How exactly does Boyle's law, Charles' law, and Avogadro's law combine to make the Ideal Gas Law?

Question

My General Chemistry 1 lecturer derived the ideal gas law as follows:

We have $V \propto \frac1{P}$ (Boyle's law), $V \propto T$ (Charles' law), and $V \propto n$ (Avogadro's law). Combining these gives $$V \propto \frac{nT}{P} \\ \implies V = \frac{RnT}{P} $$

Where $R$ is a proportionality constant.

But I'm confused as to how these proportionality expressions were combined. As far as I know, $A \propto B$ and $A \propto C$ does not imply $A \propto BC$, so how is this justified here?

Answer 1

This is not how the gas laws are derived originally. However, old physics or chemistry school books taught it this way (somewhere around 1890s or early 1900s). Your teacher over simplified the derivation this way and all the key algebraic steps are missing. Old does not mean that it is false or the facts have changed, it is still very elegant if taught properly.

$A \propto B$ and $A \propto C$ does not imply $A \propto BC$, so how is this justified here?

Yes, you are right that if $A$ is proportinal to $B$ and $A$ is also proportional to $B$, then we cannot write right away that $A \propto BC$. However, ratio and proportion have another property, that if we say that the quantity $A$ varies jointly if it varies as a product of two variables. So you can say that the volume of a gas varies jointly with amount ($n$) and absolute temperature ($T$)., i.e., if we were to plot $V$ on the y-axis and the product nT on the x-axis, we will see a straight line at constant pressure. Note the temperature must be in Kelvins or appropriately shifted.

Also, the volume of the gas is inversely proportional to pressure P i.e., if we keep nT constant, and plot $V$ on the y-axis and P on the x-axis, we get a hyperbola (which proves inverse relationship).

Note that in ratio and proportion convention, the dependent variable is written on the left. In this case, $V$ is the dependent variable.

See the gas laws in Physics for University Students: Heat, electricity, and magnetism By Henry Smith Carhart, 1894 from Google Books (legally free to download) for a more logical derivation of the gas law using ratio and proportion properties.

Answer 2

It is much easier to do the contrary, namely to start from $pV = nRT$, and to derive the other laws. Example :

1. Boyle's law. If a gas sample contains $n$ moles of gas at a constant temperature $T_0$, the pressure $p$ and the volume $V$ can vary, but the product $pV$ remains constant. So going from state $1$ to state $2$, the pressures and volumes are related by $${p_{1}*V_{1} = p_{2}*V_{2} = nRT_0 = \mathrm{constant}}$$
2. Charles' law. If a gas sample is maintained at constant pressure $p_o$, the volume $V$ is proportional to the temperature $T$. Using $pV = nRT$, the following expression can be written : $V/T = nR/p_o$ = constant. So that heating a gas volume $V_1$ at temperature $T_1$ to a higher temperature $T_2$, without changing $n, R$ and $p_o$, the volume becomes $V_2$, but the ratio $V/T$ remains constant and equal to $nR/p_o$ according to :$$\frac{V_{2}}{T_{2}} = \frac{V_{1}}{T_{1}} = \frac{nR}{p_o} = \mathrm{constant}$$
3. Avogadro's law. At fixed values of $p, V$ and $T$, the number of moles is the same for all gases. This is an immediate consequence of the law $pV = nRT$, as this law is valid for all perfect gases, and real gases are nearly perfect gases.