0.14 Ideal gas law  (Page 3/7)

We conclude that the pressure of a gas sample depends on the volume of the gas and the temperature, but not onthe composition of the gas sample. We now add to this result a conclusion from a previous study . Specifically, we recall the Law of Combining Volumes , which states that, when gases combine during a chemical reaction at a fixed pressure and temperature, the ratiosof their volumes are simple whole number ratios. We further recall that this result can be explained in the context of the atomicmolecular theory by hypothesizing that equal volumes of gas contain equal numbers of gas particles, independent of the type of gas, aconclusion we call Avogadro's Hypothesis . Combining this result with Boyle's law reveals that the pressure of a gas depends on the number of gas particles, the volume in which they are contained, and the temperature of the sample. The pressure does not depend on the type of gas particles in the sample or whether they are even allthe same.

We can express this result in terms of Boyle's law by noting that, in the equation $PV=k$ , the "constant" $k$ is actually a function which varies with both number of gas particles in thesample and the temperature of the sample. Thus, we can more accurately write

explicitly showing that the product of pressure and volume depends on $N$ , the number of particles in the gas sample, and $t$ ,the temperature.

It is interesting to note that, in 1738, Bernoulli showed that the inverse relationship between pressure andvolume could be proven by assuming that a gas consists of individual particles colliding with the walls of the container.However, this early evidence for the existence of atoms was ignored for roughly 120 years, and the atomic molecular theory was not tobe developed for another 70 years, based on mass measurements rather than pressure measurements.

Observation 2: volume-temperature measurements on gases

We have already noted the dependence of Boyle's Law on temperature. To observe a constant product ofpressure and volume, the temperature must be held fixed. We next analyze what happens to the gas when the temperature is allowed tovary. An interesting first problem that might not have been expected is the question of how to measure temperature. In fact,for most purposes, we think of temperature only in the rather non-quantitative manner of "how hot or cold" something is, but thenwe measure temperature by examining the length of mercury in a tube, or by the electrical potential across a thermocouple in anelectronic thermometer. We then briefly consider the complicated question of just what we are measuring when we measure thetemperature.

Imagine that you are given a cup of water and asked to describe it as "hot" or "cold." Even without a calibratedthermometer, the experiment is simple: you put your finger in it. Only a qualitative question was asked, so there is no need for aquantitative measurement of "how hot" or "how cold." The experiment is only slightlymore involved if you are given two cups of water and asked which one is hotter or colder. A simple solution is toput one finger in each cup and to directly compare the sensation. You still don't need a calibrated thermometer or even a temperaturescale at all.