Combination of Gas Laws?

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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
jfizzix
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We can write the relevant gas laws as:

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

If we solve the second and third equations for [itex]T[/itex], we get

[itex]V k_{3}(N,V) = P k_{2}(N,P)[/itex]

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

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

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

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

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

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

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

[itex]\frac{k(N)}{N} = \frac{P}{T}k_{4}(P,T)[/itex]

Again, each side must be independently constant.

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

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

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

Working in units of mole number [itex]n = \frac{N}{N_{A}}[/itex], we have the more common version of the ideal gas equation [itex]PV = n RT[/itex], where [itex]R=k N_{A}[/itex].
 
  • #3
Borek
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In the simplest form, combined gas law is

[tex]\frac {P_1V_1}{T_1} = \frac {P_2V_2}{T_2}[/tex]

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

[tex]{P_1V_1} = {P_2V_2}[/tex]

[tex]\frac {P_1}{T_1} = \frac {P_2}{T_2}[/tex]

[tex]\frac {V_1}{T_1} = \frac {V_2}{T_2}[/tex]
 

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