The ideal gas law  (Page 4/7)

Finally, imagine that you are given a cup of water each day for a week at the same time and are asked todetermine which day's cup contained the hottest or coldest water. Since you can no longer trust your sensory memory from day to day,you have no choice but to define a temperature scale. To do this, we make a physical measurement on the water by bringing it intocontact with something else whose properties depend on the "hotness" of the water in some unspecified way. (For example, thevolume of mercury in a glass tube expands when placed in hot water; certain strips of metal expand or contract when heated; some liquidcrystals change color when heated; etc. ) We assume that this property will have the same value when it is placed in contact withtwo objects which have the same "hotness" or temperature. Somewhat obliquely, this defines the temperaturemeasurement.

For simplicity, we illustrate with a mercury-filled glass tube thermometer. We observe quite easily that when the tube is inserted in water we consider "hot," the volume ofmercury is larger than when we insert the tube in water that we consider "cold." Therefore, the volume of mercury is a measure ofhow hot something is. Furthermore, we observe that, when two very different objects appear to have the same "hotness," they also givethe same volume of mercury in the glass tube. This allows us to make quantitative comparisons of "hotness" or temperature based onthe volume of mercury in a tube.

All that remains is to make up some numbers that define the scale for the temperature, and we can literally dothis in any way that we please. This arbitrariness is what allows us to have two different, but perfectly acceptable, temperaturescales, such as Fahrenheit and Centigrade. The latter scale simply assigns zero to be the temperature at which water freezes atatmospheric pressure. We then insert our mercury thermometer into freezing water, and mark the level of the mercury as "0". Anotherpoint on our scale assigns 100 to be the boiling point of water at atmospheric pressure. We insert our mercury thermometer intoboiling water and mark the level of mercury as "100." Finally, we just mark off in increments of $\frac{1}{100}$ of the distance between the "0" and the "100" marks, and we have a working thermometer. Given the arbitrariness of this way ofmeasuring temperature, it would be remarkable to find a quantitative relationship between temperature and any otherphysical property.

Yet that is what we now observe. We take the same syringe used in the previous section and trap in it a smallsample of air at room temperature and atmospheric pressure. (From our observations above, it should be clear that the type of gas weuse is irrelevant.) The experiment consists of measuring the volume of the gas sample in the syringe as we vary the temperature of thegas sample. In each measurement, the pressure of the gas is held fixed by allowing the piston in the syringe to move freely againstatmospheric pressure. A sample set of data is shown in and plotted here .