= Respirometer gas balance
For an original idealized <mathematical model>, let $n_{\mathrm O_2}(t)$ be the <amount of substance> of <molecular oxygen> consumed since the initial reading and $n_{\mathrm{CO}_2}(t)$ the <carbon dioxide> produced. Suppose a fraction $f$ of that <carbon dioxide> is removed from the gas, all other gas inventories remain fixed, and <temperature> $T$ and gas <pressure> $P$ stay constant. The <ideal gas law> gives the positive gas contraction
$$
D(t)=V(0)-V(t)=\frac{RT}{P}\left[n_{\mathrm O_2}(t)-(1-f)n_{\mathrm{CO}_2}(t)\right],
$$
where $R$ is the <molar gas constant> and $V$ the gas <volume>. With complete absorption, $f=1$, the contraction reflects <molecular oxygen> uptake regardless of the amount of <carbon dioxide> made. For a uniform pipette bore of <area> $A$, an inward boundary displacement $x$ gives $D=Ax$. A graduated pipette can instead supply the displaced <volume> directly.
Let $D_s$ denote sample contraction and $D_b$ the matched blank contraction. Define $D_{\mathrm{corr}}=D_s-D_b$ with both taken relative to their own initial readings. Under the preceding assumptions,
$$
n_{\mathrm O_2}(t)\simeq\frac{P D_{\mathrm{corr}}(t)}{RT},\qquad
\overline{r}_{\mathrm O_2}=\frac{P}{RT}\frac{D_{\mathrm{corr}}(t_2)-D_{\mathrm{corr}}(t_1)}{t_2-t_1}.
$$
The <slope> of corrected <volume> against <time> estimates uptake over the chosen interval. The approximate sign acknowledges that a liquid boundary also responds to hydrostatic and capillary <pressure>, while the gas can exchange <water> vapour. A fixed-<volume> apparatus would instead show a <pressure> decrease; its ideal conversion is $n_{\mathrm O_2}\simeq V\bigl(P(0)-P(t)\bigr)/(RT)$ after complete absorption and appropriate blank correction.
The <ideal gas law> also exposes drift without any biology: for fixed gas inventory, $V\propto T/P$. A one-percent rise in absolute <temperature> at fixed <pressure> produces a one-percent expansion. Matching the initial gas spaces makes such common changes easier to subtract. Subtracting unmatched blanks can leave a physical signal that looks biological.
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