Another Work Out

Work versus Energy

Work can come in many forms including mechanical, electrical, chemical, and biological work. This shouldn't be a big surprise considering that we measure it in units of Joules just like energy, and the mantra of many text books is that energy is neither created nor destroyed, it can only change forms. Why shouldn't there be different forms of work as well? In fact, the First Law of Thermodynamics, is just the Law of Conservation of Energy revamped.

However, one must admit it can be confusing to distinguish between work and energy. So here are the nitty-gritty details:

Energy is the potential for an object to produce or create work.
Work is defined as a force provided along a change in distance.

Consider a stationary bagel on a countertop. The bagel has potential energy calculated from the height of the countertop, but isn't doing any work. However, if you push said bagel to skim across the countertop (assume countertop and bagel are frictionless, this wouldn't work too well in practice!) the bagel will have kinetic energy as well as work.

Defining Work with Calculations

So here is the work equation from first year physics:

This can be represented in as an integral: 

**note that you are looking at a change in length.

Now, we are going to examine work with regards to a piston in a cylinder. I should really steal a photo of this from the web and stick it on this blog, but I'm sure many are available in your text book of choice. Pressure in this system is defined as: 

Substituting this back into the work integral gives: 

But note that area times length equals volume. So the equation is simplified to: 

And let it take place at constant pressure, so we can pull a constant from the integral. 

The equation needs to fit with our conventions that work done by the system is positive and work done on the system is negative, and so we place a negative sign in front of the equation. 

 Note that the pressure in this equation is always, always, always the external pressure, not the pressure of the gas.  At a reversible rate, the external pressure will equal the pressure of the gas. 

When is no work done? 

Interesting question. Or not. But still, the problems sets are probably going to test understanding of this concept, so let's examine it by letting the equation equal zero.

All right, so there are two options: 

No work is done when there is no external pressure so the reaction takes place in a vacuum, or when there is no volume change aka the system has constant volume. 

These two situations are irreversible paths. 

Isothermal Process: A Reversible Path

An isothermal process is where a system experiences a change under constant temperature (no temperature change). The internal energy change is zero, as the heat received is transferred directly into the work done on the gas, for example. 

How is a system like this possible without the system interacting with the temperature of the surroundings? As the Zeroth Law says heat flows from hot to cold until both bodies reach the same temperature. This situation seems far too idealistic to be true. 

First of all, remember that thermodynamics is about ideal relationships. In other words, it avoids facing reality. (Kay, maybe this was too harsh and untrue.) Secondly, various sources suggest that this type of process can occur in a carefully maintained heat bath through slow heat exchange. 

Anyway, the isothermal process is reversible.

And of course there is a handy formula derivation that goes with it; the results of which you can keep in your back pocket or arsenal supply, depending on your attitude and emotional outlook on this material.

It involves the Ideal Gas Law (with n=1) where 

So returning to the work equation

We substitute in the Ideal Gas definition for pressure

And the fun begins! (Don't forget temperature is a constant.) 

Recall that 

And applying this to the work equation gives

Recall that 

So the equation in final form can be written as 

Yipee! 

Forthcoming Posts and Plans

I wish to find an example of how to use the isothermal work equation, because I think that would be useful, so I will check my text book and various sources online, and possibly post a subsequent series of equations solved, as those in my class who can't download the prof's ebook, also can't readily download the solutions manual. The limitation to this is that equations I can't solve, I can't post. However, because typing equations is fairly time consuming with the lack-there-of programs I have, I will probably type up the problem from the book and provide a solution by means of a photograph of my notebook. 

Also, I intend to do further posts with topics and subtopics including processes at constant volume, processes at constant pressure, the enthalpy definition, calorimetry and heat capacity, so look out for those if you are interested. 

*The title of this post comes from a wonderful jazz album by the tenor saxophonist Hank Mobley. On this album is one of my favourite jazz recordings of all time entitled "I should care." This isn't related to thermodynamics, but it's pretty hot stuff.