# Path Functions

- Page ID
- 1927

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Path functions are properties or quantities whose values depend on the transition of a system from the initial state to the final state. The two most common path functions are heat and work.

## Introduction

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 is necessary to take into account. For example, a person may decide to hike up a 500 ft mountain. Regardless of what path the person takes, the starting place and the final place on top of the mountain will remain constant. The person may decide to go straight up to the mountain or decide to spiral around to the top of the mountain. 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 final state. These two functions are introduced by the equation \(\Delta{U} \) which represents the change in the internal energy of a system.

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

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

## Several conditions that could apply

**Constant Volume:**if ΔV = 0, the work is also zero since w= -P(ΔV) and substituting zero for the volume would ultimately make the entire term become zero.*So: at constant volume:*Δ*U**= q***Constant Pressure:**When a reaction takes place at constant pressure, the volume is able to expand or contract to ensure that the pressure is constant. Thus, the volume will change*so work will be done.*So, the equation for ΔU in this rxn is: Δ*U**= q + w*: note that whether a reaction is carried out under constant volume or constant pressure, Δ**Constant Volume vs. Constant Pressure***U*will ultimately 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*or the constant volume situation) because the amount of*w(work)*done will make up for it (the paths by which the reaction achieves Δ*U*can differ).**Reversible Process:**a process carried out by making really, really small changes to a component of the system without a loss of energy. Because the changes are so small, the system remains at rest throughout the entire process. A real world example of a reversible process is the measurement in efficiency of a heat engine (a device used to convert thermal energy to mechanical work). It is called a reversible process when no heat is lost or wasted and so the machine runs as efficiently as possible.

The difference in altitude that one experiences when driving from Los Angeles to Lake Tahoe is not dependent on the route they take. Whether they head north on the 101 or on I-5 to get to Tahoe they will still experience an altitude change of approximately 6,620 feet (state function). The distance traveled to get there, however, does depend on the route taken (path dependent function).

Suppose you only slept 3 hours last night because you were up all night working on a paper and that today you have a mandatory 8 am lecture. So, you decide to have a coffee before class. Your transition from being non-caffinated to caffeinated is like a change of state. The way in which you obtain that caffeine, however, is a path dependent change; the time or money you spent to buy or make coffee depends on whether you made it at home or went by starbucks on your way to class.

Suppose you work at a deli that makes world famous turkey sandwiches. For each sandwich you want exactly 4.2 grams of turkey. To measure how much meat you are putting on a sandwich you use a meat scale. Whether you measure it out by consecutively adding 0.01 ounce pieces of turkey, or you simply weigh out two 2.1 ounce pieces, you will ultimately measure out a total of 4.2 grams of turkey (reach the same final measurement: state change). However, the time and the work you put in to each method (the path you take to the same outcome) will vary (pdf). If you were adding infinitely small pieces each time, that could be considered a * reversible process.* Since the pieces you were adding were so tiny, practically immeasurable, you could add or remove a piece without disturbing the a significant change in energy.

Suppose a swimmer times themselves on swimming 50 yards. Whether they do doggie paddle or freestyle they will ultimately still travel 50 yards (the initial state and final state will be the same in each case; state function). However, the energy they expend doing doggie paddle will be much greater than that which they will use doing freestyle (the means or path which they take to complete the 50 yards will affect the amount of energy they use to complete the yardage.

## Problems

**True or False:**The heat of a reaction measured in a calorimeter with a fixed volume is equal to*delta U.***True or False:***delta H*is a path dependent function.**True or False:**When a reaction takes place in a container at constant pressure, work is being done so*delta U = q + w*

## Answers

3)true 2)false 1)true

## References

- Petrucci, et al. General Chemistry Principles and Modern Applications. 9th ed. New Jersey: Prentice Hall, 2007.

## Contributors and Attributions

- Heidi Erickson