Lecture 16

Heat and Temperature

We all know what temperature is, and how to measure it. Still it might be useful to remember that there is a certain amount of subjectivity in the perception of temperatures (ice-cream at 60 ⁰F is warm, while coffee at the same temperature is cold...). Moreover, our normal temperature scales are, like most other units, arbitrary (why should the freezing point of water be at 0⁰C or at 32⁰F?)

Note : at the beginning of Chapter 12, the textbook makes a distinction between "degree Celsius", used to indicate a temperature, and "Celsius degree", used to indicate a temperature difference !!! I have never encountered such a distinction before in my life, and I invite you to ignore it.

You are probably more familiar with Fahrenheit, but it is useful to get used to Celsius (also called Centigrades). From their definition, it is easy to derive the conversion equations :

$T_{C}~=~5/9(T_{F}-32)~\approx~(T_{F}-32)/2$

$T_{F}~=~9/5~T_{C}+32~\approx~2T_{C}+32$

$\Delta T_{F}~=~2\Delta T_{C}$

Question 1: is the Fahrenheit numerical value always bigger than the corresponding Celsius ?

Question 2 : is there a temperature at which a C and an F thermometer would give the same reading ?

Is there a way to define a non-arbitrary scale for temperatures? Almost, since Physics has taught us that Nature has set an absolute value for the lowest meanigful temperature, which is called the Absolute Zero. The need for the existence of a lowest possible temperature appears in many different Physics phenomena. Perhaps the easiest verification comes from the relation between Pressure and Temperature in a gas kept at a constant volume (you will be able to see that for yourself in a future lab session). Experience shows that, in a fixed volume of gas, pressure and temperature are proportional to each other (i.e. a plot of pressure vs. temperature would be a straight line). Measuring P and T at a few points (two would be enough in principle) we can find the point where the line intersects the horizontal axis (i.e. find the temperature corresponding to zero pressure). Temperatures below this value would be meaningless, since they would correspond to unphysical negative pressure.

When measured in Celsius, absolute zero is found to correspond to -273.15⁰. This value is then used as the zero point for the absolute temperature scale, whose unit is the kelvin.

In a naive picture, we could think of absolute zero in the following way :

it is known (as we will find out before the end of the term) that in a gas temperature is related to the average kinetic energy of its molecules. We also know that pressure is a manifestation of innumerable molecule collisions. We can think of the absolute zero as the temperature at which the molecules reach zero velocity, therefore are not colliding any more. But be aware that this picture, even though appealing, is not correct, especially when Modern Physics (as represented by Quantum Mechanics) comes into play.

From a practical point of view, temperature can be measured exploiting any suitable phenomenon that is temperature dependent (ideally, temperature-proportional). Among them are thermal expansion, gas pressure, electric resistivity, electromagnetic radiation (e.g. infrared, we will learn more about the connection between radiation and temperature later), etc.

Thermal Expansion

No need to stress the well known fact that a heated body expands and, within a fairly wide range, the expansion is proportional to the temperature change. Quantitatively, this is expressed by

$\Delta L/L~=~\alpha\Delta T$

representing the expansion in a given dimension. The value of $\alpha$, the coefficient of linear expansion depends upon the choice of units for temperature (why does it not depend upon the choice of units for length?)

Even though, for most materials, $\alpha$ is a rather small quantity (of the order of 1/100,000), for large enough L effects of expansions can be quite dramatic, and have to be accounted for in most engineering projects. And, when dealing with thermal expansion, it is better to allow for it than try to fight it (see e.g. example 4).

Thinking in terms of linear expansion is useful when dealing with "one-dimensional objects" (e.g. a long thin rod). For more ordinary shapes (and especially when dealing with fluids) it is more useful to think in terms of volume expansion, which is expressed by a similar formula:

$\Delta V/V~=~\beta\Delta T$

For a given material, $\beta$ is roughly equal to $3\times\alpha$. We can understand this, by thinking of a cube of side L. Upon heating, each dimensions will grow by $\Delta L$, so that its final volume will be

