# Exact vs. Inexact Differentials

## Exact

:
You should recall, from the previous chapter, that enthalpy is a property or state variable and its differential is therefore exact: dH. The process path that connects states 1 and 2 in no way affects the enthalpy change from state 1 to state 2.

## Inexact

:
In this lesson, we have learned that

is a

#### path variable

and its differential is

#### inexact

. We cannot evaluate

#### work

without knowing exactly how P and V changed during the process so that we can evaluate the area under the process path on the P-V Diagram.
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### Ch 4, Lesson A, Page 12 - Exact vs. Inexact Differentials

• So, what is the big difference between exact and inexact differentials ?
• The key difference, for purposes of this course, becomes apparent when we INTEGRATE these differentials.
• Consider the diagram on the left.
• When we integrate a state variable or property we just obtain the property, in this case, H, evaluated at the upper limit of integration minus H evaluated at the lower limit of integration.
• This is what you are accustomed to doing in calculus, right ?
• Here we introduce the symbol “Δ”.  Δ is used to indicate the change in a property.  So, ΔH is the final value of H minus the initial value of H.
• Remember, Δ is final minus initial and it ONLY applies to state variables, NOT path variables !
• Now, take a look at the P-V diagram on the right.
• When you integrate an inexact differential such as dW, you do NOT obtain W2 – W1 !  What would W1 mean anyway ?  Never write ΔW because it makes no sense !
• The system does not “have” work at either state 1 or state 2.  The system exchanges work with the surroundings as it moves from 1 to 2.
• We use the symbol W12 to represent the work that is exchanged as the system moves from state 1 to state 2.
• Finally, it is important to note that although there is just ONE unique ΔH for any process from 1 to 2, W12 is NOT unique !
• If the system followed a different path from 1 to 2, then W12 would have a different value !
• So how do we know what path a process follows ?  Hmmm…