[Halogen876]

Hello,

I am having trouble understanding error in transmittance. Here is  problem I am having trouble with:

A certain type of spectrophotometer exhibits one type of error which is constant with transmittance and which amounts to 0.25%T and a second type which is proportional to transmittance (relative error in T is constant) which amounts to 0.20%.

(a) At what value of transmittance will the relative error in concentration be a minimum? (Ans: 0.341)
(b) What is the realtive error in concentration at this point? (Ans: 0.71%)
(c) Between what values of T will the relative error in concentration be equal to or less than 1.0% (Ans. 0.18-0.647)

I know there are equations to use for this but I don't have a reference book that has these equations. I am also not really clear on the difference between the two types of error.  ::)Any clarification would be very much appreciated. Thank you!

[mjc123]

Have you got them the right way round? 0.25%T is not constant with T and 0.2% is not proportional to T.
What transmittance would you expect for a given concentration?
Taking account of the errors, what transmittance would you see?
How do you work out the concentration from the transmittance?
What would be the apparent concentration you would deduce from the observed transmittance?
What is the relative error in concentration?

I don't agree with the answers you give. Have you posted the problem fully and accurately? (Or have I made a mistake?)

[Halogen876]

I don't think the problem is clear either. Yes, unfortunately, I have posted it correctly. If you think there is an error in the problem and/or the answers, I think that is entirely possible. If that is the case, I don't know if there is much more I can do. I am very confused by the proportional/constant wording. It seems to me that they are both proportional the way the question reads. Even if that is the case though, I'm not sure where I would start. Any more insight would be appreciated but if there is an issue with the question and/or answer itself, I am ok with letting it go. Thank you for your time. It is much appreciated.

[mjc123]

Here's how I did it. For a "true" value of transmission T, the observed transmission T' = T(1+y) + x, where x is the constant error and yT is the proportional error.
Transmission and concentration are related by T = 10^-αCL, so C = -1/(αL)*logT
The apparent concentration C' = -1/(αL)*logT' = -1/(αL)*log[T(1+y) + x]
The error in concentration ΔC = C' - C = -1/(αL)*{log[T(1+y) + x] - logT} = -1/(αL)*log[(1+y) + x/T] ≈ -1/(2.303αL)*(y + x/T)
The relative error c = ΔC/C = 0.434(y + x/T)/logT = (y + x/T)/lnT
dc/dT = {-x/T²*lnT -(y+x/T)/T}/(lnT)² = 0  you get (1+lnT)/T = -y/x, which I think you have to solve numerically.

[Halogen876]

Thank you for all those details!

I follow all the math up to a point but I do get lost at a certain point.

I am good with:

ΔC = C' - C = -1/(αL)*{log[T(1+y) + x] - logT} = -1/(αL)*log[(1+y) + x/T]

I am confused with the next step though:

You have: -1/(2.303αL)*(y + x/T)

But I have: -1/(2.303αL)*ln(1+y+x/T)

So basically I have an extra ln and an extra 1. I thought when you converted log using 2.303 you ended up with a ln. I also don't know what happened to the number 1 in your expression. I am sure I am just forgetting some simple log/ln rules that I learned many years ago and have not used in a while but if someone could refresh my memory, I would really appreciate it! Thank you!

[mjc123]

For small x, ln(1+x) ≈ x
Actually the expansion is ln(1+x) = x - x²/2 + x³/3 - x⁴/4 ...

[Halogen876]

Thanks for explaining that ln expression.

At the endpoint, we have (1+lnT)/T = -y/x.

I know what x and y are (0.25 and 0.2) because they are the given errors so the thing to do here would be to solve for T but there is one T in the ln and another one not in the ln and I don't know how to solve for that.