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

$${P_1V_1} = {P_2V_2}$$

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

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