You are right:
Vial 1 : CaO2= HB(15)X1.34XSaO2+0.003XPao2(2)=almost 0 because of the Pa02=2 (see HgB dissociation curve)
Vial 2 CaO2=HB (15)X1.34XSa02+0.003xPaO2(98) = 100.2
Mixing them in anaerobic conditions PO2=0
CaO2=HgB(15)X1.34XSao2 +0.003xPa02(0)=20.1XSa02
It was mentioned that the blood is "normal" - HgB=15
Vial1Ca02+Vial2Ca02=0+100.2=100.2 (total content of O2)
100.2=20.1xSa02
Sa02=50% Therefore from the HgB dissociation curve PaO2=27mmHG
100.2 / 20.1 = 5 not 50, and in these equations SpO2 of 50% gets entered as 0.5 not 50 anyway.
But this is a red herring; even if perfect, the math is unnecessary and just makes the problem harder (if not impossible) to solve since ultimately you have to eyeball the end point somewhere in the steep part of the dissociation curve. What I mean is that a general solution for varying Hb would be
CaO2 vial 1 = 1.34 * 1 * Hb + .003 * 98 (in 1 mL)
CaO2 vial 2 = 1.34 * 0 * Hb + .003 * 2 (in 1 mL)
CaO2 (vial 1+2) = 1.34 * Hb + .3 (in 2 mL)
CaO2 (1 mL of the 1+2 mix) = .67 * Hb + .15
1.34 * SpO2 * Hb + .003 * PO2 = .67 * Hb + .15
PO2 = (.67 * Hb + .15 - 1.34 * SpO2 * Hb) / .003
Assuming Hb is 15 (normal blood)
PO2 = 3400 - 6700 * SpO2
And from here you need to look at the dissocation curve to find a PO2 and SpO2 that lie on it and satisfy this equation. (We can't exactly solve it, because there's no formula for the dissociation curve ... so we're stuck eyeballing it.) As it turns out this equation produces reasonable solutions with an SpO2 around 50% ... but very small changes yield very different PO2s (eg, valid pairs are SpO2 .5 & PaO2 of 50, and SpO2 of .5034 & PaO2 of 27) ... but no human can look at a graph and give that precise an answer.
But again this math is unnecessary; if you mix equal volumes of 100% saturated and 0% saturated blood, regardless of Hb and regardless of the shape/position of the dissociation curve, you're going to get a 50% saturated mix. From there it's easy to take the appropriate curve and plot a rough PO2.
If we were starting with vials that had unequal Hb concentrations, different volumes, or PO2 values NOT at the edges of the dissociation curve, you're stuck doing the math ... but you likely won't end up with as precise an answer as 27 (unless the endpoint is itself at the edge of the curve).
There, I think I've pedanted that to death.
🙂
