Could someone please check my math/logic? Conc determination

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Hello, could someone check my math/logic here for back-calculations? I am doubting myself and would love it if someone could glance this over and let me know if I am on the right track!

I am re-designing a simple HPLC method for determining concentrations of analyte in unknown samples. Currently, their back-calculations from instrument response only take into account the response factor of a single concentration of standard (100 ppm), assuming linear response through all concentrations which in my mind is not correct for this analyte.

I am using a Beckman System Gold 126 pump with 166 detector. This has been a fun experience for me having come from previously using the Agilent 1260! I am first going to create a calibration curve using concentrations of standard(s) that will correlate with real-life sample concentrations after sample preparation and subsequent dilutions. I have some work to do in this regard such as determining ideal sample mass and appropriate dilutions, but my primary concern at this time is whether or not the math involved in my proposed back-calculations is correct...

The instrument will give us data about an unknown sample which we must use to determine the % analyte in the unknown sample (mg analyte/g sample).

Paramenters/Units:
• Retention Time, rt [min]
• Concentration [ppm = µg/mL]
• Peak Area [mVolt*min]
• Response Factor [(mVolt*min)/(concentration)]
• Dilution Factor - constant

Given the calibration curve for a given analyte in the form y=mx+b, we can use the instrument response (y) for an unknown sample to determine concentration (x) of the supernatant post-dilution. The concentration of said supernatant is used in conjunction with extraction solvent volume, dilution factor, and original sample mass [g] to determine the concentration [%] of analyte in the unknown sample.

Ex) 2.500 g of sample is solvated with 50 mL ACN using gentle heat and stirring to ensure homogeneity. A 500 µL aliquot is removed from the resulting supernatant and diluted to a final volume of 10 mL (20-fold dilution). After analysis via HPLC, the resulting peak area count(s) are used to back-track to the original sample concentration.
• The resulting area count for a given peak of interest is plugged into the calibration curve associated with said analyte to solve for the concentration of supernatant prior to injection [ppm]. The concentration of analyte in said supernatant undergoes unit conversion [mg/mL] and is multiplied by the extraction solvent volume [mL], the dilution factor, and finally divided by the original sample mass used [g] to obtain a final concentration of analyte in sample [mg/g = %].

Long story short, would this be the proper way to back-calculate to the % analyte in the original sample?

1. First determine concentration of supernatant post-dilution by plugging response (y) of unknown into the appropriate calibration curve and solving for concentration (x); lets call the resulting concentration value "Q". Take the resulting value for Q and convert the units such that it is [mg/mL].

2. % Analyte in Sample = [(Q)(50mL)(20)]/2.5 g = [mg/g]

... where 50 mL comes from the original extraction solvent volume, and 20 is the dilution factor of the resulting supernatant prior to injection.

Am I forgetting something? I am doubting whether or not it is proper to first use extraction solvent volume, then the subsequent dilution factor OR if I should be taking into account the ENTIRE dilution which I believe would be (in this proposed scenario) 400-fold?

I really appreciate anyone who took the time to read through this long post! I hope it wasn't too troublesome to read... Thank you so much for your time and have a nice day!
Dear jhal

I think that your calculations are OK! The only doubt is: your calibration curve plot is response versus mg/ml?

Best of luck

Fernando
Fernando wrote:
Dear jhal

I think that your calculations are OK! The only doubt is: your calibration curve plot is response versus mg/ml?

Best of luck

Fernando


Thanks so much for the reply!

Yes sir, I tend to think of my concentrations in ppm but for the purpose of my back-calulations I just make the quick conversion from ug/mL to mg/mL as I need my answer in terms of mg/g (%). Long story short yes the calibration curve will indeed be response [mVolt*min] versus concentration! I am sorry if I am not quite understanding your concern/question but I hope this clears it up! Please let me know if I didn't answer you properly.

Have a good rest of your day!
Dear jhal

Yes you answer is quite right, I question you because I have a doubt, because I always plot µg/ml and I was not sure about it.

Best regards

Fernando
Thank you so much for the assistance Fernando! I really hope we are correct ^^
You are correct. All of this assumes that 100% of your analyte moves from the sample (I presume it's a solid because you talk about a supernatant solution where the 0.5 mL aliquot is taken) to the ACN in the first step. In this case, the dilution factor is 400.

This is the dimensional analysis with units:

Q µg Analyte/mL S1 x 10 mL S1/0.5 mL ACN x 50 mL ACN/2.5 g Sample = µg Analyte/g Sample

S1 = the solution that is injected into the HPLC

Multiply what the calibration model gives you by 400 and you have the concentration in the sample.
Thank you very much rb6banjo!
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