# Physical Chemistry Lesson of the Day – State Functions vs. Path Functions

Today’s lesson may seem mundane; despite its subtlety, it is actually quite important.  I needed to spend some time to learn it and digest it, and it was time well spent – these concepts are essential for understanding much of thermodynamics.  For brevity, I have not dived into the detailed mathematics of exact differentials, though I highly recommend you to learn it and review the necessary calculus.

Some thermodynamic properties of a system can be described by state variables, while others can be described by path variables.

A state variable is a variable that depends only on the final and initial states of a system and not on the path connecting these states.  Internal energy and enthalpy are examples of state functions.  For example, in a previous post on the First Law of Thermodynamics, I defined the change in internal energy, $\Delta U$, as

$\Delta U = \int_{i}^{f} dU = U_f - U_i$.

State variables can be calculated by exact differentials.

A path variable is a variable that depends on the sequence of steps that takes the system from the initial state to the final state.  This sequence of steps is called the path.  Heat and work are examples of path variables.  Path variables cannot be calculated by exact differentials.  In fact, the following quantities may seem to have plausible interpretations, but they actually do not exist:

• change in heat ($\Delta q$)
• initial heat ($q_i$)
• final heat ($q_f$)
• change in work ($\Delta w$)
• initial work ($w_i$)
• final work ($w_f$)

There is no such thing as heat or work being possessed by a system.  Heat and work can be transferred between the system and the surroundings, but the end result is an increase or decrease in internal energy; neither the system or the surroundings possesses heat or work.

A state/path variable is also often called a state/path function or a state/path quantity.

### 7 Responses to Physical Chemistry Lesson of the Day – State Functions vs. Path Functions

1. Surya says:

plz give some daily examples for state functions and path functions

• Hi Surya,

Could you please clarify what you mean by “daily” examples? I already mentioned a few in the post.

• Shubham says:

Surya meant to say…… daily means examples of state function in day to day life

• Altitude is a state function. It doesn’t matter how you reached a certain altitude as you ascend or descend in your path in the atmosphere.

2. ABED MUTHOMI says:

good work explain what isolated system.

3. ela says:

Heat and work are path functions, but difference between change in heat and change in work is called state function, why?

• Hi ela,

As I wrote in this blog post, there is no such thing as “change in heat” or “change in work”. Thus, I’m don’t fully understand your question. Please clarify.

Eric