Path Function vs Point Function

Updated: Aug 5

We come across a lot of properties and functions in Thermodynamics. These are divided mainly into path functions and point functions. All the Thermodynamic properties like volume, temperature, pressure, etc are point functions. Whereas the path functions are not properties of the system and are boundary phenomena. Path functions are path-dependent and are inexact differentials. Let's find out what these terms mean one by one.


What is a boundary phenomenon?

The quantities like heat and work are transferred between the system and the surroundings at the boundary during the interaction, in other words, the amount of heat and work transferred depends on the kind of process the system follows. But in the case of the state functions or point functions like volume, pressure, temperature, entropy, etc the change in the amount of these quantities is fixed between two states no matter what kind of processes they follow.

The system does not possess the quantities which are boundary phenomena. On the other hand, properties like internal energy, temperature, pressure, etc are possessed by the system.


What is an exact differential and what is an inexact differential?

The exact differentials are functions whose integration is nothing but the difference between values at the endpoints or limits.

For example, when the volume is integrated between two states V1 and V2, it equals the difference between V2 and V1.

But, in the case of the inexact differentials, the integration or differentiation is not straightforward subtraction between endpoints. For example, work transfer is a path function hence an inexact differential. Its integral over a process can not be a simple difference between values at the endpoints.

Inexact differential

Rather,

Inexact differential

Also, I would like to mention here about the cyclic integral of the Thermodynamic properties.

The properties are point functions and exact differentials hence when the cyclic integral is taken over the thermodynamic cycle the system goes back to the initial state hence the cyclic integral of the Thermodynamics properties is zero.

Path function: Analogy with vectors

Path function is a directional phenomenon that requires magnitude as well as direction for its complete description. They are analogous to vector quantities in some sense. A point function is analogous to a scalar quantity which only requires endpoint values for a complete description.


Why do the heat and work are path functions?

The heat and work can not be defined at a particular state point. They are always defined as quantities in transit (boundary phenomenon). The amount of heat transfer or work transfer does depend on how the process happens. Heat and work are not intrinsic properties of the system.


Example 1: Suppose you want to go to the 10th floor of the building and you have two options for reaching there, either you take a lift or you take the stairs. The amount of work done by your body during this process is completely different in each case. If you take a lift your body needs to do a small amount of effort on the other hand if you take a staircase route, your body will do so much work that it will get exhausted by the time you reach the 10th floor.


Let's understand these concepts in a little more technical sense by using P-v and T-s diagrams.


Example 2: Suppose you have already decided on initial and final state points and you are trying to find out the best suitable path to reach the final state point from the initial state point.

There are multiple possibilities through which this can happen. I have shown 3 possible paths through which the system can change from initial state point 1 to final state point 2.

We know that the areas under the curve in P-v and T-s diagrams represent Work and Heat respectively. As you can see from the following figure, in each case the area under the curve is different, that is the work done is different. This can also be proved using equations by adding up the work done in each process of the path.


Work Done in 3 different cases, path function
Work interaction in 3 different cases between 2 same state points

Similarly,

Heat interaction in 3 different cases between 2 same state points
Heat interaction in 3 different cases between 2 same state points

The heat transferred in each case is also different. Hence it is safe to say that heat and work are path functions and they depend on the path followed.

Heat interaction in 3 different cases between 2 same state points
Heat interaction in 3 different cases between 2 same state points


On the other hand, quantities like pressure, temperature, and volume are independent of the path. The pressure and volume values at state 2 are always exactly the same in all 3 cases above.


What is the difference between path function and point function?

Path Function

Point Function

Boundary Phenomenon

Depends only on the end states of the process

Inexact differentials

Exact differentials

Cyclic integral is not zero

Cyclic integral is zero

Depends on the type of process followed by the system

Independent of the type of process followed by the system

Examples: Work Transfer, Heat Transfer, Entropy generation

Examples: Temperature, Pressure, Volume, Entropy, Internal Energy, etc

Can you make a path function change to the point function?

No, it can not be done.


What can you do with all this information? What is the significance of path function and point function?

It becomes easier when analyzing systems to choose the quantities that do not depend on the path.


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