= Two-point thermometer calibration example
Consider an original worked <thermometer calibration>, using the height $X$ of a <liquid> column responding through <thermal expansion>. Stipulate two equilibrated reference baths whose corrected <thermometer> readings are $0\,{}^\circ\mathrm C$ and $40\,{}^\circ\mathrm C$, in <degree Celsius> units. Suppose the corresponding column heights are $12\,\mathrm{mm}$ and $52\,\mathrm{mm}$. These are constructed inputs, not observed measurements. Assume the response is linear over this interval, with the same immersion and reading procedure throughout. Applying <thermometer calibration> gives
$$
\frac{t_C(X)}{{}^\circ\mathrm C}
=40\frac{X-12\,\mathrm{mm}}{52\,\mathrm{mm}-12\,\mathrm{mm}}.
$$
For $X=37\,\mathrm{mm}$, first subtract the zero-reference height, then divide by the full reference span: $(37-12)/(52-12)=25/40$. Multiplying by the $40\,{}^\circ\mathrm C$ span gives $t_C=25\,{}^\circ\mathrm C$, or $T=298.15\,\mathrm K$ in <kelvin>. Thus equal $37\,\mathrm{mm}$ readings in separately equilibrated samples test equal <temperatures>, regardless of how much <internal energy> those samples contain. The Celsius offset and modern absolute-scale definition are described in <degree Celsius> and <kelvin>.
\Image[/zeroth-law-thermometer-bath-calibration.png]
{title=A reference-bath <thermometer> comparison and a worked two-point <thermometer calibration>}
{id=zeroth-law-thermometer-bath-calibration}
{description=Left: two probes sample gently stirred <water>, with sensing tips immersed away from the walls and bottom; the heater is off for the ambient comparison. Right: an assumed linear column response maps $12\,\mathrm{mm}$ to $0\,{}^\circ\mathrm C$, $52\,\mathrm{mm}$ to $40\,{}^\circ\mathrm C$, and $37\,\mathrm{mm}$ to $25\,{}^\circ\mathrm C$. The schematic and all plotted values are illustrative, not experimental data.}
The two endpoints determine the assumed line, but do not establish linearity. An independent middle reference and a return to the initial bath can test curvature and drift. If the height has an error of $0.2\,\mathrm{mm}$ while reference values and endpoints are held exact in this calculation, the inferred <temperature> error is $0.2\,{}^\circ\mathrm C$; that is only the height contribution, not a full measurement error budget. A boiling-water reference needs a measured <atmospheric pressure> correction, rather than an automatic $100\,{}^\circ\mathrm C$ assignment. The modern <kelvin> is not defined by either bath.
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