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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?

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2 Answers 2

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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.

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    $\begingroup$ Thank you for the reference! Do you know where this derivation can be found in the book (i.e. which chapter, pages)? $\endgroup$
    – Mailbox
    Feb 13, 2023 at 18:44
  • $\begingroup$ @Mailbox, I intentionally did not mention the page numbers so that you can download and explore the book. Search gas laws inside the book. $\endgroup$
    – AChem
    Feb 13, 2023 at 19:26
  • $\begingroup$ I see, thanks! I just wasn't exactly sure what to search for. Do you mind if I comment the page range for anyone else who might come across this post (and so you can verify that I'm looking at the right information, or suggest a different page range for further reading)? $\endgroup$
    – Mailbox
    Feb 13, 2023 at 21:20
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    $\begingroup$ It is pages 37-39. This is where the author uses ratios and proportions. $\endgroup$
    – AChem
    Feb 13, 2023 at 22:42
  • $\begingroup$ @Mailbox, Did you see those book pages? $\endgroup$
    – AChem
    Feb 13, 2023 at 23:58
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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.
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  • $\begingroup$ But is it possible to derive $pV = nRT$ without having known those three laws beforehand? $\endgroup$
    – Mailbox
    Feb 13, 2023 at 21:45
  • $\begingroup$ @Mailbox, Historically, Boyle's, Charles, laws were known before the pV=nRT law. $\endgroup$
    – AChem
    Feb 13, 2023 at 23:07
  • $\begingroup$ @AChem I thought so, but I didn't want to just discard this answer just because of that. If there is a way to derive $pV = nRT$ without the aforementioned laws, not only would I be interested to see that, but it would solve the potential circular reasoning issue here. $\endgroup$
    – Mailbox
    Feb 13, 2023 at 23:17
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    $\begingroup$ @Mailbox, Yes indeed. There is a rigorous way of deriving the ideal gas law using the kinetic theory of gases. Search this terms and you will find the derivations. $\endgroup$
    – AChem
    Feb 13, 2023 at 23:57
  • $\begingroup$ Like this? $\endgroup$
    – Mailbox
    Feb 14, 2023 at 0:30

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