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


*

*From these how to prove that $pV=NkT$ for some constant $k$?

*How to show that it is the unique solution?

*Do we really also need Avogadro’s law, $V\propto f_1(p,T)\cdot N$?

 A: 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 $R$).

A: 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.
