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Understanding the law of Boyle

Here is a discussion on the physics of Boyle's law and its relationsihp to the ideal gas equation.

First, the law of Boyle makes sense only for gases. In order to introduce the law of Boyle, consider a gas storaged at a given pressure inside a container such as in Fig. 01 A below,

Fig.  01 A sketch to show pressure and volume relationship according to Boyle's law

In Fig. 01, A the gas is kept at a given pressure, say $P_1$ as indicated in the manometer, and temperature, say $T_1$ as indicated in the thermometer. The volume of the gas, say $V_1$, is determined from the geometry of the container and let us say that there is known mass represented as the moles $n_1$.

Consider now a mechanism inside the container shown in Fig. 01 A, using a piston, that allows you to compress the gas as presented in Fig. 01 B to $V_2$. Since the gas has been compressed its volume has been reduced as well. The manometer in Fig. 01 B indicates a pressure increase to $P_2$ but the temperature remains the same as in Fig. 01 A. The number of moles is unchanged too.

Let us now take the product of pressure $P_1$ and volume $V_1$ in Fig. 01 A resulting in a given value, say,

$P_1V_1=k_1$        Eq. (01)

and the product of pressure $P_2$ and volume $V_2$ in situation Fig. 01 B which gives $k_2$,

$P_2V_2=k_2$        Eq. (02)

According to Boyle's law, if the temperature $T$ and the number of moles $n$ are the same for both cases, Fig. 01 A and Fig. 01 B, then we should find that $k_1=k_2$.

What are the physical implications of Boyles' law?

In other words, what is it that you need to guarantee for Boyle's law to hold in a given experiment or situation? This question can be addressed as follows,
  • since the number of moles $n$ remains constant, no chemical reaction may take place inside the container. If no reaction takes place, you should guarantee that the container remains completely sealed so that no gas leaves or enters,
  • since te temperature $T$ remains constant, no heat exchange between the gas and the surroundings should be allowed. One way of doing this is by thermally, insulating, the walls of the container so that no heat is exchanged.
You should be aware that keeping/having the above conditions in a system may not always be possible. This, as you may notice, shall introduce some error or fault in your conclusions.

Another way of looking at Boyle's law

From the above presentation of Boyle's findings we may write his law as,

$PV=k$        Eq. (03)

However, there is another popular expression of Boyle's law which connects two situations, as shown in Fig. 01 for example.

If the experiments in Fig. 01 fulfill the Boyle's law conditions then $k_1=k_2$. Also, we may use this fact to state,

$k_1=k_2$

$P_1V_1=P_2V_2$        Eq. (04)

Equation (04) is another form of Boyle's law referring to two states of the same system but a little more tractable. Equation (04) may also be more useful since it relates the two parameters changing in the system.

Boyle's law and the state equations

You may also ask: how do I connect the changes in pressure and volume to the number of moles and temperature?

Well, that is easy. You need to introduce a state equation. You may also further ask: which state equation?

And that question is answered very easily too: anyone you like. Connecting Boyle's law to a state equation is useful to estimate temperature $T$ or moles $n$ at any given state regarding you already know one of these parameters.

You may also use the fact that $T$ and $n$ remain constant to estimate the new values of $P$ or $V$ for another state. In this way, the state equation you may select is irrelevant.

Any question? Write in the comments and I shall try to help.

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