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Zeroth Law of Thermodynamics.
There is a law concerning thermal intensity which, though obvious, needs to be spelled out in a little detail: if two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other (Zeroth Law).

Say that all we can observe and measure of a system concerns its mechanical parameters, such as pressure and volume. If two systems are put into thermal contact, the magnitude of these parameters will adjust in each system. in equilibrium, each system will be in a definite state defined by the magnitude of the parameters. Now, if we have two such systems independently in equilibrium with a third, then when these two are brought into thermal contact with each other, they will remain in the same state.

We say that all systems in thermal equilibrium with each other are at the same temperature. The thermal intensity of a system is non-observable and has to be "measured" indirectly by using the observable parameters, such as pressure and volume. This is where thermometers come in. These instruments have a small thermal capacity, so that putting them into thermal contact with a system disturbs the system only slightly. The thermometric property used, such as the length of a mercury column, is then the indicator of the system temperature.

Empirical temperature.
A thermometric property of a system is simply some observable property that varies with thermal intensity. For example, the pressure of a gas can be used if the volume is kept constant. At the other extreme, there are pigments which change colour at certain critical thermal intensities.

The "metric part" of "thermometric" comes in only when we put numbers to variations in the property that is supposed to indicate different thermal intensities. The following are the basic moves in constructing a centigrade scale of temperature.

Choose two easily reproducible thermal intensities; for example, those of the melting point of pure ice (ice point) and of the boiling point of water (steam point) under normal atmospheric pressure of 76 cm of mercury. We call the thermometric property X. Label the magnitude of X at the ice point "0 degrees centigrade", and label the magnitude of X at the steam point "100 degrees centigrade". The ice point and steam point are examples of what are called "fixed points". Changes in the magnitude of X are registered by some visible movement-e.g. a pointer on a surface. This we can call the "movement operative part of a thermometer".

The intermediary positions between "0 degrees" and "100 degrees" are numbered by dividing the whole distance into 100 equal parts, and calling each part "one degree centigrade". Note the assumption of a linear dependence of X on the system parameter, specifying the thermal intensity.

Temperature is the number given to indicate thermal intensity. We have constructed a centigrade scale for what is called an empirical temperature one based on the particular thermometric property and instrument that is used. Here is the structure behind a thermometer reading:

We can in practice construct thermometers so that once we have calibrated them to indicate 0°C at the ice point and 100°C at the steam point, they will then continue to do the same in the future. In other words, we can construct reliable indicators of thermal intensities. But, if we compare two empirical temperatures, and ', as measured by two different kinds of thermometer for the same thermal intensity, the two values will not in general agree. It would be surprising if they did. A graph of against ' might look like this:

Not only will there be differences between the temperatures due to two clearly distinct thermometric properties, such as the expansion of mercury and the resistance of platinum, but even small changes in the instrument construction will change the readings. Take for illustration the mercury-in-glass thermometer. The movement operative part of that thermometer is the movement of the mercury column in the glass capillary tube. If one thermometer were made from Jena 16" glass and read 50°C, a thermometer made from Jena 59" glass would read 49.92°C under the same thermal conditions.

Kelvin's abstract definition of temperature.
We want a definition of temperature that does not depend on any particular thermometric property. The effects of heat transfer depend on the entropic changes produced, that is, on the disordering involved-which is, in its turn, a function of the internal state of the system. This suggests that we look for a situation in which no entropy changes are involved. The heat engine can approximate to such conditions, since in the ideal case:

d S = 0

That is, the entropy produced during the working cycle in zero. The heat engine is discussed in Study Unit 3. We have a quantity of thermal energy (Q) taken from a source. Part is converted into work (W); the rest is lost to a sink (Q). The ratio W / Q has an upper limit (when the cycle is perfectly reversible - see Study Unit 3) in which the maximum possible work is got out of a given input of thermal energy. This ideal limit depends only on the thermal intensities of the source and sink.

For convenience we take the ratio (Q) / (Q) (which will also be a maximum in the ideal case) as a function of TQ and TQ - which are measures of the thermal intensities of the source and sink respectively. We are free to choose almost any function whatsoever. In fact, Kelvin eventually chose the simplest of all (for reasons we shall see later):

( ( Q1 / Q2 ) = ( T1 / T2 ) ) when d S = 0

This corresponds to:

Q1 = T1 S and Q2 = T2 S

(It is a good practice to invent other functions, such as ( Q / Q) = log ( T / T ), and determine how the resulting scale would look against what we call the "absolute scale", resulting from Kelvin's choice of function.)

Now, how do we turn this function into a scheme of actual numbers? We have in T / T a ratio of absolute thermodynamic temperatures, and there must be one absolute temperature that is assigned a number in order that the rest may be calculated. What is done is to take the triple point of water, where water, ice and water co-exist in equilibrium, and assign to it the temperature 273.l6°K (degrees Kelvin). When that is done, the interval between the ice point (275.15°K) and the steam point (373.15°K) is maintained at almost exactly 100 degrees. (There are other solutions. An important alternative is to adopt the starting point that there are exactly 100 degrees between the ice point and the steam point. Then it becomes possible by measurement to calculate the absolute temperature of the ice point and consequently any other one. However, results obtained in this way by different groups of researches tended to differ by as much as one percent.)

