 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 $V_b$ and temperature $T_b$. Adding the particle inventories givesThus 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 $100\mathrm{c}{\mathrm{m}}^3$, a connecting volume of $1\mathrm{c}{\mathrm{m}}^3$ maintained at $300\mathrm{K}$, and bulb reference and test temperatures of $300$ and $400\mathrm{K}$, conservation of $N$ givesThese 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.