Class content > Thermodynamics and Statistical Physics
Prerequisites:
In our work so far we've looked at several different forms of energy:
- Kinetic -- the energy associated with the motion of an object;
- Potential -- the energy associated with interactions between objects including gravitational, electric, and spring potential energies;
- Thermal -- the energy (both kinetic and potential) associated with the incoherent random motions of molecules;
- Chemical, which is the kinetic and electric potential energy of the electrons and nuclei that form molecules.
At the most fundamental level, the First Law of Thermodynamics is just a statement of the conservation of energy, a principle you're familiar with from every science class you've ever taken. Even if mechanical energy is only conserved under specific circumstances, the total amount of energy in the universe is always conserved, as long as you take into account all objects and every type of energy. But we can get more precise than that, and look at the energy of a system: a particular subset of the universe that we're interested in (which is going to be more useful than looking at the whole universe!).
The first law in a system
Let's say we have a system: some object or set of objects, with no matter entering or leaving the system during the period of time we're interested in. Examples include: a ball flying through the air; a tank full of fish; the whole Earth.
If our system is smaller than the whole universe, then the energy of the system is not necessarily conserved! Energy might enter or leave the system, or both (even if matter doesn't!). But what we know about conservation of energy tells us that this energy doesn't just appear out of nowhere or disappear to nowhere; it has to come from or go to somewhere else. This is the basic idea of the First Law of Thermodynamics. There are many different ways of stating this law, but one way is:
(1) The change in the total energy of a system is equal to the net input (= input minus output) of energy into the system.
This includes all forms of energy, both macroscopic and microscopic.
Energy of a system
In our Newtonian framework of macroscopic objects we identified the sum of two kinds of energy as mechanical energy: the (coherent) kinetic energy (KE) of motion of the object and its potential energy (PE) arising from interactions with other objects. The coherent kinetic energy is associated with an accompanying momentum -- the KE is ½mv2 (as a vector). Although the potential energy may be associated with objects outside our system (such as the earth), if those objects are fixed and carry no KE themselves, we may treat the PE as belonging to our system.
The objects in our system also have a thermal energy due to the incoherent kinetic (and potential) energies associated with the motion of their molecules, and the chemical energy associated with the binding of atoms into molecules. We will write the total result of these internal energies as Uinternal. With this, the total energy of our system can be written
E = KE + PE + Uinternal
You may already have noticed that we are using the symbol "U" for both the potential energy of the macroscopic object and the thermal energy. This is not a problem if we always put the appropriate subscript on the U -- gravity, electric, spring, or thermal. But there are two problems. First, the internal energy of the object is not just a potential energy -- it is part KE as well. Second, when we are just talking about thermodynamic systems that do not move as a whole (have no momentum), the subscript "internal" is often dropped. This doesn't matter if you are working in chemistry and are not going to create momentum; but if you are calculating how your internal energy transforms via chemical reactions in order to help you climb a tree, then both kinds of "U" will be involved. Be careful!
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Exchange of energy with a system
Since our system can interact with other objects, it can, in principle, exchange energy with the rest of the universe. How? You already know about two major ways already from our study of Newtonian mechanics and the macroscopic properties of matter: work and heat.
- Work is energy transfer at the macroscopic level, caused by a force acting over a displacement. Examples include: the expansion or compression of a gas, air resistance acting on a flying ball (here, friction does negative work and slows the ball down).
- Heat is energy transfer at the microscopic level. Heat is the transfer of energy due to a difference in temperature (the average energy of a molecule), and deep down it is caused by collisions between molecules.
Therefore, you might come across other expressions of the First Law, which spell out the types of energy transfer. You might see the First Law written as:
(2) ΔE = Q + W
or
(3) ΔE = Q - W
where E is the total energy of the system (so ΔE is the change in the total energy of the system), Q is the net heat transfer into the system (= heat in minus heat out), and W is work.
Fussing about signs
Wait. There are two forms of the law with different signs? Why is the work sometimes written with a plus and sometimes written with a minus?
The answer is that it depends whether we're talking about the net work done on the system, in which case it's positive, since you're putting energy into the system, so this work leads to an increase in the total energy, or the net work done by the system, in which case it's negative, since the system is doing work on other things, and energy is taken away.
