3 Explain the difference between point and path functions. Define thermodynamic work and prove that it is a path tubesson?
Answers
Answer:
state function is a property whose value does not depend on the path taken to reach that specific value. In contrast, functions that depend on the path from two values are call path functions. Both path and state functions are often encountered in thermodynamics.
Introduction
Whenever compounds or chemical reactions are discussed, one of the first things mentioned is the state of the specific molecule or compound. "State" refers to temperature, pressure, and the amount and type of substance present. Once the state has been established, state functions can be defined. State functions are values that depend on the state of the substance, and not on how that state was reached. For example, density is a state function, because a substance's density is not affected by how the substance is obtained. Consider a quantity of H2O: it does not matter whether that H2O is obtained from the tap, from a well, or from a bottle, because as long as all three are in the same state, they have the same density. When deciding whether a certain property is a state function or not, keep this rule in mind: is this property or value affected by the path or way taken to establish it? If the answer is no, then it is a state function, but if the answer is yes, then it is not a state function.
Mathematics of State Functions
Another way to think of state functions is as integrals. Integrals depend on only three things: the function, the lower limit and the upper limit. Similarly, state functions depend on three things: the property, the initial value, and the final value. In other words, integrals illustrate how state functions depend only on the final and initial value and not on the object's history or the path taken to get from the initial to the final value.
Here is an example of the integral of enthalpy, HH, where t0t0 represents the initial state and t1t1 represents the final state.
∫t1toH(t)dt=H(t1)−H(to)(1)(1)∫tot1H(t)dt=H(t1)−H(to)
This is equivalent to a familiar definition of enthalpy:
ΔH=Hfinal−Hinitial(2)(2)ΔH=Hfinal−Hinitial
As represented by the solution to the integral, enthalpy is a state function because it only depends on the initial and final conditions, and not on the path taken to establish these conditions. Therefore, the integral of state functions can be taken using only two values: the final and initial values. On the other hand, multiple integrals and multiple limits of integration are required take the integral of a path function. If an integral of a certain property can be calculated using just the property and it's initial and final value, the property is a state function.
State Functions vs. Path Functions
State functions are defined by comparing them to path functions. As stated before, a state function is a property whose value does not depend on the path taken to reach that specific function or value. In essence, if something is not a path function, it is probably a state function. To better understand state functions, first define path functions and then compare path and state functions.
Path functions are functions that depend on the path taken to reach that specific value. For example, suppose you have $1000 in your savings account. Suppose you want to deposit some money to this account. The amount you deposit is a path function because it is dependent upon the path taken to obtain that money. In other words, the amount of money you will deposit in your savings account is dependent upon the path or way 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 would receive two different amounts of money at the end of the week. Thus, a path function is a property or value that is dependent on the path taken to establish that value.
State functions do not depend on the path taken. Using the same example, suppose you have $1000 in your savings account. You withdraw $500 from your savings account. It does not matter whether you withdraw the $500 in one shot or whether you do so at a rate of $50. At the end when you receive your monthly statement, you will notice a net withdrawal of $500 and will see your resulting balance as $500. Thus, the bank balance is a state function because it does not depend on the path or way taken to withdraw or deposit money. In the end whether you do so in one lump or in multiple transactions, your bank balance will stay the same. The figure below illustrates state functions in the form of enthalpy:
Figure 1: Two different paths from same initial and final states results in the state variables values
In this figure, two different steps are shown to form NaCl(s)NaCl(s).
Path one: The first path takes only a single step with an enthalpy of formation of -411 kJ/mol:
Na+(g)+Cl−(g)→NaCl(s)(3)(3)Na(g)++Cl(g)−→NaCl(s)
Path two: The second path takes five steps to form NaCl(s)NaCl(s)
Na(s)+1/2Cl(g)→Na(g)+1/2Cl(g)(1: sublimation)