Inexact differential

An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another function and is therefore path independent. Consequently, an inexact differential cannot be expressed in terms of its antiderivative for the purpose of integral calculations i.e. its value cannot be inferred just by looking at the initial and final states of a given system.[1] It is primarily used in calculations involving heat and work because they are not state functions.

Definition

An inexact differential is commonly defined as a differential form dx where there is no corresponding function x such that: $x = \int dx \,$. More precisely, an inexact differential is a differential form that cannot be expressed as the differential of a function. In the language of calculus, for a given vector field F, $\delta F = F \, dr$ is an inexact differential if there is no function f such that

$F = \nabla f$

The fundamental theorem of calculus for line integrals requires path independence in order to express the values of a given vector field in terms of the partial derivatives of another function that is the multivariate analogue of the antiderivative. This is because there can be no unique representation of an antiderivative for inexact differentials since their variation is inconsistent along different paths. This stipulation of path independence is a necessary addendum to the fundamental theorem of calculus because in one-dimensional calculus there is only one path in between two points defined by a function.

First law of thermodynamics

Inexact differentials are known especially for their presence in the first law of thermodynamics:

$\mathrm{d}\,U = \delta\,Q -\delta\,W$

The symbol δ instead of the plain d, which originated from the work of German mathematician Carl Gottfried Neumann[2] indicates that Q and W are path-dependent.

Internal energy U is a state function, meaning its change can be inferred just by comparing two different states of the system (not its transition path), which we can therefore indicate with U1 and U2. Since we can go from state U1 to state U2 either by providing heat Q = U2 − U1 or work W = U2 − U1, such a change of state does not identify uniquely the values of provided W and Q, but only the change in internal energy ΔU.

Examples

Although difficult to express mathematically, the inexact differential is very simple conceptually. There are many everyday examples that are much more relevant to inexact differentials in the actual context in which it is used.

The easiest example is the difference between net distance and total distance. For example, in walking from Point A to Point B one covers a net distance B-A that is equal to the total distance. If one then returns to Point A, however, net distance is now 0 while total distance covered is 2*(B-A). This example captures the essential idea behind the inexact differential in one dimension.

Precisely, the differential of net distance is simply the exact one form $df = 1 \, dx$ with corresponding function $\Delta f = x_B - x_A$. It is exact because 1 has antiderivative x everywhere on the real line. On the other hand, the differential of total distance is the inexact one form $\delta g = \sgn(dx) dx$ (i.e. the sign function). It is inexact because sgn(x) has antiderivative |x| which is not differentiable at x =0. Therefore $\Delta g \ne x_B - x_A$ and instead we must look at the path dependence. In our example, in the first leg of the journey, sgn(dx) is 1 since x is increasing. In the second leg, sgn(dx) is -1 since x is decreasing. We can then evaluate the total distance as:

$\Delta g = \int_A^B dx + \int_B^A (-dx) = 2 \int_A^B dx = 2(B-A)$

It is known that, with some skill, it is possible to start a fire only using friction and tinder. This is a way to convert mechanical energy (work, W) into an increase of internal energy, ΔU, which finally results into an increase in the local temperature of wood, its gasification and combustion, thereby creating a fire.

It is also possible to start a fire by adding heat using a lighter. This is a way to convert heat (Q) into an increase of internal energy, ΔU, but it has the same result as in the example involving work.

Both friction and heat transfer increase the internal energy of the system, since work and heat are both form of energy transform. Therefore, the sum of exchanged heat and work is an exact differential (dU), but since they are equivalent and the lack of one can be compensated by the presence of the other, singularly they are inexact differentials. In other words, specifying the change in internal energy alone is insufficient to determine the heat evolved or the work done because there is no differentiation between the two forms of energy.

Integrating factors

It is sometimes possible to convert an inexact differential into an exact one by means of an integrating factor. The most common example of this in thermodynamics is the definition of entropy:

$\mathrm{d}\,S = \frac{\delta Q_\text{rev}}{T}$

In this case, δQ is an inexact differential, because its effect on the state of the system can be compensated by δW. However, when divided by the absolute temperature and when the exchange occurs at reversible conditions (therefore the rev subscript), it produces an exact differential: the entropy S is also a state function.