0.10 The ideal gas law  (Page 6/7)

The ideal gas law

We have been measuring four properties of gases: pressure, volume, temperature, and "amount",which we have assumed above to be the number of particles. The results of three observations relate these four propertiespairwise. Boyle's Law relates the pressure and volume at constant temperature and amount of gas:

Charles' Law relates the volume and temperature at constant pressure and amount of gas:

The Law of Combining Volumes leads to Avogadro's Hypothesis that the volume of a gas isproportional to the number of particles ( $N$ ) provided that the temperature and pressure are held constant. We can expressthis as

We will demonstrate below that these three relationships can be combined into a single equation relating $P$ , $V$ , $T$ , and $N$ . Jumping to the conclusion, however, we can more easily show that these threerelationships can be considered as special cases of the more general equation known as the Ideal Gas Law :

where $R$ is a constant, $n$ is the number of moles of gas, related to the number of particles $N$ by Avogadro's number, $N_A$

In Boyle's Law, we examine the relationship of $P$ and $V$ when $n$ (or $N$ ) and $T$ are fixed. In the Ideal Gas Law, when $n$ and $T$ are constant, $nRT$ is constant, so the product $PV$ is also constant. Therefore, Boyle's Law is a special case of the Ideal Gas Law. If $n$ and $P$ are fixed in the Ideal Gas Law, then $V=\frac{nR}{P}T$ and $\frac{nR}{P}$ is a constant. Therefore, Charles' Law is also a special case of the Ideal Gas Law. Finally, if $P$ and $T$ are constant, then in the Ideal Gas Law, $V=\frac{RT}{P}n$ and the volume is proportional the number of moles or particles. Hence, Avogadro's hypothesis is a special case of the IdealGas Law.

We have now shown that the each of our experimental observations is consistent with the Ideal Gas Law. Wemight ask, though, how did we get the Ideal Gas Law? We would like to derive the Ideal Gas Law from the three experiementalobservations. To do so, we need to learn about the functions $k_1(N, T)$ , $k_2(N, P)$ , $k_3(P, T)$ .

We begin by examining Boyle's Law in more detail: if we hold $N$ and $P$ fixed in Boyle's Law and allow $T$ to vary, the volume must increase with the temperature in agreement withCharles' Law. In other words, with $N$ and $P$ fixed, the volume must be proportional to $T$ . Therefore, $k_1$ in Boyle's Law must be proportional to $T$ :

where $k_4$ is a new function which depends only on $N$ . then becomes

Avogadro's Hypothesis tells us that, at constant pressure and temperature, the volume is proportional tothe number of particles. Therefore $k_4$ must also increase proportionally with the number of particles:

where $k$ is yet another new constant. In this case, however, there are no variablesleft, and $k$ is truly a constant. Combining and gives

This is very close to the Ideal Gas Law, except that we have the number of particles, $N$ , instead of the number of the number of moles, $n$ . We put this result in the more familiar form by expressing the number ofparticles in terms of the number of moles, $n$ , by dividing the number of particles by Avogadro's number, $N_{\mathrm{A}}$ , from . Then, from ,

The two constants, $k$ and $N_{\mathrm{A}}$ , can be combined into a single constant, which is commonly called $R$ , the gas constant. This produces the familiar conclusion of .