Understanding State Functions and Reversible Paths
Thermodynamics is the corner stone of physical chemistry. Two of the more difficult concepts for people to understand is a state function and a reversible path. I hope I can clarify what these mean and give some examples.
In the Thermodynamics of Weight Loss I mentioned that the energy content of cookies is found by burning them and measuring the heat of combustion. Whether burned in your body or in a bomb calorimeter, the energy change is the same because energy is a state function. A state function depends only on the initial state (cookies) and the final state (burnt cookies). Straight combustion and metabolism produce the same amount of energy even though the two combustion paths differ.
State functions play a special role in thermodynamics because no matter how you go from the initial to final states, a state function always has the same value.
Some quantities are not state functions, like heat and work. Getting work from a system depends upon the efficiency of your equipment so for the same energy change, you get more work from a more efficient path and less work from a less efficient path. Let’s stay with state functions for now.
Imagine a ball on a table. When it is pushed off, it falls and ends up at a lower location.
We say the state of the ball has changed from having a higher potential energy before it is pushed (the initial state), and a lower potential energy after it has reached the bottom (the final state). So we write:
It does not matter how the ball falls or is raised, when it is at a height h above the ground, it has more energy than after it has fallen. On the way down it can do work,
Not all the potential energy can change into work. Some goes into sound, some into deformation of the plank, and some kinetic energy stays with the original ball.
When the ball falls, it gives energy to the surroundings. When the ball is raised up, the ball gains energy from the surroundings. That is conservation of energy.
Reversible and Irreversible paths
In later entries I will talk about different paths a system can follow, but of all the paths, there is a special one called the reversible path. Let us clarify the meaning of reversible. First it is never possible to obtain a true reversible path. It is an idealized case in which at every step along the way the system (the ball) is in equilibrium with its surroundings. Clearly if something is at equilibrium there is no noticeable change. Hence one says that a reversible process happens so slowly that equilibrium is always maintained.
I want to illustrate this by moving the ball back up to the top. Watch the movie and notice that in every case, a lot more energy is expended than the energy gained by the ball which is
The following movie shows some of the many irreversible paths to take the ball back up to its original position.
This is always the case. Think of the energy in a liter of gasoline. The energy you are interested in when you fill up the tank is “How many kilometers per liter (or miles per gallon)” is your main question. However it took a lot of work to deliver that gasoline to the station, and that energy is not counted in the energy content of the gas (but it is in the price). It is the same with the cookies. It has a certain energy content, let us call it the internal energy, but those cookies went through a whole process to get to your plate. Let us tack on that energy saying it is due to the equipment needed to produce the system (a cookie, a liter of gas, etc.)
These are irreversible paths because the work done by the equipment not only raises the ball but also drives the apparatus. We always have to pay a higher price to do something. For a reversible path we ignore that extra energy, so we can focus only on the system (the ball). In the movie below think of the ball being raised up ideally so that no extra energy is used.
This is the idealized reversible path. All we want to know is how much energy is in the ball, not the energy expended in raising the ball with inefficient methods.
Now there is an important property of reversible paths. That is you can always repeat the process: push the ball off, raise it back up reversibly, and you get to the same spot. Then repeat of the process is called a reversible cycle. You can always have irreversible cycles too, but reversible ones are special because they only relate to the system (the ball).
Many examples of thermodynamics are done with an ideal gas as the system, rather than a ball. So let us suppose we have one mole of an idea gas. Consider an isothermal (constant temperature) process, which is Boyles’ law, then the pressure of the gas is inversely proportional to the volume,
In the movie, the reversible process maintains the external pressure on the piston equal to the gas pressure
Reversible path: and the isotherm is followed.
The irreversible paths show two cases: first the external pressure is suddenly dropped to zero as the gas expands. In the second, the pressure is suddenly raised to 5 atmospheres and the gas is compressed.
Irreversible path: the path deviates from the isotherm.
The unique reversible path follows the isotherm. There is only one reversible path. The two irreversible paths do not follow the isotherm. There are an infinite number of irreversible paths.