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To calculate the pressure, we divide by the area $A$ , to find that
or, rearranged for comparison to Boyle's Law ,
Since we have assumed that the particles travel with constant speed $v$ , then the right side of this equation is a constant. Therefore the product of pressure times volume, $PV$ , is a constant, in agreement with Boyle's Law . Furthermore, the product $PV$ is proportional to the number of particles, also in agreement with the Law of Combining Volumes . Therefore, the model we have developed to describe an ideal gas is consistent with ourexperimental observations.
We can draw two very important conclusions from this derivation. First, the inverse relationship observedbetween pressure and volume and the independence of this relationship on the type of gas analyzed are both due to the lackof interactions between gas particles. Second, the lack of interactions is in turn due to the great distances between gasparticles, a fact which will be true provided that the density of the gas is low.
The absence of temperature in the above derivation is notable. The other gas properties have all beenincorporated, yet we have derived an equation which omits temperature all together. The problem is that, as we discussed atlength above, the temperature was somewhat arbitrarily defined. In fact, it is not precisely clear what has been measured by thetemperature. We defined the temperature of a gas in terms of thevolume of mercury in a glass tube in contact with the gas. It is perhaps then no wonder that such a quantity does not show up in amechanical derivation of the gas properties.
On the other hand, the temperature does appear prominently in the Ideal Gas Law . Therefore, there must be a greater significance (and less arbitrariness) to the temperaturethan might have been expected. To discern this significance, we rewrite the last equation above in the form:
The last quantity in parenthesis can be recognized as the kinetic energy of an individual gas particle, and $N\frac{1}{2}mv^{2}$ must be the total kinetic energy ( $\mathrm{KE}$ ) of the gas. Therefore
Now we insert the Ideal Gas Law for $PV$ to find that
This is an extremely important conclusion, for it reveals the answer to the question of what property is measuredby the temperature. We see now that the temperature is a measure of the total kinetic energy of the gas. Thus, when we heat a gas,elevating its temperature, we are increasing the average kinetic energy of the gas particles, causing then to move, on average, morerapidly.
We are at last in a position to understand the observations above of deviations from the Ideal Gas Law . The most important assumption of our model of the behavior of an idealgas is that the gas molecules do not interact. This allowed us to calculate the force imparted on the wall of the container due to asingle particle collision without worrying about where the other particles were. In order for a gas to disobey the Ideal Gas Law , the conditions must be such that this assumption isviolated.
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