Chapter 19

Temperature

Temperature is a concept with which we are very familiar, even though it involves a certain amount of subjectivity (ice cream at room temperature is warm, while room temperature coffee is cold). In Physics, we need to define temperature in a somewhat more rigorous way. Soon we will find out that what we perceive as temperature of an object is a macroscopic manifestation of the motion of the object's molecules, and in fact we will see that temperature is proportional to the average kinetic energy of the molecules. But for now we will not consider the molecular motion but we will introduce the idea of temperature in the following way:

• take two objects, a "cold" one and a "hot" one and put them in contact with each other. As experience tells us, after a while the cold one will warm up and the hot one cool down
• this process occurs since there is a transfer of (thermal) energy from the hot one to the cold one
• after a while, the objects are in thermal equilibrium, i.e no net energy transfer occurs any more
• when two objects are in thermal equilibrium , we say that they are at the same temperature. Conversely, we could say that temperature is the property that determines whether two objects are in thermal equilibrium.

Note that, in order for the whole definition and relative comparison of temperatures to make sense, we need to assume the validity of the zeroth law of Thermodynamics

if both objects A and B are in thermal equilibrium with a third object C, then A is in thermal equilibrium with B

From a practical point of view, we can assign a (relative) numerical value to temperatures by exploiting one of many temperature-dependent properties of matter, viz.

1.

the length of a thin rod (or any dimension of a solid)

2.

the volume of a liquid

3.

the volume of a gas (at constant pressure)

4.

the pressure of a gas (at constant volume)

5.

the electrical resistance of a conductor

6.

the colour of an object (more precisely, the spectrum of electro-magnetic radiation emitted by an object)

We can exploit property 4 to realize that temperature values are not completely relative, i.e. depending on our arbitrary choice of temperature scales, but are absolute, at least in the sense that there is a lowest possible temperature, the absolute zero. Experimentally, we could measure the pressure of a gas kept at constant volume while varying the temperature, and notice that there is a simple proportionality between temperature (measured e.g. in ⁰C with a mercury thermometer) and pressure (measured with any arbitrary barometer or manometer). The (pressure vs. temperature) curve is then a straight line

P = m T_C + q

showing that P=0 for T_C=-q/m . We can already guess that such a temperature is the lowest one could get, since below that value one would have (meaningless) negative pressure. Moreover, we could see that any other gas behaves the same way, i.e. its pressure s. temperature behaviour is given by

P = m' T_C + q'

with q/m = q'/m' = 273.15 ⁰C. This temperature is what is called the absolute zero. As mentioned before, we will eventually relate temperature to molecular motion, therefore it would be appealing to think of the absolute zero as the temperature where all the molecules are at rest. This picture, while meaningful in classical physics, is not valid within the realm of Quantum Mechanics. Still, it is legitimate to sate that absolute zero corresponds to the state where all elementary constituents are at their lowest possible energy level. Abolute temperatures are measured in kelvins, T_K = T_C + 273.15 .

Thermal Expansion Most substances (with some notable exceptions) increase their dimensions at the increasing of temperatures. Quantitatively, the growth $\Delta L$ in a given direction for a temperature change $\Delta T$ can be expressed by

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

where $\alpha$ is the coefficient of linear expansion. Note that, as long as one is dealing with temperature differences, $\Delta T$ has the same value whether measured in kelvins or celsius. A similar expression is valid to represent changes in volume

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

and, typically, $\beta = 3\alpha$ (you can easily prove this by considering the growth in volume in a cube of side L, and neglecting terms containing $\alpha^2$ and $\alpha^3$).

Water has a (very important) anomalous behaviour, since its volume decreases with decrasing temperature down to 4⁰C, at which point it starts increasing again (this is why ice floats and large bodies of water only freeze at the surface).

Ideal Gas Equation of State

We have already mentioned the proportionality between pressure and temperature (at constant volume), as well as of volume and temperature (at constant pressure). These two properties can be summarized as

PV = nRT

wehre T is the absolute temperature, n is the number of moles in the gas sample under exam and R is a universal constant, i.e a constant valid for any ideal gas . In SI units, R=8.315 J/(mol K). What is an ideal gas? We could cheat and say that an ideal gas is a gas for which the PV=nRT expression is valid.... More usefully, a real gas will to a very good extent behave like an ideal gas if it is far enough from its condensation point, and the (infrequent) interactions among molecules can be interpreted purely in terms of billiard ball-like elastic collisions.

Reminder : what is a mole? The definition given in the book is quite convenient : a mole of a substance is an amount containing $6.022\times 10^{23}=N_{A}$ particles (atoms or molecules) of the substance (N_A= Avogadro's number). You might have encountered different definitions, e.g. " a mole is an amount in grams numerically equal to the molar mass of the substance", or " one mole is the number of atoms in a 12 gram sample of Carbon-12". All these definitions are equivalent, due to the fact that one mole of a substance always contains the same number N_A of particles (atoms or molecules). If N is the total number of particles in a sample, we also have

n = N/N_A

Moreover, if M_m is the molar mass (i.e. the mass of one mole) of our substance, and m the mass of one molecule,

M_m = mN_A

Finally, if M is the total mass of the sample

n = M/M_m=M/(mN_A)