# 2.3: Thermal Energy and Heat

- Page ID
- 84589

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The question of what makes objects hot in some circumstances, and cold in some other situations, had intrigued philosophers for millennia.^{1} In the XVIII Century, a well-established theory was that the entity responsible for such effects was a mysterious *caloric fluid* -an invisible, weightless and odorless substance, capable of penetrating matter. An object with high content of caloric flu id would get hot, and an object depleted of it would become cold. Only in 1799 a British scientist Humphry Davy performed a famous experiment showing that bodies may be heated up by *performing work on them*, which some time later led to the formulation of modern theory of thermal phenomena, usually referred to as *thermodynamics*. As we know now, it's not the caloric fluid that makes bodies hot -it's the *internal energy*, or *thermal energy*. It's a special category^{2} of energy, usually denoted as \(U\) for distinction from pure kinetic energy or potential energy.

Only in diluted gases their internal energy can be identified with pure kinetic energy -here \(U\) is the sum of the kinetic energy of all individual molecules comprising the gas. In solids, in contrast, all constituent atoms are at fixed positions, they cannot move freely. However, each atom is coupled to several neighboring atoms by psysiological of hot and coldbonds", which act as springs of sub-nanometer lengths. Therefore, each individual atom behaves like a miniature spring oscillator. Then, the \(U\) of a given solid is simply the sum of the oscillatory energy of all individual vibrating atoms (more often, we call it *vibrational energy*.

So, it is the energy which decides of whether a body is hot or is cold. The internal energy of a body, call it A, can be changed in two ways. One is, as Humphry Davy discovered, by performing work on the body. For the amount of work performed, we use the symbol \(\Delta W\). The other way is by *heat transfer *from another body call it B, brought in contact with body A. If B is psysiological of hot and coldwarmer" than A, some energy flows into A, and if B is psysiological of hot and coldcooler" than A, some energy exits A and flows into B. The portion of energy entering or exiting body A in such a way is called *the heat transfer*, and conventionally denoted as \(\Delta Q\). What we have stated above by words, can be expressed in a mathematical form:

\[\Delta U = \Delta W + \Delta Q \]

which is known as the **First Law of Thermodynamics**.

Two comments concerning the above. One is simply about the terminology. According to the psysiological of hot and coldofficial language of thermodynamics, we cannot say: *It's the heat content that decides of whether the body is warmer or cooler*. The correct way is to say: *It's the amount of internal energy in a body which decides of whether it is warmer or cooler*. The term *heat* should be used only for describing the amount of thermal energy transferred into the body, or out of the body, via contact with another body.

The other comment is about physics. Namely, consider a situation in which there is no heat transfer, \(\Delta Q = 0\), only work \(\Delta W\) \textbf{is performed on the body.} Then, according to the First Law, the content of the internal energy in the body increases by:

\[ \Delta U = \Delta W\]

Seems in perfect agreement with the Work-Energy Principle? Yes, it is!

But now one can think simple mindedly: if so, then the First Law may work in the other way, namely:

\[ \Delta W = \Delta U \]

Meaning that the body can deliver work of \(\Delta W\) by diminishing its internal energy by \(\Delta U\) -which also seems to be in perfect agreement with the Work-Energy principle!

But the bad news are that the Fist Law *does not* work the other way ;-( ... Thermal energy cannot be converted to mechanical work directly... The troublemaker here is the Entropy which emerges only in the **Second Law of Thermodynamics**.

We will talk about the Second Law somewhat later --because now we have to introduce one more essential thermodynamic parameter -namely, the *temperature*

## Temperature

We all have an intuitive concept of temperature, based on -quoting the words of Herbert Callen^{3}^{ }-*psysiological of hot and cold. *It's also a common knowledge that "temperature is what you read from a thermometer". It is not silly -it's something we call an "operational definition". Sometimes it is really difficult to find something better than such a definition for notions in physics. For instance: "Time is what we read from a clock" -can you think of a better definition of time? It's certainly not easy!

So, in low-level courses of physics instructors or textbooks often decide that such an "operational definition" of temperature is sufficient -with additional information about "standard temperature points" needed for calibrating the thermometer. In the Celsius scale, created in 1742, such points are the water freezing temperature, taken as 0 degrees, and water boiling temperature, taken as 100 degrees. Mr. Fahrenheit, who created a temperature scale even earlier, in 1724, took the temperature of human body as 100 F, and that of the mixture of ice and ammonium chloride as 0 F. Those points were not very precise, so later people started using the same points as in the Celsius scale: water freezes at 32 F, and boils at 212 F. Therefore, the conversion formulae between the two scales are:

\[ t_{Fahr.} = 1.8 \times t_{Cels.} + 32 F\]

and

\[t_{Cels.} = \frac 59 \times (t_{Cels.} 32 F)\]

However, I can bet that you don't want to think that you are taking only a low-level physics course" -and you'll certainly like to know more, what is the **underlying physics** in the notion of temperature. Well, to satisfy your wish may not be an easy task for an instructor...

