Proving ideal gas equation from Boyle’s, Charles’ and Gay-Lussac’s laws

Question

Assuming the empirical laws by Boyle, Charles and Gay-Lussac, which respectively say that \begin{align} p&\propto f(T,N)\cdot {1\over V}\\ V&\propto g(p,N)\cdot T\\ p&\propto h(V,N)\cdot T\\ \end{align}

Questions:

1. From these how to prove that $pV=NkT$ for some constant $k$?
2. How to show that it is the unique solution?
3. Do we really also need Avogadro’s law, $V\propto f_1(p,T)\cdot N$?

Answer 1

Too long for a comment.

Being myself a thermodynamicist who enjoys the history of science, what did happen is

• In $1662$, Boyle showed experimentally that, at constant $T$, the product $P\,V$ is almost a constant
• In $1787$, Charles showed experimentally that, at constant $P$, the change of volume $\Delta V$ is proportional to the change of temperature $\Delta T$
• In $1802$, Gay-Lussac verified Charles's law, quantified the effect of temperature and proposed $V=V_0(1+k T)$
• In $1834$, Clapeyron combined these results into the first statement of the so-called ideal gas law as $P\,V=k (T+267)$
• Later work showed that the number should be $273.2$ when temperature is in Celsius and that $k/n$ ($n$, number of moles) is substance independent (it became $R$).

Answer 2

First and third equation say that $$T\cdot h(V, N) \propto V^{-1} \cdot f(T, N)$$ i.e. $$V\cdot h(V, N)\propto T^{-1}\cdot f(T, N)$$ Both sides can depend only on $N$, i. e. $f(T, N)=T\cdot F(N)$. From this we recover: $$pV\propto F(N)\cdot T$$ Now we use the Avodagro's law to get $F(N)\propto N$.

Notice that without it, every equation $pV=F(N)T$ is a solution to first three equations.