Observation 2: Volume-Temperature Measurements on Gases

We now want to measure what happens to the pressure and volume of a sample of gas when the temperature is allowed to vary. An interesting rst 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. However, we measure temperature in a variety of ways that do not seem to be hot or cold, such as by examining the length of mercury in a tube or by measuring the electrical potential across a thermocouple in an electronic thermometer. It is an important but complicated question of just what we are measuring when we measure the temperature.

Imagine that you are given a cup of water and asked to describe it as "hot" or "cold." Even without a calibrated thermometer, the experiment is simple: you put your nger in it. Only a qualitative question was asked, so there is no need for a quantitative measurement of "how hot" or "how cold" the water is. The experiment is only slightly more involved if you are given two cups of water and asked which one is hotter or colder. All we would have to do is put one nger in each cup and directly compare the dierences in sensation. You still do not need a calibrated thermometer or even a temperature scale at all.

Finally, imagine that you are given a cup of water each day for a week and are asked to determine 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 dene a temperature scale. To do this, we make a physical measurement on the water by bringing it into contact with something else whose physical properties depend on the "hotness" of the water in some unspecied way. For example, the volume of mercury in a glass tube

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144 CHAPTER 14. PHYSICAL PROPERTIES OF GASES expands when placed in hot water, certain strips of metal expand or contract when heated, some liquid crystals change color when heated, and so on. We assume that this physical property will have the same value when it is placed in contact with two objects which have the same hotness, or temperature. This allows us to make comparisons in a quantitative way, which denes the temperature measurement. Keep in mind that, when we use a thermometer to measure how hot or cold and object is, we are actually measuring a physical property which varies with how hot or cold the object is.

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

"hotness" or temperature based on the volume of mercury in a tube.

All that remains is to make up some numbers that dene the scale for the temperature, and we can literally do this in any way that we please. This arbitrariness is what allows us to have two dierent but perfectly acceptable temperature scales, such as Fahrenheit and Centigrade. The latter scale simply assigns zero to be the temperature at which water freezes at atmospheric pressure. We then insert our mercury thermometer into freezing water, and mark the level of the mercury as "0". Another point on the Centigrade scale assigns 100 to be the boiling point of water at atmospheric pressure. We insert our mercury thermometer into boiling water and mark the level of mercury as "100." Finally, we just mark o in increments of 1/100 of the distance between the "0" and the "100" marks, and we have a working thermometer. Given the arbitrariness of this way of measuring temperature, it is not all obvious what physical property we are measuring, and as such it would be remarkable to nd a quantitative relationship between temperature and any other physical property.

Yet that is what we now observe for volumes and pressures of gases. We take the same syringe used in the previous experiment and trap in it a small sample of air at room temperature and atmospheric pressure.

(From our observations above, it should be clear that the type of gas we use is irrelevant since the pressure and volume relationship is the same for all gases.) The experiment consists of measuring the volume of the gas sample in the syringe as we vary the sample's temperature, which is measured by a mercury thermometer.

In each measurement, the pressure of the gas is held xed by allowing the piston in the syringe to move freely against atmospheric pressure. A sample set of data is shown in Table 3 and plotted in Figure 4.

Sample Data from Volume-Temperature Measurement

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Figure 14.5

This is an amazingly simple relationship. The volume of the gas at xed pressure correlates exactly with the volume of the mercury in the glass cylinder, both of which correlate with the temperature. There is a simple linear (straight line) relationship between the volume of a sample of gas and its temperature. We can express this in the form of an equation for a line:

V = α t + β

where V is the volume and t is the temperature in ^◦C. A and B are the slope and y-intercept of the line, respectively. For this sample of gas at this temperature, our measurements give us A = 0.335 mL/^◦C and B= 91.7 mL.

These numbers alone don't mean much to us, because these are the specic numbers for a specic amount of gas at a specic pressure. But we can rewrite this equation in a slightly dierent form:

V = α (t + β/α)

This is the same equation, except that it reveals that the quantity β/α must be a temperature, since we can add it to a temperature and it has units of temperature. If we extrapolated the straight line in Figure 4 all the way to the x-axis, we nd that the x-intercept of the graph, or the quantity -β/α equals -273 ^◦C.

At that temperature, the volume of the gas would be expected to be zero. This assumes that the equation can be extrapolated to that temperature. This is quite an overly optimistic extrapolation, since we haven't made any measurements anywhere near to -273 ^◦C. In fact, our gas sample would condense to a liquid or solid before we ever reached that low of a temperature.

We know that the volume of a gas sample depends on the pressure and the amount of gas in the sample.

This means that the values of α and β also depend on the pressure and amount of gas and carry no particular signicance. However, when we repeat our observations for many values of the amount of gas and many values of the xed pressure, we nd a wonderful result: the ratio -β/α = -273 ^◦C does not vary from one sample to the next. Every set of observations produces a graph like Figure 4 with a straight line that extrapolates to an x-intercept of -273 ^◦C.

Although we do not yet know the physical signicance of this exact temperature, we can assert that it is a true constant, independent of any choice of the conditions of the experiment. We refer to this temperature as

absolute zero, since a temperature below this value would be predicted to produce a negative gas volume.

Evidently, then, we cannot expect to lower the temperature of any gas below -273 ^◦C.

One immediate consequence of this observation is that it provides us an absolute temperature scale

with a zero that is not arbitrarily dened. Since V = α (t + β/α)

then the gas volume is proportional to t + β/α where t is in ^◦C. This means it would be useful to dene a temperature scale T = t + β/α = t + 273. This new scale is called the absolute temperature scale and the units are in Kelvin (K). Note that the size of the unit K is the same as the size of the unit ^◦C; for example,

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146 CHAPTER 14. PHYSICAL PROPERTIES OF GASES the temperature change of 1 K between 373 K and 374 K is the same as the temperature change of 1 ^◦C between 100 ^◦C and 101 ^◦C.

In this new temperature scale, V = αT. This means that the volume is proportional to the absolute temperature in Kelvin, provided that the pressure and amount of gas are held constant. This result is known as Charles' Law, dating to 1787.

We can see this result in the data in Table 4, which are now recalibrated to the absolute temperature scale, and in Figure 5.

Table 4

Analysis of Volume-Temperature Data in Table 3

Temperature (^◦C) Temperature (K) Volume (mL)

11 284 95.3

25 298 100.0

47 320 107.4

73 350 116.1

159 432 145.0

233 506 169.8

258 531 178.1

Table 14.4

Figure 14.6

As with Boyle's Law, we must now note that the constant α is not really constant, since the volume also depends on the pressure and quantity of gas. Also as with Boyle's Law, we note that Charles' Law does not depend on the type of gas on which we make the measurements, but rather it depends only the number of particles of gas. Therefore, we slightly rewrite Charles' Law to explicitly indicate the dependence of α on the pressure and number of particles of gas:

V = α(N,P) T

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