We now have the abstract definition; but how to realize it in practice? Do we attempt to build ideal heat engines and measure quantities of heat?

The "ideal gas" as a bridge between theory and practice.
With regard to this problem, there are two significant features of the thermometric behaviour of gases.

(i) Nearly all gas thermometers, if their pressures are made very small, produce almost exactly the same temperature readings of a given thermal intensity. The pressure of the gas at constant volume is the usual thermometric property.

(ii) The working apparatus of the heat engine can consist of a mass of gas enclosed in a cylinder with a piston. A cycle can be worked out in which the gas expands and contracts, gains thermal energy and loses it, and produces an output of work. Now if we have PV = constant x where is the gas thermometer temperature, we can work out the quantities Q, Q and W for a cycle of the heat engine. It then turns out that

( Q1 / Q2 ) = ( G 1 / G 2 )

so that the gas thermometer temperature is identical with the absolute thermodynamic temperature.

But the gas thermometer temperature is that temperature which is the common limit of the empirical temperatures of a large range of gases. The limit is that in which the pressure tends towards zero. In fact, we have that:

Limit PV = constant x T
P 0

In that limit, we have the existence of the ideal gas. For any actual gas at high pressures, the temperature readings that can be obtained from it will not be in accordance with the absolute thermodynamic or Kelvin scale. Since, however, it is possible in practice to get very near to the ideal of the ideal gas, we have a way of coming very close to Kelvin temperatures.

The actual use of gas thermometers.
Gas thermometers are very cumbersome and tend to be unreliable. Because of this, they are used only to calibrate convenient thermometers. How is this done?

The temperatures of any one temperature scale can be translated into temperatures ' of any other scale by using a polynomial formula:

' = a 0 + a 1 + a 2 2 + a 3 3 + ...

where a0, a1, a2, etc. are constants for any given pair of temperature scales. Now, between O°C and 63O.5°C (the melting point of gold) the standard instrument is the platinum resistance thermometer. If we put the resistance as R and the ideal gas thermometer temperature for that resistance as T, then the following equation is used:

RT = R0 + aT + bT2

R, a and b are constants that are found by experiment, using gas thermometers.

Beyond the range of standard instruments.
However, what are we to do about temperatures outside the working range of gas thermometers? How can we possibly calibrate, with respect to the Kelvin scale, any thermometers which operate below 1°K? An important case is that of the magnetic thermometer, in which magnetic temperatures T* are defined by

Curie's Law: T* = ( constant / ) where is the magnetic susceptibility.

We have to turn to theoretical operations in order to find an equation which can enable us to translate magnetic temperatures into absolute Kelvin temperatures. A fundamental step must be made, and we turn to the fundamental equation of thermodynamics (discussed in Study Unit 3):

TdS = dU + dW,

which gives us when no work is done: T = ( dU / dS ). This can be written as: T = ( dU / dT* ) . ( dT* / dS ), where is some empirical temperature.

In the present case: T = ( dU / dT* ) . ( dT* / dS ).

If we can measure the two differential coefficients, ( dU / dT* ) and ( dT* / dS ), we can find T using magnetic temperature (T*) measurements. ( dU / dT* ) is just an 'empirical' specific heat, corresponding to conditions in which no work is done, and can be measured.

( dS / dT* ) is more complicated to measure. What we have to do is to measure differences in entropy with respect to differences in magnetic temperature. This can be done, but the method is described in many books and the details are not of immediate concern. However, it is interesting to note that, when the magnetic thermometer registers 0.03°, the corresponding Kelvin temperature turns out to be 0.004°K - almost a whole difference in magnitude!

At very high temperatures, yet other techniques have to be used in order to obtain Kelvin readings. These techniques involve the use of radiation, an example of which we shall study in Study Unit 7.

What is "one degree centigrade"?
We can add together the intensities of stores of mechanical energy. A set of three batteries of say 1 volt, 2 volts and 3 volts, can be connected in series to make a battery of 6 volts. A collection of compressed springs can exert a combined force which is the arithmetical sum of the forces of each spring taken separately. A mass raised to a certain height can be raised by another equal height by the same amount of energy, whether it starts from its previously raised position or from the ground; and so on. But there is no such way in which we can add together thermal intensities. If we have two systems, each at a certain thermal intensity, we cannot add the intensity of the one to the other. If they are placed in contact, each will try to assume the same thermal intensity as the other and the outcome will be somewhere between the original two intensities.

You should also notice that, in the definition according to Kelvin, we deal in ratios of absolute temperatures. This means that if we have a heat engine with an interval of X°K between source and sink, the work-output/heat-input ratio will be different depending on the temperature of the source. Will the ratio be greater or less the greater the temperature of the source?