Inexact differential

An inexact differential or imperfect differential is a type of differential used in thermodynamics to express changes in path dependent quantities. In contrast, an integral of an exact differential (a differential of a function) is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential; i.e. its value cannot be inferred just by looking at the initial and final states of a given system.[1] Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions.

Definition

An inexact differential is commonly defined as a differential form $δu$ for which there is no corresponding function $f$ such that $f=\int \delta u$ . More precisely, an inexact differential is a differential form that cannot be expressed as the differential of a function. In the language of vector calculus, for a given vector field $F$ , $\delta f=\mathbf {F} \cdot \mathrm {d} \mathbf {r}$ is an inexact differential if there is no function $f$ such that

\mathbf {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

Instead of the differential symbol d, the symbol δ is used instead, a convention which originated in the 19th century work of German mathematician Carl Gottfried Neumann,[2] indicating that Q (heat) and W (work) are path-dependent, while U (internal energy) is not.

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 ΔQ = U₂ − U₁ or work ΔW = U₂ − U₁, such a change of state does not uniquely identify the amount of work W done to the system or heat Q transferred, 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.

Total distance

The easiest example is the difference between net distance and total distance. For example, in walking from point $A$ to point $B$ along a straight line, 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 $2\times(B - A)$ . 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 $d f = 1 d x$ with corresponding function $Δ f (x) = x$ . 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 $δg = |d x |$ . It is clear that if at any time the change in the $x$ position is negative, then $\Delta g|_{A}^{B}\neq x_{\text{B}}-x_{\text{A}}$ so instead we must look at the path dependence. In our example, in the first leg of the journey, $sgn(d x)$ is 1 since $x$ is increasing. In the second leg, $sgn(d x)$ is −1 since $x$ is decreasing. We can then evaluate the total distance as:

\Delta g=\int _{A}^{B}\mathrm {d} x+\int _{B}^{A}(-\mathrm {d} x)=2\int _{A}^{B}\mathrm {d} x=2(B-A)

Heat and work

A fire requires heat, fuel, and an oxidizing agent. The energy required to overcome the activation energy barrier for combustion is transferred as heat into the system, resulting in changes to the system’s internal energy. In a process, the energy input to start a fire may comprise both work and heat, such as when one rubs tinder (work) and experiences friction (heat) to start a fire. The ensuing combustion is highly exothermic, which releases heat. The overall change in internal energy does not reveal the mode of energy transfer and quantifies only the net work and heat. The difference between initial and final states of the system’s internal energy does not account for the extent of the energy interactions transpired. Therefore, internal energy is a state function (i.e. exact differential), while heat and work are path functions (i.e. inexact differentials) because integration must account for the path taken.

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.

Example

Consider the inexact differential form,

\delta u=2y\,dx+x\,dy

This must be inexact by considering going to the point (1,1). If first increase $y$ and then increase $x$ , then that corresponds to first integrating over $y$ and then over $x$ . Integrating over y first contributes ${\textstyle \int _{0}^{1}x\,dy|_{x=0}=0}$ and then integrating over $x$ contributes ${\textstyle \int _{0}^{1}2y\,dx|_{y=1}=2}$ . Thus, along the first path we get a value of 2. Likewise, along the second path we get a value of ${\textstyle \int _{0}^{1}2y\,dx|_{y=0}+\int _{0}^{1}x\,dy|_{x=1}=1}$ . We can make $δu$ an exact differential by multiplying it by $x$ , yielding

x\,\delta u=2xy\,dx+x^{2}\,dy=d(x^{2}y)

And so $x δu$ is an exact differential.

References

Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4.CS1 maint: multiple names: authors list (link)
Neumann, Carl G. (1875). Vorlesungen über die mechanische Theorie der Wärme [Lectures on the Mechanical Theory of Heat]. Leipzig: Teubner.

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

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[Laider-1] Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press. ISBN 0-19-855919-4.CS1 maint: multiple names: authors list (link)

[2] Neumann, Carl G. (1875). Vorlesungen über die mechanische Theorie der Wärme [Lectures on the Mechanical Theory of Heat]. Leipzig: Teubner.