= Thermometer
{wiki}
A thermometer measures <temperature> through a physical response whose relation to <temperature> is known. Examples include the length change from <thermal expansion>, a change in <electrical resistance>, and the <pressure> of a <gas thermometer>. If the response <function> $X=f(T)$ is an <injective function> over the working range, its measured value determines $T=f^{-1}(X)$ through the <inverse function>. <Thermometer calibration> establishes that relation for the particular instrument and measurement conditions. Two matching readings are meaningful only after calibration to a common scale and with the probes sampling comparable equilibrium states. https://openstax.org/books/university-physics-volume-2/pages/1-2-thermometers-and-temperature-scales[Ling, Moebs and Sanny, University Physics Volume 2, section 1.2: physical responses used in thermometers]{ref}
For a contact thermometer, the sensing part must reach <thermal equilibrium> with the region being measured. That region may differ from the rest of the object: an immersed probe near a heater can read a local <temperature> while the surrounding <water> remains cooler. A stable reading can also be a <steady state> set by competing <heat transfer> paths to the sample and the room. Good contact, adequate immersion, mixing where appropriate, and checks for spatial differences help establish which <temperature> the reading represents. These are measurement controls inferred from the <thermal equilibrium> requirement, not a guarantee that every flat trace is equilibrium.
A thermometer also changes the sample it touches. Consider an original toy calculation with an initially uniform sample at $T_o=350\,\mathrm K$, a probe at $T_p=300\,\mathrm K$, and constant whole-body <heat capacities> $C_o=100\,\mathrm{J\,K^{-1}}$ and $C_p=1\,\mathrm{J\,K^{-1}}$. Suppose together they form an <isolated system>, with no <work> or <phase transition> and negligible contact <energy>. The <first law of thermodynamics> and <thermal equilibrium> give
$$
T_f=\frac{C_oT_o+C_pT_p}{C_o+C_p}=349.50495\ldots\,\mathrm K,\qquad
T_f-T_o=\frac{C_p}{C_o+C_p}(T_p-T_o).
$$
The final <thermometer> reading underestimates the sample's original <temperature> by about $0.495\,\mathrm K$, although it correctly measures their new common <temperature>. The disturbance becomes small when the participating probe <heat capacity> is small compared with the sample's. A large controlled <heat bath> is a different limit: it supplies or removes <heat> while approximately maintaining its own <temperature>.
Waiting time is a separate issue from disturbance. Under a toy <Newton's law of cooling> model for a uniform probe in a maintained <heat bath>, a constant contact conductance $G$ gives
$$
C_p\frac{dT_p}{dt}=G(T_b-T_p),\qquad
T_p(t)=T_b+[T_p(0)-T_b]e^{-t/\tau},\qquad \tau=\frac{C_p}{G}.
$$
The <time constant> $\tau$ controls this <exponential decay>. For stipulated values $T_b=350\,\mathrm K$, $T_p(0)=300\,\mathrm K$ and $\tau=5\,\mathrm s$, the probe reads about $331.61\,\mathrm K$ after one <time constant> and $347.51\,\mathrm K$ after three. These are calculated values, not an instrument specification or measured data. Real probes may have multiple relaxation times and additional <heat transfer> paths; choose an equilibration criterion from the required accuracy, rather than assuming that a fixed waiting time works universally.
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