Understanding the Combined Gas Law

[uestions]

What do textbooks mean when the gas laws are "combined" to make the ideal gas law?
I think that if the equations were combined, the result would look something like this:

P = k(T)T P = k(V)/V P = k(n)n
P^3 = (k(T)T * k(V)*k(n)n)/V

or

P/T = k(T) PV = k(V) P/n = k(n)
(P^3*V)/nT = k(T)*K(V)*K(n)

(k(n), k(T), k(V) = constants)
I get pressure raised to the third power instead of plain pressure. Any pointers or insights into what I am not understanding? I find it hard to believe the very top three equations can just be mushed together without multiplying their respective pressures to get pressure to the third, as what PV = nRT suggests in my mind.

 

[jfizzix]

We can write the relevant gas laws as:

PV = k_{1}(N,T) (Boyle's law)
\frac{V}{T} = k_{2}(N,P) (Charles's Law)
\frac{P}{T} = k_{3}(N,V) (Gay-Lussac's Law)

If we solve the second and third equations for T, we get

V k_{3}(N,V) = P k_{2}(N,P)

Since this equation must be true for all values of (P,V,N), each side must equal a constant; the same constant.

In order for V k_{3}(N,V) =const, for all values of V, we require that
k_{3}(N,V) =\frac{k_{a}(N)}{V}

The same thing is true for P k_{2}(N,P)=const, so that
k_{2}(N,P) =\frac{k_{b}(N)}{P}

Substituting k_{2}(N,P) into our second equation, or k_{3}(N,V) into our third equation, we find that

\frac{PV}{T} =k_{a}(N) =k_{b}(N)\equiv k(N) (combined law)

This gets us almost all the way. the last law we need is Avogadro's law
\frac{V}{N} = k_{4}(P,T) (Avogadro's law)

Solving both the combined law and Avogadro's law for the volume, we find

\frac{k(N)}{N} = \frac{P}{T}k_{4}(P,T)

Again, each side must be independently constant.

Where
\frac{k(N)}{N} = const
we find that
k(N) = k_{c} N
which gives us (from combined law)
\frac{PV}{NT} =k_{c}

Where
\frac{P}{T}k_{4}(P,T)=const
we find that
k_{4}(P,T)=k_{d}\frac{T}{P}
which gives us (from Avogadro's law)
\frac{PV}{NT} = k_{d}=k_{c}\equiv k

In either case, we arrive at the final result:
PV = k NT
where k is a constant of proportionality independent of P,V,N, or T (i.e. a universal gas constant). More rigorous theoretical investigations show that k is Boltzmann's constant.

Working in units of mole number n = \frac{N}{N_{A}}, we have the more common version of the ideal gas equation PV = n RT, where R=k N_{A}.

 

[Borek]

In the simplest form, combined gas law is

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

It nicely follows as a generalization (although not in a strict way) from partial results

{P_1V_1} = {P_2V_2}

\frac {P_1}{T_1} = \frac {P_2}{T_2}

\frac {V_1}{T_1} = \frac {V_2}{T_2}