= Gas thermometer
{title2=$P\propto T\quad(N,V\text{ fixed})$}
A gas thermometer infers <temperature> from the <equation of state> of a confined <gas>. The constant-<volume (thermodynamics)> version measures <pressure>; the constant-<pressure> version measures <volume (thermodynamics)>. For an <ideal gas>,
$$
PV=Nk_BT=nRT,\qquad
\frac{T}{T_0}=\frac{P}{P_0}\quad(N,V\text{ fixed}),\qquad
\frac{T}{T_0}=\frac{V}{V_0}\quad(N,P\text{ fixed}).
$$
Here $N$ is particle number, $n$ is <amount of substance>, $k_B$ is the <Boltzmann constant>, and $R$ is the <molar gas constant>. The reference and test states must be internally equilibrated, and $T_0$ must be known on the absolute <thermodynamic temperature> scale. A pressure ratio cancels the unknown gas inventory and bulb size in the ideal constant-volume limit. This cancellation is the method's appeal: a mechanical measurement transfers an absolute <temperature> scale between different materials. The <Maxwell speed distribution> and <ideal gas pressure from wall collisions> explain the microscopic origin of the response. <University Physics Volume 2 by Samuel J. Ling and others/2.1 Molecular Model of an Ideal Gas>[OpenStax, University Physics Volume 2, section 2.1: the ideal gas law]{ref}
For an original ideal-model calculation, choose $T_0=300\,\mathrm K$, $P_0=120\,\mathrm{kPa}$ and $P=160\,\mathrm{kPa}$. Then $T=400\,\mathrm K$. These are stipulated values, not observations. Both <pressures> must be absolute, referred to vacuum. If the surrounding <atmospheric pressure> were $100\,\mathrm{kPa}$, the corresponding <gauge pressures> would be $20$ and $60\,\mathrm{kPa}$; their ratio is three, which would incorrectly predict $900\,\mathrm K$. A pressure difference from the atmosphere cannot be substituted into the <ideal gas law>.
\Image[/temperature-thermometry-gas-apparatus.png]
{title=An original schematic of constant-volume <gas thermometer> measurements}
{description=Simplified measurement layout inspired by the NBS apparatus, not a replica or a report of measurements. A controlled bath thermalizes the bulb, separate calibrated <thermometers> check its surroundings, and a diaphragm transmits <pressure> to an absolute-pressure standard without mixing gases. The connecting tube contains gas at other <temperatures>; its inventory belongs in the <gas thermometer dead-space correction>.}
\b[A motivating calibration experiment.] Schooley's account of the NBS/NIST work describes a <helium>-filled metal bulb, a capillary, a <capacitance diaphragm gauge>, and a mercury pressure standard. Circulating bath liquid passed the bulb assembly and platinum resistance <thermometers>. The resistance probes checked the surrounding <temperature>; gas <pressure> supplied the independent thermodynamic comparison. In an early series, workers filled the cleaned bulb with purified <helium> at the highest test <temperature>, checked <pressure> over three hours, then measured at lower <temperatures> and near a reference at $0\,{}^\circ\mathrm C$. Six filling pressures allowed extrapolation toward zero pressure. https://nvlpubs.nist.gov/nistpubs/jres/095/jresv95n3p255_A1b.pdf[J. F. Schooley (1990), NBS/NIST Gas Thermometry From 0 to 660 °C, sections 4–5]{ref}
To turn this principle into a readable measurement protocol, use the following controls. These steps are a proposed teaching interpretation of the cited apparatus, rather than additional reported NBS observations:
* \b[Define the inventory.] Evacuate and clean the bulb and connecting tube, fill with a known pure <gas>, close the filling valve, and keep the same total particle number during each reference/test comparison. Monitor <pressure> at a maintained <temperature> before and after a run. A leak changes $N$; a slow drift can also reflect contamination, sorption, or imperfect equilibration, so pressure stability alone does not identify its cause.
* \b[Measure the reference.] Place the bulb in a controlled environment with independently assigned $T_0$. Record $P_0$ after thermalization, together with the independent <thermometer> readings and their spread. A reference near $0\,{}^\circ\mathrm C$ has an absolute <temperature> near $273.15\,\mathrm K$, rather than zero <temperature>. Repeating reference readings tests whether the inventory or apparatus changed during the comparison.
* \b[Measure the test state.] Change the bath setting, allow the bulb and <gas> to settle, and record $P$ only when the surrounding <thermometers> and <pressure> are sufficiently stable for the intended accuracy. Compare readings at different positions and vary the settling time. Record the connecting tube's <temperature> profile too: uniform bath <temperature> does not make the entire connecting tube isothermal.
* \b[Measure the volume changes.] Determine the bulb's <thermal expansion> and its deformation from differences between inside and outside <pressure>. Measure the connecting tube's dimensions and the effective valve volume. “Constant volume” is the ideal model; a real vessel expands. These measurements supply corrections, even though the absolute bulb volume cancels in the simplest pressure ratio.
