In thermodynamics, the concepts of state and path functions are crucial. A state function is a property whose value remains unchanged regardless of the path taken to reach that specific value. On the other hand, a path function is dependent on the route taken to attain a particular value. For instance, if you walk 25 paces forward and then 15 back, your total displacement is still 10, similar to a state function. In contrast, the distance travelled is a path function as it relies on the path taken. This classification is essential when examining various thermodynamic equations and calculations.

Characteristics | Values |
---|---|

Definition | A property whose value depends on the path taken to reach a specific function or value |

Dependence on Path | Depends on the path taken to reach a specific value |

Examples | Work, distance travelled, heat |

Opposite | State function |

## What You'll Learn

**Distance travelled is a path function**

For example, a person may decide to hike up a 500-foot mountain. Regardless of the path taken, the starting and ending points will remain the same. The person may opt to climb straight up or spiral around to the summit. There are numerous ways to reach the final state, but the final state will remain unchanged. The distance travelled to reach the summit is a path function because it depends on the route taken.

Another example is driving from Los Angeles to Lake Tahoe. The difference in altitude, approximately 6,620 feet, is not dependent on the route taken. However, the distance travelled does depend on the chosen route (a path-dependent function).

In contrast, a state function is a property whose value does not depend on the path taken to reach a specific value. Instead, state functions are based on the established state of a system, including temperature, pressure, amount, and identity of the system. For instance, volume is a state function because it only depends on the final and initial values, regardless of the path taken to establish those values.

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**State functions are independent of the path taken**

State functions are values that are independent of the path taken to reach a specific function or value. In other words, state functions are not affected by how a certain state is obtained. For instance, density is a state function as the density of a substance remains the same regardless of how it is obtained. For example, the density of a quantity of H2O will be the same whether it is obtained from the tap, a well, or a bottle. As long as the substance is in the same state, its density will be the same.

State functions are often defined in contrast to path functions. Path functions are functions that depend on the path taken to reach a specific value. For example, if you have $1000 in your savings account and you decide to deposit some money into it, the amount you deposit is a path function because it depends on the path taken to obtain that money. If you work as a CEO of a company for a week versus working at a gas station for a week, you will receive two different amounts of money at the end of the week. Thus, a path function is dependent on the path or way taken to establish that value.

State functions are commonly encountered in thermodynamics. Many of the equations involved in thermodynamics, such as changes in internal energy (∆U) and enthalpy (∆H), are state functions. State functions are crucial in thermodynamics because they simplify calculations and allow for the determination of data that could otherwise only be obtained through experiments. For example, state functions facilitate the use of Hess's Law, which allows for the manipulation of the enthalpies of half reactions when adding multiple half reactions to form a full reaction.

Mathematically, state functions can be thought of as integrals. Integrals depend only on the function, the lower limit, and the upper limit. Similarly, state functions depend only on the property, the initial value, and the final value. This means that integrals illustrate how state functions are dependent only on the final and initial values, and not on the path taken to get from the initial to the final value.

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**Path functions depend on the path taken**

Path functions are functions that depend on the path taken to reach a specific value. They are often encountered in thermodynamics, and the two most common path functions are heat and work.

Consider a scenario where you have $1000 in your savings account and you want to deposit some money. The amount you deposit is a path function because it depends on the path taken to obtain that money. For instance, the amount of money you will deposit depends on whether you work as a CEO of a company or at a gas station for a week.

Another example is climbing a mountain. If someone asks how many meters you travelled on your route, you would need to take into account the bends and distances between the bends to calculate the distance travelled. This is a path function because it depends on the path taken.

In the context of chemical reactions, the formation of NaCl from Na^+_(g) and Cl^-_(g) can occur through different paths. The first path is a single step with an enthalpy of formation of -411 kJ/mol:

**> Na^+_(g) + Cl^-_(g) → NaCl_(s)**

**The second path takes five steps to form NaCl:**

> 1. Na_(s) + 1/2 Cl_(g) → Na_(g) + 1/2 Cl_(g) (sublimation)

> 2. Na_(g) + 1/2 Cl_(g) → Na_(g) + Cl_(g) (atomization)

> 3. Na_(g) + Cl_(g) → Na^+_(g) + Cl_(g) (ionization)

> 4. Na^+_(g) + Cl_(g) → Na^+_(g) + Cl^-_(g) (electron affinity)

> 5. Na^+_(g) + Cl^-_(g) → NaCl_(s) (lattice formation)

When the enthalpies of all these steps are added, the enthalpy of formation of NaCl is still -411 kJ/mol. This demonstrates that the enthalpy of formation is a state function, as the same value is obtained regardless of the path taken.