The truth is that with the progress in physics, when more and more experimental facts were collected, and more and more understanding of them was achieved, the definition of temperature also had to be consistent with all the accumulated knowledge of thermal phenomena. And presenting the current definition would be difficult, for two reasons. First, because it involves the notion of** Entropy**, which is a highly abstract concept. And second, because the definition in question is given by a mathematical equation including **partial derivatives**. Well, my intention is to make the present text digestible even for students who have not necessarily taken any **calculus classes**, and therefore I try to seldom use even the **simple derivatives**, only when there is really no other option -but the theory of partial derivatives, it's a much more advanced math than the theory of simple derivatives!

Therefore, I will not even try to discuss this super-duper definition here, but I will give you a definition that is perhaps not perfect, but definitely is good enough in the temperature region around room temperature, and for higher temperatures. Which is OK, because in this course we won't be talking of any phenomena that occur in the region of very low temperatures! And one more advantage is that the definition that will follow is primarily a conceptual one", with not much math.

The definition is based on the so-called ”kinetic theory of gases", a theory that matured in the second half of the XIX Century. Please recall what was said above about the internal energy \(U\) of diluted gases: it's simply the sum of the kinetic energies of all molecules comprising the gas considered. And what we said about the the energy \(U\) in solids? That it is the sum of **vibrational** energies of all atoms in the solid.

Now, remember what happens in vibrational (oscillatory) motion of a spring oscillator: the potential energy of the spring \(V_{\rm stretch.}\) changes to kinetic energy \(K\), \(K\) changes to potential energy of the compressed spring \(V_{compr.}\), the latter back to \(K\), \(K\) back to \(V_{stretch.}\), and then the cycle repeats, and so on, and so on. From the above it follows that on average, half of the time the oscillator energy is potential, and half of the time it is kinetic. And averaged over a long time, the potential energy contribution \(\langle V \rangle\)^{4} and the kinetic energy contribution" \(\langle K \rangle\) are equal.

So, we can write that the total energy of a spring oscillator is:

\[ E_{Oscill.} = \langle V \rangle + \langle K \rangle \]

But also:

\[

\langle V \rangle = \langle K \rangle \]

So that:

\[E_{Oscill.} = 2\langle K \rangle \]

In view of the above, since a solid may be thought of as an assembly of miniature atomic oscillators", we can conclude that the thermal energy \(U\) of the solid is:

\[ U_{solid} = 2\langle K_{all\;atoms} \rangle \]

In conclusion, we can say that both for a gas and a solid:

\[ U \propto \langle K_{all\;atoms} \rangle \]

where the meaning of the mathematical symbol \(\propto\) is proportional to.

Now, it can be further reasoned:

\more \(U\) \(\Rightarrow\) warmer body \(\Rightarrow\) higher temperature;

and

less \(U\) \(\Rightarrow\) cooler body \(\Rightarrow\) lower temperature,

which lead to the final conclusion:

**The temperature of a body is a measure of the average kinetic energy of constituent atoms.** We are nearly done with temperature! But before we finish, I need to tell you about two important conclusions emerging from the considerations in this sub-sub-sub-section. The first conclusion is that the internal energy \(U\) is a linear function of the temperature, which we will denote now as \(T\). "Linear function" simply means that the dependent value (\)U\) in the present case) is equal to the independent value (here, the \(T\)) multiplied by a constant factor. So:

\[ U_{\rm given~body} = {\rm constant~factor} \cdot T\) \]

\[ = C_{\rm given~body}\cdot T \]

where the \(C\) coefficient is called the "heat capacity" of a given body.

The second conclusion is that that if all internal energy is removed from the body, i.e., when \(U=0\), then it must be \(T = 0\). Indeed, zillions of experiments carried out since the birth of modern thermodynamics have made it possible to determine that such a temperature point does exist, at -273.15 \(^{\circ}\)C, or -459.67 \(^{\circ}\)F. This point is called the "absolute zero". The SI system introduces yet another temperature scale, called the "Absolute Temperature Scale", or the "Kelvin Scale", in which the absolute zero is the fundamental thermometric point. It was decided, for simplicity, that one degree in the Kelvin scale, 1 K in short (without the \( "^\circ\)") would be equal to 1 \(^{\circ}\) C, one degree in the Celsius scale. So, the conversion from the Celsius scale to Kelvins is pretty straightforward:

\[ T [{K}] = t[{^\circ}{C}] + 273.15 {K}\]

( [...] means in the units of what is written between the two brackets).

Hence, water freezes at 273.15 K, and boils at 373.15 K.

For distinction, capital \(T\) should be used only for expressing temperatures in Kelvins, and lower-case \(t\) for temperatures in the Celsius or Fahrenheit scales. Please always pay attention to in which scale the temperature is given, confusing \(^\circ\)C with K is one of the most common errors made by students in homeworks and exams in the Ph313 course!

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1. in the old days, there were no psysiological of hot and coldscientists" of different categories, all people doing any kind research were called psysiological of hot and coldphilosophers", what in ancient Greek language meant psysiological of hot and coldthey who love wisdom"

2. Also, a very important category in the context of energy usage, because most of the electric energy consumed today comes from \textit{thermal power plants}, i.e., from big facilities in which thermal energy is converted to electric energy.

3. Herbert Callen, 1919-97, was a distinguished American physicist, author of the textbook T*hermodynamics and an Introduction to Thermostatistics*, the most frequently cited thermodynamic reference in physics research literature.

4. In physics, writing a quantity in between angular brackets: \(\langle {\rm quantity} \rangle\) is an often used symbol of averaging.