* \b[Measure pressure against a reference.] For a liquid-column pressure standard, measure the vertical column height, the liquid's <temperature>-dependent <mass density>, local gravitational acceleration, and the reference-side <pressure>. Include diaphragm offsets and the gas's <hydrostatic pressure> change between the sensor and bulb. The pressure needed in the <equation of state> is at the bulb, not automatically at the remote gauge.
The corresponding NBS apparatus used a protective case with an outer <helium> layer held close to the bulb <pressure>, reducing pressure deformation while aiding thermal contact. <Thermal expansion> was measured on bulb material. Gas in the capillary remained partly outside the hot region, requiring a measured tube <temperature> profile and geometry. Pure working <gas> also mattered: contaminants that sorb on surfaces can change the freely circulating inventory. These are physical effects to measure or bound, rather than scatter that disappears by averaging. https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nbsir85-3133.pdf[J. F. Schooley, Thermometry, NBS IR 85-3133, chapter V, gas thermometry and the NBS experiment]{ref}
For illustration, neglect connecting-tube gas and describe a uniform bulb with a dilute <virial expansion> in particle <number density> $\rho_N=N/V$:
$$
PV=Nk_BT\left[1+B_2(T)\rho_N+O(\rho_N^2)\right],\qquad
\frac{T}{T_0}=\frac{P}{P_0}\frac{V(T)}{V(T_0)}\frac{Z(T_0)}{Z(T)}.
$$
Here $Z=PV/(Nk_BT)$ and $B_2$ is the <second virial coefficient> in the particle-density convention; a molar-density convention rescales the coefficient. The second relation follows exactly from the definition of $Z$ for the same $N$, and shows the signs of the corrections. If the bulb expands by $0.1\%$ and $Z=1$ at both states, the toy pressure ratio above gives $400.4\,\mathrm K$, rather than $400\,\mathrm K$. Improved expansion and <helium> nonideality models really changed the temperatures inferred from the 1989 NIST data: Harvey's 2023 reanalysis found expansion updates of roughly $1$–$3\,\mathrm{mK}$ and nonideality updates of roughly $-0.1$ to $-0.5\,\mathrm{mK}$. Those are reported corrections to historical results, not the much larger illustrative correction just calculated. https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=956523[Allan H. Harvey (2023), Small Corrections to 1989 NIST Constant-Volume Gas Thermometry Data, sections 2–3]{ref}
With equal corrected bulb <volumes>, negligible or corrected connecting-tube gas, fixed reference and test <temperatures>, and a dilute classical single gas phase,
$$
\frac{P}{P_0}=\frac{T}{T_0}\left[1+\bigl(B_2(T)-B_2(T_0)\bigr)\rho_N+O(\rho_N^2)\right],\qquad
\lim_{\rho_N\to0}\frac{P}{P_0}=\frac{T}{T_0}.
$$
Repeat the reference/test comparison for several sealed fills, reducing $N$ between comparisons but keeping it fixed within each pair. The extrapolation varies density at fixed endpoint <temperatures>; it does not cool one fill toward zero <temperature>. Real gases have different finite-density corrections but share the classical <ideal gas> intercept under these assumptions. The limit does not remove leaks, bulb expansion, gauge offsets, or the <gas thermometer dead-space correction>. At very low <pressures>, <thermal transpiration> can make remote sensor and bulb <pressures> differ through a nonisothermal capillary; this must be corrected too. The NBS work measured this pressure correction using a separate comparison of narrow and wider tubes. https://nvlpubs.nist.gov/nistpubs/jres/095/jresv95n3p255_A1b.pdf[Schooley (1990), sections 7.3–7.6: pressure, volume, dead space and gas nonideality]{ref}
\Image[/temperature-thermometry-pressure-extrapolation.png]
{title=Ideal pressure ratios and a dilute-limit <gas thermometer> comparison in two toy gas models}
{description=Left: the stipulated ideal constant-volume calibration $P=0.4\,\mathrm{kPa\,K^{-1}}T$. Right: apparent <temperature> $T_{\rm app}=300\,\mathrm K\,P/P_0$ for invented dimensionless fill $x=\rho_N/\rho_*$, with $P/P_0=(4/3)(1+0.03x)/(1+0.01x)$ for model A and $(4/3)(1-0.02x)/(1+0.01x)$ for model B. Both approach the stipulated $400\,\mathrm K$ intercept; neither curve represents a named gas or measured data.}
At $x=1$, the toy models give $T_{\rm app}=407.92\,\mathrm K$ and $388.12\,\mathrm K$, respectively, although their imposed true test <temperature> is the same. Their zero-density intercept is the reusable quantity. Measuring several low-density points permits a fit and a check for curvature; asserting a straight line from one finite-density point supplies neither. A practical fit must balance nonideality against increasing relative pressure error and the capillary corrections, and keep its extrapolation uncertainty.
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