# Overview: Heat and temperature

#### Prerequisite

Our sensory experience of the world responds to a property of matter that we've mostly ignored in our discussion of motion — heat and temperature. We all have a familiarity with things being hot or cold and we have a rich vocabulary to describe it.  As scientists have figured out what it means for an object to be hot or cold, the explanation has resolved a number of challenges to the Newtonian explanation of motion in a strange and interesting way. Although you probably know most of the results we will develop in this section, you may not have appreciated how strange some of them are!

## Where does the energy go?

A big challenge to the Newtonian theory of motion was the loss of mechanical energy due to non-conservative forces. Energy conservation seems like a very appealing idea. Why should some forces seem to destroy mechanical energy and others not? Why can we think of what some forces do as creating a potential energy and others not?

It took more than a century of hard work by many people (and it's a fascinating story) to figure out what that's all about. The answer is: it's all about motion. Friction, viscosity, and drag steal coherent mechanical energy — energy associated with the motion of all the molecules of an object moving together in the same way — and move it into incoherent mechanical energy — energy associated with the motion of the molecules of an object moving randomly in every which way. We call the latter thermal energy and we describe the increase in this energy as a rise in temperature.

## Hidden Motion

Molecules in a gas, a liquid or a solid are always in chaotic, random motion. And it's not just a little bit of wiggling! The random kinetic energy of the molecules in a baseball at rest is 100 times greater than the additional kinetic energy the baseball has when it is thrown as a 100 miles/hour fastball. Ordinary matter has immense hidden energies that we don't notice in our ordinary interactions with the world.

This topic is another place where a careful analysis of the physics of what's going on leads us to see that we live on the small fringes of immense energies! Remember that we learned that matter that seems relatively inert — it just sits there — in fact consists of positive and negative charges which attract and repel each other  with immense forces. Understanding those forces has led to the mastery of electrical energy and immense changes in the lives of human beings. Similarly, we will discover that the well down which our mechanical energy disappeared is a storehouse of huge thermal energies contained in every object. Understanding these energies led to the first industrial revolution (in the 19th century) and immense changes in the lives of human beings.

So we are going to have to conclude that macro motion runs down because the energy of motion can be hidden in the temperature of objects. Now temperature appears to "run down" too, because hot things get cooler, but cold things warm up too. The real result, we shall find, is that thermal energy tends to get shared evenly — but it never goes away. (It can even sometimes be transformed back to coherent mechanical energy.) There is a new "total energy" -- coherent mechanical plus thermal; and we can restore a conservation of energy theorem!

## Basic thermal concepts

In developing an understanding of hot and cold we have to refine our experiences and pay attention to three characteristics of an object: the internal thermal energy contained in an object at rest, its temperature (the average energy of motion per molecule or mole), and heat — the flow of energy from one object to another.

• temperature ($T$) — the average energy of motion of an individual molecule in an object;
• internal thermal energy ($U$) — the total energy, potential and kinetic, of the molecules of an object; and
• heat ($Q$) — the flow of internal energy from an object of one temperature to an object in contact with it at a different temperature (sometimes heat flow).

## Kinds of internal energies

We have said that internal energy, $U$, is the total amount of energy in an object. This is a tricky concept since there are many kinds of energies that we can identify inside an object: the kinetic energy of the motion of its molecules and the potential energy of their interactions (its thermal energy), the kinetic energy of the electrons inside the atoms and the potential energies of the electron-nucleus interactions (its chemical energy), and even the mass energy of the atoms through $E = mc^2$!

What we pay attention to depends on what transformations we are making. If we are not doing nuclear reactions we can ignore the mass energy. If we are not considering chemical reactions, we can ignore the internal energy of the electrons inside the atoms and molecules. All the different kinds of energy hidden inside a chunk of matter can be quite confusing. For this reason, it is a good idea to focus on changes in internal energy, $ΔU$. Then in any particular example, we can be clear on what part of the internal energy matters in that case.

## Language, language, language!

The language we use to describe thermal energy is tricky and is used in contradictory ways in different contexts. In physics, especially when we are doing the mathematics of thermal energy, we try to be very careful to distinguish the three. They have different mathematical properties. $U$ is extrinsic (proportional to the size of the object), $T$ is intrinsic (density-like: independent of the size of the object); and $Q$ is a given amount of energy — but it is a transfer of thermal energy. Furthermore, you can't think of $U$ as the "sum of all the $Q$s" since $Q$ doesn't doesn't play nicely with temperature, each kind of object translating thermal energy into temperature in its own idiosyncratic way.* So in physics, you have to be careful not to confuse these three.

Understanding the way the  term "heat" is used across disciplines is particularly tricky. In physics, we use the term to me thermal energy transferred between objects. In chemistry, when energy released in an exothermic reaction it is also referred to as heat and is often treated as a part of the enthalpy (which includes the work needed to make more room for any gases that might be produced at constant pressure). And in biology, where thermal calculations are less common, heat is often used to mean internal energy.  (We admit that even in physics, when we are not in the throes of an actual calculation, we might use that language as well — but you should try to avoid this until you are confident you can clearly distinguish all three concepts.) We will deal with these issues in more detail in our chapter on statistical mechanics where we analyze how molecular motion connects with thermodynamics.

## The structure of the chapter

Since for the first half of the 19th century, the molecular model of matter was not developed yet, most of the work of developing thermodynamics — the science of heat and temperature — was developed phenomenologically: by studying the phenomena macroscopically and developing laws to describe the observed behavior.

In this chapter we focus on the macroscopic laws of thermodynamics and what can be inferred from everyday observations and experiments. We will study how thermal energy is distributed in matter and how it relates to temperature, in particular, how objects at different temperatures exchanges thermal energy and how those objects translate gain or loss of thermal energy into changes of their temperature. The key concepts are:

• Measuring temperature: Introducing a new dimensionality with an operational definition
• Heat capacity and specific heat: How objects translate thermal energy into temperature
• Heat transfer: How different objects exchange thermal energy by conduction, convection, and radiation
• Biological implicationSince temperature has a strong influence on chemical reaction rates, the thermal properties of matter have significant biological implications.

## Starting thermodynamics

The most important consequence of our study of heat and temperature is that it will be the start of the science of thermodynamics — the framework for much more detailed understanding of how and when energy flows spontaneously, including the first law of thermodynamics (energy conservation) and the second law of thermodynamics (entropy increases / free energy decreases). These provide powerful tools for quantifying the flow of energy in biological systems and describes not only the energy of random motion, our starting point, but how chemical energies interact with other energy systems — topics of critical importance to understanding biological phenomena!

* Technically, the mathematical point is that "heat" is not a state function, so small bits of transferred thermal energy cannot be integrated to give a unique result. This is one of the reasons for introducing entropy. Small bits of heat divided by the temperature (dQ/T) can be added uniquely and yields the entropy.

Julia Gouvea & Joe Redish 12/24/14

Article 548