$V+\Delta V~=~(L+\Delta L)^{3}~=~L^{3}+\Delta L^{3}+3L^{2}\Delta L+3L\Delta L^{2}~\approx~L^{3}+3L^{2}\Delta L ~=~V+3L^{2}\Delta L$

we then have

$\Delta V~=~3L^{2}\Delta L, \longrightarrow \Delta V/V~=~ 3L^{2}\Delta L/L^{3}~=~3\Delta L/L$

It is also useful to think how density depends upon temperature. Density is defined as the ratio m/V. When heating a body, its mass doesn't change, but its volume increases, therefore the density decreases (and obviously the density increases when the object cools down). This is true for practically all situations, with the extremely important exception of water, whose behaviour is the opposite below 4⁰C. The next time freezing temperatures cause your pipes to burst, don't be too mad, since the same effect also prevents lakes and oceans from freezing through, with rather important consequences on both the origin and the continuous development of life.....

HEAT

For a long time, scientists did not have a clear idea of what heat, or, more precisely, heat transfer really was. In fact, the most common belief was that heat was carried by some sort of intangible, mysterious fluid, called phlogiston. The eventual realization that heat was in reality yet another form of energy was a major milestone, that effectively opened up the road to the industrial revolution, the development of steam engines, the science of Thermodynamics, etc. Interpreting heat as a form of energy closed the loop in most mechanical processes, showing that, whenever mechanical energy is lost (e.g. due to friction) in any given process, an equivalent amount of energy is generated in the form of thermal energy, and this in turn gives to the Principle of Energy Conservation a much much broader scope.

When studying heat transfers, it is more straightforward to deal with the unit originally introduced to describe such phenomena, i.e. the calorie. 1 calorie is defined as the amount of heat required to increase by one degree Celsius 1 gram of water (the exact defintion would state ....to bring 1 g of water from 14.5 to 15.5 degrees). A related unit is the Calorie (notice the capital C), which is equal to 1000 calories, therefore it is the amount of heat required to increase by one degree the temperature of 1 kilogram of water.

In the next step, we define the specific heat capacity of any substance as the amount of heat (in Calories) required to increase by 1⁰C 1 kg of the substance. From these definitions it follows immediately that the heat capacity of water, measured in Cal/(kg ⁰C), is 1. The heat capacity of water is among the highest of most material, it takes lots of heat to warm up water and conversely, hot or warm water can store and release lots of heat.

From these definitions, it follows immediately that the quantity Q of heat involved in changing the temperature of a body of mass m by an amount $\Delta T$ is given by

Q = m c $\Delta$T

where c is the specific heat of the particular substance

When, through the work of scientists like Benjamin Thompson Count Rumford (a rare case of an American emigrating to Europe) and James P. Joule, it was realized that heat was just another form of energy, it was also possible to obtain the exact correspondence between the traditional units for measuring heat and the standard units for energy (you will do that yourself in the lab). And the answer was:

1 cal = 4.186 joules, or 1 Cal = 4186 joules

This result is extremely useful, since it allows us to work out how much weight lifting we need to do in order to burn off a two-doughnut snack ...

Problems 12.47, 12.51

Phase Changes
Adding or removing heat to/from a substance will change its temperature up to a certain point : we all know what happens to water when we heat it or cool it too much. The most important thing to remember when dealing with phase changes is that, throughout the change the temperature of the phase-changing substance remains constant. When we say that water freezes at 0⁰C, we mean that, if we have some water and lower its temperature, when it reaches 0 it will stay at that value until all the water has frozen, and only then removing more heat will further cool the sample. Conversely, if we start with very cold ice and warm it up, it will start melting when it reaches 0⁰, and the ice/water mixture will remain at that temperature until all the ice have melted (and the same behaviour occurs when water boils away, i.e it turns into its gaseous phase, at 100⁰C).

The amount of heat that is involved in the phase change (the latent heat of fusion or of vaporization) is considerable: the latent heat of fusion for water is 79.7 cal/g (or Cal/kg), which means for instance that it takes the same amount of heat to melt 1 kg of ice originally at 0⁰C as to increase its temperature from 0 to $\approx$ 80⁰ !!!. Similarly, the latent heat of evaporation for water is 539 cal/g .

Remember that the process goes both way: if e.g. lots of heat is required to vaporize liquid water at 100⁰, condensing steam will release an equivalent amount of heat: a burn can be much more severe when caused by steam rather than water at 100⁰C.