Question

Imagine that you are climbing a mountain. (a) Is the distance you travel to the top a state function? (b) Is the change in elevation between your base camp and the peak a state function? [Section 5.2\(]\)

Short answer

(a) The distance traveled to the top of the mountain is not a state function, as it depends on the chosen path which can vary between individuals. (b) The change in elevation between the base camp and the peak is a state function, as it depends only on the initial and final states (elevations) and not on the path taken.

Step-by-step solution

(a) Determining if the distance traveled is a state function.

To determine if the distance traveled to the top of the mountain is a state function, we must consider if the distance depends only on the initial and final states, or if it depends on the path taken. In this case, the distance traveled to the top of the mountain can vary depending on the path chosen. For example, one person may take a longer, less steep route, while another may take a shorter, steeper route. Therefore, the distance traveled is not a state function, as it depends on the chosen path.

(b) Determining if the change in elevation is a state function.

Now, we will determine if the change in elevation between the base camp and the peak is a state function. The change in elevation depends solely on the initial elevation at the base camp and the final elevation at the peak. Regardless of the path taken to climb the mountain, the difference in elevation is solely determined by these two points. Since the change in elevation does not depend on the path taken and only on the initial and final states, it is a state function.

Key concepts

Path Dependency

When discussing state functions, it is crucial to understand the concept of path dependency. A property or variable is path-dependent if its value changes based on the pathway taken to reach a particular state. In essence, path dependency implies that multiple pathways can result in different end results because the steps in between matter.

Consider the scenario of climbing a mountain. Let's say one hiker takes a zigzag path, while another opts for a direct straight path. Both paths lead to the peak, but the distance covered differs. This variation in distance based on chosen routes indicates that the distance traveled is path-dependent. It's not just the starting and ending points that matter, but how you get there. This is why distance traveled is not considered a state function.

To simplify:

• Path Dependency involves multiple routes with different outcomes.
• A path-dependent property is affected by how you move from start to finish.

Initial and Final States

State functions are those properties that solely depend on the initial and final states of a system, not on how you arrived there. Think of any system having a start and a finish, attracting your focus only on these two points, regardless of the journey in between.

In the case of the mountain climbing example, if we consider only the change in elevation from the base camp to the peak, it is solely determined by these two points — the base camp elevation and the peak elevation. It doesn't matter if you find a long windy trail or a straight climb. The starting and ending elevations dictate the change, highlighting why elevation change qualifies as a state function.

Key thoughts:

• State Functions depend only on the start and end states.
• No influence from the pathway taken.
• They offer a simplified way to analyze systems, focusing on net changes.

Elevation Change

Elevation change, as discussed, is an excellent example of a state function. Its uniqueness lies in its independence from the path taken between two points. Only the altitude at the start (initial state) and the altitude at the end (final state) matter to determine how much you've ascended.

Imagine starting at a base camp with an elevation of 1000 meters and reaching a peak at 3000 meters. Regardless of the trail—be it winding, steep, or direct—the change in elevation remains 2000 meters. This consistent value conflicts with the notion of path dependency and firmly categorizes elevation change as a state function.

To summarize:

• Elevation change relies strictly on the altitude difference between two points.
• The journey in between does not impact this change.
• It is a straightforward metric offering predictability and simplicity.