0.14 Ideal gas law  (Page 5/7)

Volume vs. temperature of a gas

We find that there is a simple linear (straight line) relationship between the volume of a gas and itstemperature as measured by a mercury thermometer. We can express this in the form of an equation for a line:

where $V$ is the volume and $t$ is the temperature in °C. $\alpha$ and $\beta$ are the slope and intercept of the line, and in thiscase, $\alpha =0.335$ and, $\beta =91.7$ . We can rewrite this equation in a slightly different form:

This is the same equation, except that it reveals that the quantity $\frac{\beta }{\alpha }$ must be a temperature, since we can add it to a temperature. This is aparticularly important quantity: if we were to set the temperature of the gas equal to $-\left(\frac{\beta }{\alpha }\right)=-273\mathrm{°C}$ , we would find that the volume of the gas would be exactly 0! (Thisassumes that this equation can be extrapolated to that temperature. This is quite an optimistic extrapolation, since we haven'tmade any measurements near to $-273\mathrm{°C}$ . In fact, our gas sample would condense to a liquid or solid beforewe ever reached that low temperature.)

Since the volume depends on the pressure and the amount of gas (Boyle's Law), then the values of $\alpha$ and $\beta$ also depend on the pressure and amount of gas and carry no particular significance. However, when we repeat our observationsfor many values of the amount of gas and the fixed pressure, we find that the ratio $-\left(\frac{\beta }{\alpha }\right)=-273\mathrm{°C}$ does not vary from one sample to the next. Although we do not know the physical significance of this temperature at this point, we canassert 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 this temperature.

This provides us an "absolute temperature scale" with a zero which is not arbitrarilydefined. This we define by adding 273 (the value of $\frac{\beta }{\alpha }$ ) to temperatures measured in °C, and we define this scale to be in units of degrees Kelvin (K). Thedata in [link] are now recalibrated to the absolute temperature scale in [link] and plotted here .

Volume vs. absolute temperature of a gas

Note that the volume is proportional to the absolute temperature in degrees Kelvin,

provided that the pressure and amount of gas are held constant. This result is known as Charles' Law , dating to 1787.

As with Boyle's Law, we must now note that the "constant" $k$ is not really constant, since the volume also depends on the pressure andquantity 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 depends only the number of particles ofgas. Therefore, we slightly rewrite Charles' Law to explicit indicate the dependence of k on the pressure and number ofparticles of gas