# ICP-OES data treatment: correct for dilution and inweight, the units

Let's say the ICP-OES software gives me a concentration of 186,49 ppm (mg/L) of Mn. I diluted the acid digested sample to 14.2 mL. I weighed in 10.17 mg of the powder I analyzed. I need to calculate the actual concentration.

It is quite logical in my head what I need to do: multiply by diluted volume and divide by inweight. However, I don't understand my units.

When I multiply 186.49 mg/L with 0.0142 L, I get 22,648 mg. When I then divide this by 10.17 mg, I end up with 0.260. But this is unitless. How do I know what unit I have? I thought I converted the w/V concentration to w/w by this procedure, but the units does not follow.

Is it so, that I end up with a ratio (not sure what kind of ratio)? Then I think I understand. 0.26 would then represent 26 %, meaning my sample contains 26 % Mn. But how do I know that this is in w/w, and not w/V as I started with. I just don't get the units to add up.

$$\left(186.49 \frac{\mathrm{mg}}{\mathrm{L}}\right)\left(0.0142\mathrm{\ L}\right) = 2.648 \mathrm{\ mg}$$ gives the mass of the Mn in your sample, so when you divide by the sample mass:
$$\frac{2.648 \mathrm{\ mg}}{10.17\mathrm{\ mg}}=0.260$$ you get the mass fraction of Mn in your sample, and you know it's a mass fraction because you have mass units in both the numerator and the denominator, leaving you dimensionless. You can express it just like that, or you can convert it to 26.0% (and this is a dimensionless ratio, so it actually makes sense to use % here), or else express it as the mass of Mn in an arbitrary sample mass: 26.0 g/100 g.