How can we combine all the three gas laws into a single ideal gas equation?

Question

In all texts that I have read, it has been stated that the combined gas law for ideal gases was derived from the individual gas laws proposed by Boyle, Charles and Avogadro.

My confusion is this is that, in each individual law, some variables of the system's state are to be kept constant. However, none of these conditions are considered in the combined gas equation. Also, how is it possible for us to suggest that for two states with variables $P$, $V$ and $T$:

$$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

Answer 1

It is indeed the case that the rearrangement of the different gas laws is not very satisfactory as it employs the different variables without any constraints. This issue has been addressed by S. Levine in J. Chem. Educ., 1985, 62 (5), p 399, but in the case you do not have access to this reference, I'll shortly repeat his arguments here.

As $P$, $V$ and $T$ are thermodynamic sate functions, we may write the total differential of $T$ as

$$\mathrm{d}T=\left (\frac{\partial T}{\partial V} \right )_P \mathrm{d}V + \left (\frac{\partial T}{\partial P} \right )_V\mathrm{d}P \equiv X\mathrm{d}V+ Y\mathrm{d}P$$

since this differential must be exact, the following relation must hold

$$\left (\frac{\partial X}{\partial P} \right )_V=\left (\frac{\partial Y}{\partial V} \right )_P$$

We can find simple functions satisfying this condition $$X=cP\text{ and }Y=cV$$ where $c$ is a constant that we may choose to be $c=1/R$ for later convenience. Substitution of these functions in the total differential gives $$\mathrm{d}T=\frac{1}{R}\left (P\mathrm{d}V + V\mathrm{d}P\right )$$ and integrating on both sides from 0K to T yields $$T-T_0=\frac{1}{R}\left [(PV)_T-(PV)_{T_0} \right ]$$ and since $T_0=0$ and $PV=0$ at $\pu{0K}$ we arrive at the ideal gas law.

Answer 2

An easier way to combine the simple gas laws is the following. We take Boyle's, Charles' and Avogadro's laws: $$PV=k_1(n,T),\quad \frac{V}{T}=k_2(n,P),\quad \frac{V}{n}=k_3(T,P),$$ where $k_1,k_2,k_3$ are constants that depends on the fixed value of $n$, $T$, or $P$.

Now, from Boyle's and Charles's laws we have $$V=\frac{k_1(n,T)}{P}=Tk_2(n,P)\iff \frac{k_1(n,T)}{T}=Pk_2(n,P)$$ In the left hand side of the last equation, all the terms are function of $n$ and $T$ only, and in the right hand side all the terms are function of $n$ are $P$ only. Thus, both terms have to be equal to a constant $C_1(n)$: $$ C_1(n)=\frac{k_1(n,T)}{T}\iff k_1(n,T)=C_1(n)T\implies V=\frac{C_1(n)T}{P}$$ With the same argument, we can combine this last equation for $V$ with Avogadro's law: $$V=\frac{C_1(n)T}{P}=nk_3(T,P)\iff \frac{C_1(n)}{n}=\frac{P}{T}k_3(T,P)$$ and both sides of the equation have to be equal to another constant $C_2$: $$\frac{C_1(n)}{n}=C_2\iff C_1(n)=nC_2\implies V=\frac{nC_2T}{P} $$ Changing the name of $C_2$, we have the magnificent: $$PV=nRT$$