Why is Work and Heat path function? What different paths can they take?

Question

I've seen some interesting analogies explaining why exactly Work and Heat are path function. One where the floors of a building describe the gravitational potential energy, while stairs and elevator are the two paths one can take to reach between the two states of energy.
But my question is how would the same analogy apply to thermodynamics? what would be the different paths taken by the system to go from one state to another? What are the "stairs and elevator" here?
the only guess I can give is that the path is dependent on the volume, pressure, temperature and other state functions, am I right?

Bob D · Accepted Answer

the only guess I can give is that the path is dependent on the volume,
pressure, temperature and other state functions, am I right?

Essentially yes, you are right. To illustrate, see the PV diagram below.
Internal energy ($U$), pressure ($P$), volume ($V$) and temperature ($T$) are all thermodynamic properties. For each equilibrium state each of these properties has a unique value.
In the diagram, the two points 1 and 2 are initial and final equilibrium states. The internal energy $U$ is shown for points 1 and 2. The work done in going from state 1 to 2 is $int_1^2 pdV$, or the area under the path going from state 1 to 2. There are an infinite number of paths (processes) that can connect the two states. Two possible paths are shown, 1-1a-2 and 1-1b-2.  Clearly the work done for path 1-1a-2 (the area under the path) is greater than the work done for path 1-1b-2. So work is path dependent. Heat is also path dependent.
Work $W$ and heat $Q$ are related to internal energy through the first law which is, for a closed system,
$$Delta U=Q-W$$
Where $Q$ is positive if heat is added to the system and work is positive if work is done by the system.
Hope this helps.

robert patrick · Answer

Path function is anything which happens to the system through the journey.
We have the formula $External  pressure * (Initial volume - Final volume $ in case of Pressure volume work.
For understanding the formula with a picture in your minds.
You can simply take a container with gas filled with a piston on the top.When you give external pressure by the piston on it,Initial volume changes to final volume.
This formula can used to find the Work done on the system by some external pressure.
Now W is a path function because you see , We can say that through the journey from Initial volume to final volume.There is a path.You can also say it as how was the process through which the system went.
It a property of path in that way because its value depends on the way the system reached that point (Initial to final volume).Similarly, P,V and T are state functions since their value depend on the state of the system.
Hope this helps.

Why is Work and Heat path function? What different paths can they take?

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