I am not slacking! You should see my notebook. And now I have company.
Here's something to be going on with for now, the meaning of the title and I think the central metaphor in the poem for blockage and inefficiency of exchange
en.wikipedia.org
great thanks
This will be made up of blood that bypassed the lungs ( Q s {\displaystyle Q_{s}}
) and that which went through the pulmonary capillaries ( Q c {\displaystyle Q_{c}}
). We can express this as
Q t = Q s + Q c {\displaystyle Q_{t}=Q_{s}+Q_{c}}
.
We can solve for Q c {\displaystyle Q_{c}}
:
Q c = Q t − Q s {\displaystyle Q_{c}=Q_{t}-Q_{s}}
.
If we add the oxygen content of Qs to Qc we get the oxygen content of Qt:
Q t ⋅ C a O 2 = Q s ⋅ C v O 2 + ( Q t − Q s ) ⋅ C c O 2 {\displaystyle Q_{t}\cdot Ca_{O_{2}}=Q_{s}\cdot Cv_{O_{2}}+(Q_{t}-Q_{s})\cdot Cc_{O_{2}}}
Substitute Qc as above, CcO2 is the oxygen content of pulmonary (alveolar) capillary blood.
Q t ⋅ C a O 2 = Q s ⋅ C v O 2 + Q t ⋅ C c O 2 − Q s ⋅ C c O 2 {\displaystyle Q_{t}\cdot Ca_{O_{2}}=Qs\cdot Cv_{O_{2}}+Q_{t}\cdot Cc_{O_{2}}-Qs\cdot Cc_{O_{2}}}
Multiply out the brackets.
Q s ⋅ C c O 2 − Q s ⋅ C v O 2 = Q t ⋅ C c O 2 − Q t ⋅ C a O 2 {\displaystyle Q_{s}\cdot Cc_{O_{2}}-Qs\cdot Cv_{O_{2}}=Q_{t}\cdot Cc_{O_{2}}-Qt\cdot Ca_{O_{2}}}
Get the Qs terms and the Qt terms on the same side.
Q s ⋅ ( C c O 2 − C v O 2 ) = Q t ⋅ ( C c O 2 − C a O 2 ) {\displaystyle Q_{s}\cdot (Cc_{O_{2}}-Cv_{O_{2}})=Q_{t}\cdot (Cc_{O_{2}}-Ca_{O_{2}})}
Factor out the Q terms.
Q s Q t = C c O 2 − C a O 2 C c O 2 − C v O 2 {\displaystyle {\dfrac {Q_{s}}{Q_{t}}}={\dfrac {Cc_{O_{2}}-Ca_{O_{2}}}{Cc_{O_{2}}-Cv_{O_{2}}}}}
Divide by Qt and by (CcO2 - CvO2).