Path functions can also be understood in terms of integrals. Integrals depend on the function, the lower limit, and the upper limit. Similarly, path functions depend on the path taken to reach the final value from the initial value. Multiple integrals and multiple limits of integration are required to integrate a path function.

In summary, path functions are functions that depend on the specific path taken to reach a certain value. The value of a path function is influenced by the various segments or steps that comprise the path.

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**State functions depend on the final and initial states**

State functions are properties whose values do not depend on the path taken to reach a specific value. Instead, they rely solely on the initial and final states of a substance. For instance, consider a quantity of H2O. The way in which the water is obtained, be it from the tap, a well, or a bottle, is irrelevant. As long as the state of the water remains the same, its density will also remain unchanged. This illustrates that density is a state function.

Mathematically, state functions can be thought of as integrals, which depend on the function, the lower limit, and the upper limit. Similarly, state functions rely on the property, the initial value, and the final value. Integrals demonstrate how state functions are dependent only on the initial and final values, rather than the path taken to get from one to the other.

State functions are often defined in contrast to path functions. Path functions are dependent on the path taken to reach a specific value. For example, the amount of money deposited into a savings account is a path function because it depends on how the money is obtained. Working as a CEO for a week will result in a different amount of money than working at a gas station for the same period.

State functions, on the other hand, do not depend on the path taken. Using the savings account example, withdrawing $500 can be done in one transaction or multiple smaller transactions. Regardless of the method, the net withdrawal will be $500, and the resulting balance will be $500. Therefore, the bank balance is a state function.

State functions are commonly encountered in thermodynamics and are crucial for simplifying calculations. They allow for the determination of data that would otherwise require experiments to obtain. For example, state functions facilitate the use of Hess's Law, which enables the manipulation of enthalpies of half reactions when forming a full reaction.

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**Path functions are properties or quantities**

For path functions, the path from an initial state to the final state is crucial. Each part, or segment of the path to the final state, needs to be taken into account. For example, a person may decide to hike up a 500-foot mountain. Regardless of the path the person takes, the starting and ending points will remain constant. The person may decide to go straight up the mountain or decide to spiral around to the top. There are many different ways to get to the final state, but the final state will remain the same.

Two important examples of a path function are heat and work. These two functions are dependent on how the thermodynamic system changes from the initial state to the final state. These two functions are introduced by the equation ΔU, which represents the change in the internal energy of a system.

**The equation for ΔU is as follows:**

\[ \Delta{U} = q + w\]

Here, U is a state function (it does not depend on how the system got from the initial to the final state).

**Several conditions could apply to this equation:**

- Constant Volume: If ΔV = 0, the work is also zero since w = -P(ΔV) and substituting zero for the volume would make the entire term become zero. So, at constant volume: ΔU = q.
- Constant Pressure: When a reaction takes place at constant pressure, the volume can expand or contract to ensure the pressure remains constant. Thus, the volume will change, so work will be done. So, the equation for ΔU in this reaction is: ΔU = q + w.
- Constant Volume vs. Constant Pressure: Note that whether a reaction is carried out under constant volume or constant pressure, ΔU will be the same (because it is a state function). However, in the constant pressure situation, q (heat) can be slightly lower or higher (depending on the situation) than q in the constant volume situation because the amount of w (work) done will make up for it. The paths by which the reaction achieves ΔU can differ.

Path functions can also be understood by contrasting them with state functions. A state function is a property whose value does not depend on the path taken to reach that specific value. State functions depend on three things: the property, the initial value, and the final value. In other words, state functions depend only on the final and initial values and not on the path taken to get from the initial to the final value.

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**Frequently asked questions**

A state function is a property whose value does not depend on the path taken to reach that specific value. It depends only on the final and initial states. For example, if you walk 25 paces forward and then 15 back, your total displacement is still 10, the same as if you had just walked 10 paces forward in the first place.

A path function is a function that depends on the path taken to reach a specific value. For example, if you have $1000 in your savings account and want to deposit some money, the amount you deposit is a path function because it depends on how you obtained that money.

Distance traveled is a path function. This is because the distance traveled depends on the path taken. For example, the distance from Los Angeles to Lake Tahoe is not dependent on the route taken, but the distance traveled to get there does.