# The 1st law of thermodynamics

#### Prerequisites

- Overview: Thermodynamics and Statistical Physics
- Kinetic theory: The ideal gas law
- Kinetic energy and the work-energy theorem
- Heat and temperature

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 — 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.

### The first law in the system schema

The energy of a system is not necessarily conserved. It depends on what system we are considering and how it is connected to the rest of the universe outside the system. (See our discussion of System Schema.) Energy might enter or leave the system, or both enter and leave the system at the same time. Let's for the moment consider only *closed systems* — systems for which no matter leaves or enters the system. We know already that this energy that enters or leaves the system must come from interactions that cross the system boundaries. This is the basic idea of the First Law of Thermodynamics. There are many different ways of stating this law, but one way is:

*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.

While this seems rather straightforward, in practice there are a lot of subtleties. We have to consider many levels of energy and what we get can depend on how we choose to look at what's happening.

### Energy of a system

In our Newtonian framework of macroscopic objects we identified the *mechanical energy* of the system as the sum of two kinds of energy: the coherent kinetic energy (KE) of motion of all macroscopic objects in the system and the potential energy (PE) of interactions between those objects.* *

The objects in our system also have a *thermal energy* associated with the motion and interactions of the molecules that make up the objects in the system. We can associate a kinetic energy with each molecule, and a potential energy with each interaction between molecules. Note that under thermal energy, each molecule moves in a random direction — the prime example being the atoms/molecules in a gas. Thus even though each molecule in a gas has kinetic energy, a macroscopic object made from such gas molecules (e.g. a balloon full of gas) will not have macroscopic kinetic energy and will not spontaneously move at speeds anywhere near the hundreds of meters/second speed of typical air molecules. Macroscopic kinetic energy requires all molecules to move in the same direction, i.e., exhibit coherent motion. Finally the objects in our system also have *chemical energy* associated with the internal kinetic and potential energies of the electrons in atoms and the binding of atoms into molecules. We will write the total result of both of these internal energies as *U*_{internal}. With this, the total energy of our system can be written

$$E = KE + PE + U_{internal}$$

You may already have noticed that we are sometimes using the symbol "$U$" for both the potential energy of the macroscopic object (denoted PE above) and the thermal energy (part of $U$ above). 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 net 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!

### Exchange of energy with a system

When our system has interactions that cross the system boundary and connect to other objects, it can, in principle, exchange energy with those other objects. 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 (where friction does negative work and slows the ball down).**Heat**is the transfer of thermal energy from one system to another. Heat is transferred if there is a difference in temperature between the two systems. Unless something happens to prevent it, heat will spontaneously flow until the temperatures of both interacting systems is the same. This is similar to the motion of molecules which naturally randomly diffuse to fill all space. Thermal energy will move randomly until the average energy of each molecule and of each interaction in both systems is the same, i.e. if the temperature of both systems are the same.

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:

$$ΔE = Q + W$$

or

$$ΔE = Q - W$$

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.

Wait. What? 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 on what work we mean. You need to tell the story of the situation and map physical meaning to the equation. If the work we're talking about in the equation is the net work done **on** the system. The sign in the equation is positive. If you do positive work, you're putting energy into the system, so this leads to an increase in the system's total energy. If the work we're talking about is the net work done **by** the system, the sign in the equation is negative, since the system is doing work on other things, and a positive work means energy is taken away from the system.

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 contact 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 physically! 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 you 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

$$ΔU = Q + W$$

or

$$ΔU = Q - W$$

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 only 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 ½*mv*^{2}'s of all the molecules plus their potential energies of binding into solids or liquids. (The stronger and more complex KE's and PE's of electron motion in atoms and molecular binding we choose to treat separately.) While we can't actually measure velocities and interaction potentials for every single molecule and interaction at the microscale, we can say something about the average energy associated with each motion and interaction: it will be proportional to the temperature! (This is the same law of averages that allowed us to make statements about the diffusion of molecules even though we cannot predict the motion of individual molecules!)

So we can actually observe changes in internal motion simply by measuring changes 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 an *isolated 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, $ΔU_{internal}$ 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 use again our concepts about probability and randomness.

Ben Dreyfus 11/8/11, Wolfgang Losert 12/3/12

Last Modified: January 7, 2022