Help in deducing a equation for Neutral red

[demander]

Hi to all, hope you can help me with a problem that took me almost all the week

Homework Statement

To find The Pka of Neutral Red, i had to use this expression \frac{A-A_{HNR}}{A_{NR}-A}
So Now I Have to show that \frac{[NR]}{[HNR^{+}]}=\frac{A-A_{HNR}}{A_{NR}-A}
i tried backwards and i did it, but starting from the concnetrations is far more dificult

Homework Equations

i Have to prove that equallity using only this equations
C_{HNR^{+}} + C_{NR}=C_{Total}(1)

A_{HNR^{+}}=\epsilon_{HNR^{+}}.b.C_{total}(2)

A_{NR}=\epsilon_{NR}.b.C_{total}(3)

A=\epsilon_{HNR^{+}}.b.C_{HNR^{+}} + \epsilon_{NR}.b.C_{NR} (4)

The Attempt at a Solution

I tried to solve 4 in order to C_{HNR^{+}} first and then in order to C_{NR}.
I arrive to
C_{HNR^{+}}=\frac{A-\epsilon_{NR}.b.C_{NR}}{\epsilon_{HNR^{+}}.b}
C_{NR}=\frac{A-\epsilon_{HNR^{+}}.b.C_{HNR^{+}}}{\epsilon_{NR}.b}

then i did the following, added and subtracted the same value in the fraction numerator, like this:
C_{NR}=\frac{A+\epsilon_{HNR^{+}}.b.C_{NR}-\epsilon_{HNR^{+}}.b.C_{HNR^{+}}-\epsilon_{HNR^{+}}.b.C_{NR}}{\epsilon_{NR}.b}

so i could do:
C_{NR}=\frac{A+\epsilon_{HNR^{+}}.b.C_{NR}-\epsilon_{HNR^{+}}.b.(C_{HNR^{+}}+C_{NR}}{\epsilon_{NR}.b}
and using equation 1 it came
C_{NR}=\frac{A+\epsilon_{HNR^{+}}.b.C_{NR}-\epsilon_{HNR^{+}}.b.(C_{t})}{\epsilon_{NR}.b}
then using equation 2:
C_{NR}=\frac{A+\epsilon_{HNR^{+}}.b.C_{NR}-A_{HNR^{+}}}{\epsilon_{NR}.b}

doing the same thing to HNR it came:
C_{HNR}=\frac{A+\epsilon_{NR}.b.C_{HNR}-A_{NR}}{\epsilon_{HNR^{+}}.b}

So doing the reason
\frac{[NR]}{[HNR^{+}]}=\frac{\frac{A+\epsilon_{HNR^{+}}.b.C_{NR}-A_{HNR^{+}}}{\epsilon_{NR}.b}}{\frac{A+\epsilon_{NR}.b.C_{HNR}-A_{NR}}{\epsilon_{HNR^{+}}.b}}
and it's here where i can't see how can i arrive to the final equation
am i going for the worst way? i thought about this all week and can't find a way to prove what i should


hope someone can help me, it's kind of urgent

 

[demander]

The backwards trying was easy... but if i try to do the reverse... the multiplying term is not to clear how it showed off from the equations i give cause backwards the final term is:
\frac{C_{NR}(\epsilon_{NR}-\epsilon_{HNR})}{C_{HNR^{+}}(\epsilon_{NR}-\epsilon_{HNR})}}
so the thing i don't understand is that term:
(\epsilon_{NR}-\epsilon_{HNR}) where can it come from from the equation i have?

 

[Ygggdrasil]

A_{HNR^{+}}=\epsilon_{HNR^{+}}.b.C_{total}(2)
A_{NR}=\epsilon_{NR}.b.C_{total}(3)

Are you sure these are correct? Shouldn't it be:

A_{HNR^{+}}=\epsilon_{HNR^{+}}.b.C_{HNR^+}
A_{NR}=\epsilon_{NR}.b.C_{NR}

 

[demander]

well i really had a doubt about that... cause i think In the two case, the absorvance is given in function of total conectration... these equations are giving in a protocol of the journal of chemistry, but they simnply say... we used this 4 equations and arrive to this :S
I couldn't seem to find that relation even trying very hard this week

 

[Ygggdrasil]

Do you have the reference to the journal article? It's hard to figure out what the equations mean without knowledge of the details of the experiment and what is being measured.

 

[demander]

http://jchemed.chem.wisc.edu/Journal/Issues/2001/Mar/PlusSub/JCESupp/JCE2001p0349W.pdf
that's the link to the article... i put the pages needed in the attchments if you can't acess the journal