Proving the neutral red absorbance ratio equation

[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