n = N/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 = (nR/P) T and 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 = (RT/P) n and the volume is proportional the number of moles or particles. Hence, Avogadro’s Law is a special case of the Ideal Gas Law.
We have now shown that the each of our experimental observations is consistent with the Ideal Gas Law. We might ask, though, how did we get the Ideal Gas Law? We would like to derive the Ideal Gas Law from the three experimental observations. To do so, we need to learn about the functions k A (N,T), k B (N,P), and k C (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 with Charles’ Law. In other words, with N and P fixed, the volume must be proportional to T. Therefore, k in Boyle’s Law must be proportional to T:
k B (N,T) = k B2 (N) × T
where k B2 is a new function which depends only on N. The first equation above then becomes
P × V = k B2 (N) × T
Avogadro’s Law tells us that, at constant pressure and temperature, the volume is proportional to the number of particles. Therefore k B2 must also increase proportionally with the number of particles:
k B2 (N) = k × N
where k is yet another new constant. In this case, however, there are no variables left, and k is truly a constant. That is why it has no subscript to distinguish it. Combining these last two equations gives
P × V = k × N × T
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 of particles in terms of the number of moles, n, by dividing the number of particles by Avogadro’s number, N A :
n = N / N A
Then, from above,
P × V = k × N A × n × T
The two constants, k and N A , are usually combined into a single constant, which is commonly called R, the gas constant. This produces the familiar conclusion:
PV = nRT
Observation 3: partial pressures
We briefly referred above to the pressure of mixtures of gases, noting in our measurements leading to Boyle’s Law that the total pressure of the mixture depends only on the number of moles of gas, regardless of the types and amounts of gases in the mixture. The Ideal Gas Law reveals that the pressure exerted by a mole of molecules does not depend on what those molecules are, and our earlier observation about gas mixtures is consistent with that conclusion.
We now examine the actual process of mixing two gases together and measuring the total pressure. Consider a container of fixed volume 25.0 L. We inject into that container 0.78 moles of N 2 gas at 298 K. From the Ideal Gas Law, we can easily calculate the measured pressure of the nitrogen gas to be 0.763 atm. We now take a second identical container of fixed volume 25.0 L, and we inject into that container 0.22 moles of O 2 gas at 298 K. The measured pressure of the oxygen gas is 0.215 atm. As a third measurement, we inject 0.22 moles of O 2 gas at 298 K into the first container which already has 0.78 moles of N 2 . (Note that the mixture of gases we have prepared is very similar to that of air.) The measured pressure in this container is now found to be 0.978 atm.