= Gas thermometer dead-space correction
Gas outside the uniform-temperature bulb of a <gas thermometer> contributes to its conserved particle inventory. An idealized calculation divides the connecting tube into small regions with <volumes> $V_j$ and local <temperatures> $T_j$. Neglect <hydrostatic pressure> offsets and <thermal transpiration>, assume a common <pressure> $P$ and a local <ideal gas law> in each region, and let the bulb have <volume (thermodynamics)> $V_b$ and <temperature> $T_b$. Adding the particle inventories gives
$$
N=\frac{P}{k_B}\left(\frac{V_b}{T_b}+\sum_j\frac{V_j}{T_j}\right).
$$
Thus even an <ideal gas> in a perfectly rigid sealed apparatus need not obey $P\propto T_b$: some of its particles remain in regions with other <temperatures>. For a stipulated bulb of $100\,\mathrm{cm^3}$, a connecting volume of $1\,\mathrm{cm^3}$ maintained at $300\,\mathrm K$, and bulb reference and test <temperatures> of $300$ and $400\,\mathrm K$, conservation of $N$ gives
$$
\frac{P}{P_0}=\frac{100/300+1/300}{100/400+1/300}
=\frac{101}{76}\approx1.328947,\qquad
T_{\rm app}=300\,\mathrm K\frac{P}{P_0}\approx398.684\,\mathrm K.
$$
These are toy dimensions and a calculated bias, not measured data. Reducing $N$ reduces both <pressures> proportionally and leaves this ratio unchanged: extrapolating away gas nonideality does not eliminate the connecting-volume error. Actual apparatus also requires the tube's <thermal expansion>, a measured <temperature> profile, and the pressure corrections omitted in this model.
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