Guide 18-1a. The First Law of Thermodynamics

Note: Study this guide before reading any of Chapter 18.

In Chapter 15, you saw that Bernoulli's equation was just the law of conservation of energy applied to fluids. You'll see in what follows that what we call the first law of thermodynamics is just the law of conservation of energy applied to thermal processes. Thermal processes include the possibility of energy transfer into or out of a system due to a temperature difference between the system and the external environment. This form of energy transfer is called heat. We'll get back to that after a bit of review.

The general equation you've used for conservation of energy throughout the course is the following:

W_ext = ΔE_sys

W_ext is the work done by external forces on the system. We italicize the phrase on the system, because that will be important to keep in mind throughout the discussion.

In thermodynamics, we apply conservation of energy to systems of large numbers of molecules, typically gases. The energy E_sys of such a system is in general both kinetic and potential. The kinetic energy of a system of molecules is the sum of the kinetic energies of all the molecules, which, of course, we can't calculate directly because we couldn't keep track of all the molecules. The potential energy is that arising from any electrical forces of interaction between the molecules. We haven't studied electrical forces yet; suffice it to say that this is the fundamental force responsible for molecular bonding as well as the force between electrons and protons in atoms.

Let's look closer at the W_ext term. How could work be performed on a gas by an external force? An example is the internal combustion engine. The gas is enclosed in cylinders in the engine. There is a piston in each cylinder which, in one part of the cycle, compresses the gas to a smaller volume. In doing so, the piston does work on the gas. This is external work. In this case, the work is positive, because the force exerted by the piston and the displacement of the piston are in the same direction. In another part of the cycle, the piston is displaced in the opposite direction; the external work of the piston on the gas is, in this case, negative. Alternatively, one could say the work done by the gas on the piston is positive. In one case, the environment (piston) does work on the gas (system); in the other case, the gas (system) does work on the piston (environment). That is,

W_{environment on system} = -W_{system on environment}

Let's emphasize that the symbol W_ext represents work done by the environment on the system by changing the subscript to W_es, which will be shorthand for work done by the environment on the system. With this change in notation, we have so far:

W_es = ΔE_sys

Note that the following would be equivalent:

-W_se = ΔE_sys

We now have to bring into the discussion the energy source of heat which we mentioned earlier. When there's a temperature difference between a system and its environment, there's the possibility of heat moving into or out of the system. The conventional symbol for heat is Q, and the sign convention is that Q is positive when heat is added to the system. In order to include heat as an energy term in the conservation equation, we write:

Q +W_es = ΔE_sys   (or Q - W_se = ΔE_sys)

In the field of thermodynamics, the term ΔE_sys is called the internal energy of the system. We don't divide it up into kinetic and potential energy terms. There's a good reason for that; we're not going to add up all the kinetic and potential energies for the molecules (remember, there are 6 x 10²³ of them in a mole). We have other ways of determining internal energy. In Chapter 17, for example, you saw that the internal energy of an ideal gas was 3nRT/2. Now we come to a point of ambiguity in terminology. This ambiguity is the fault of physicists who sometimes insist on using the same symbol for different things. In settling on a single term to represent internal energy, the community of physicists seems to have settled on U. You see the confusing thing here. You're used to having U represent potential energy. Now we're telling you that for thermodynamics, U represents the sum of kinetic and potential energies. Sorry, but we're stuck with it. So that means our conservation of energy equation for thermodynamics becomes the following:

Q +W_es = ΔU   (or Q - W_se = ΔE_sys)

Now let's compare these results to the equation given in the approved AP exam list. The latter equation is Q +W = ΔU. (We'll call this Equation A.) The AP people don't include a subscript on the work term; they assume W = W_es. That assumption, by the way, is stated together with the equation in the list of AP exam equations.

Now let's compare to the textbook. The textbook writes the formula as Q - W = ΔU. (We'll call this Equation B). The textbook doesn't use a subscript either, and they assume W = W_se! So you see how important it is to know what W you're talking about. You'll make sign mistakes if you're not careful.

To summarize, here are the two expressions of conservation of energy for thermodynamics:

Q +W_es = ΔU   [Eq. A]

Q - W_se = ΔU    [Eq. B]

From now on, however, we will use only Equation B in order to be consistent with the textbook readings and problems. We will not leave off the subscript.

Now you can read sections 18.1, 18.2 and 18.3 up to but not including the section on isothermal processes.

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