v$ . Suppose that the state space is two dimensional and any of the five quantites are exact differentials. Then by the chain rule In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

$(1)~~~~~ dz = left(frac{partial z}{partial x}right)_y dx+ left(frac{partial z}{partial y}right)_x dy = left(frac{partial z}{partial u}right)_v du +left(frac{partial z}{partial v}right)_u dv$

but also by the chain rule:

$(2)~~~~~ dx = left(frac{partial x}{partial u}right)_v du +left(frac{partial x}{partial v}right)_u dv$

and

$(3)~~~~~ dy= left(frac{partial y}{partial u}right)_v du +left(frac{partial y}{partial v}right)_u dv$

so that:

$(4)~~~~~ dz = left[ left(frac{partial z}{partial x}right)_y left(frac{partial x}{partial u}right)_v + left(frac{partial z}{partial y}right)_x left(frac{partial y}{partial u}right)_v right]du$

$+ left[ left(frac{partial z}{partial x}right)_y left(frac{partial x}{partial v}right)_u + left(frac{partial z}{partial y}right)_x left(frac{partial y}{partial v}right)_u right]dv$

which implies that:

$(5)~~~~~ left(frac{partial z}{partial u}right)_v = left(frac{partial z}{partial x}right)_y left(frac{partial x}{partial u}right)_v + left(frac{partial z}{partial y}right)_x left(frac{partial y}{partial u}right)_v$

Letting $v = y$ gives:

$(6)~~~~~ left(frac{partial z}{partial u}right)_y = left(frac{partial z}{partial x}right)_y left(frac{partial x}{partial u}right)_y$

Letting $u = y$ , $v = z$ gives:

$(7)~~~~~ left(frac{partial z}{partial y}right)_x = - left(frac{partial z}{partial x}right)_y left(frac{partial x}{partial y}right)_z$

using ( $partial a/partial b)_c = 1/(partial b/partial a)_c$ gives:

$(8)~~~~~ left(frac{partial z}{partial x}right)_y left(frac{partial x}{partial y}right)_z left(frac{partial y}{partial z}right)_x =-1$

References

Perrot, P. (1998). A to Z of Thermodynamics. New York: Oxford University Press.
Zill, D. (1993). A First Course in Differential Equations, 5th Ed. Boston: PWS-Kent Publishing Company.

External links

Inexact Differential – from Wolfram MathWorld
Exact and Inexact Differentials – University of Arizona
Exact and Inexact Differentials – University of Texas
Exact Differential – from Wolfram MathWorld

Categories: Thermodynamics | Multivariate calculus

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Overview GA_googleFillSlot("encyclopedia_square");

Some useful equations derived from exact differentials in two dimensions

See also

References

External links

Overview