Since the work done is a force times a distance, we know from Newton's 3rd Law that the work done on a system (by a force exerted on the system by something else) and the work done by a system (by the force the system exerts on that something else) have the same magnitude and differ only by a minus sign. N3 says that if A exerts a force on B, then B exerts an equal force on A in the opposite direction. So we can extend this and conclude that if A exerts a force on B in the direction of the displacement (and thereby does some amount of positive work W), then B must be exerting an equal force on A at the same time, in the direction opposite to the displacement (and thereby does some amount of negative work -W). So if the system does work W on its surroundings, then work -W is done on the system, and vice versa.
Much ink has been spilled attempting to argue for whether equation 2 or equation 3 is a better way of representing the First Law symbolically. Instead of getting dragged into that fight, we're going to stay above the fray and advise you instead to think about what's really happening! Is energy being added to the system, or taken away from the system? As a result, is the total energy in the system increasing or decreasing? Thinking this through will lead to more success than trying to plug numbers into an equation.
Dropping the macro motion
But that's not all! Sometimes you'll see the First Law written as
(4) ΔU = Q + W
or
(5) ΔU = Q - W
where U is only the internal energy, the energy associated with what's happening inside the system. So ΔU includes the chemical and thermal energy of the molecules inside a system, but not the kinetic or potential energies associated with the motion and position of the system as a whole. So these representations of the First Law are valid if the overall kinetic and potential energy of the system are not changing (e.g. if the system is staying in one place), and therefore any transfer of energy into or out of the system results in changes to internal (chemical and/or thermal) energy.
The change in internal energy is literally the change in the sum of all the ½mv2's of all the molecules plus their potential energies of binding into solids or liquids. (The stronger and more complex PE's of molecular binding we choose to treat separately.) Since we can't actually measure this for every single molecule, we can observe this as a change in temperature.
Some examples and biological relevance
In the universe as a whole, Q = 0 and W = 0 since the universe can't transfer heat or work to or from anything outside the universe. The First Law therefore tells us that ΔE for the whole universe is 0, so the total energy of the universe is constant. (That was what we started with, so if we don't get that result, something is wrong!)
More generally, if we have a closed system -- one that isn't exchanging either matter OR energy with anything else, we can make the same argument that Q = W = 0 for that system. We can conclude similarly that ΔE = 0 for that system. Does this mean that nothing is changing in that system? No! Even though the total energy stays constant, energy can be transforming from one form to another. For example, if you have a chemical reaction taking place inside an insulated container (so no energy can get in or out), it could be an exothermic reaction (so chemical energy is converted into thermal energy) or an endothermic reaction (so thermal energy is converted into chemical energy).
Let's look at a biological example, and choose your body as the system of interest. You're metabolizing right now: chemical reactions are taking place in your body, and releasing energy. On the whole, these reactions are exothermic, so the net effect of all these chemical reactions is to decrease the total chemical energy in the system (your body). You're a warm-blooded mammal, so you're maintaining a constant body temperature (or at least trying to), so the total thermal energy in your body is staying constant. Therefore, ΔUinternal for your body is negative. So the First Law tells us there must be energy leaving your body, via either heat or work (or both). So maybe you're doing work on other objects (e.g. kicking a soccer ball or pushing a cart), or maybe there is heat transfer from your body to the surroundings (if you have a lot of people packed into a room, the room will get noticeably hotter).
The next step: one way street!
So far, everything here is completely reversible. Whichever way you choose to express the First Law, you could stick minus signs on both sides of the equation. That means that if you had all the energy transfers go in the reverse direction, this wouldn't violate the First Law at all. For example, one of the examples discussed above is that you have cellular respiration going on in your cells (a series of chemical reactions with a net release of energy), and there is heat transfer from your body to the outside. Could this happen backwards? Can you take a heating pad, use it to transfer heat to your skin, and thereby make respiration run backwards, so that you end up with more chemical energy than you started with? Can you take in energy through heat and store this energy for later? The First Law doesn't say no! Based on conservation of energy alone, there is no reason why this shouldn't work.
Of course it doesn't work. But to explain why, we'll need the Second Law of Thermodynamics, and to understand the Second Law, we'll need to start developing some concepts about probability and randomness.
Follow-ons:
Ben Dreyfus 11/8/11
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