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Combination of Gas Laws?

  1. Oct 14, 2013 #1
    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.
     
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  3. Oct 15, 2013 #2

    jfizzix

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    Gold Member

    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].
     
  4. Oct 15, 2013 #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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