Gas Thermometer Scale Builder

Physics Thermodynamics • Temperature and Zeroth Law

Written by STEM Calculators Team Published January 2, 2026 Updated February 24, 2026

9. Gas Thermometer Scale Builder

Build a temperature scale from gas-thermometer data using a linear model (ideal-gas limit): at constant volume, \(P \propto T\); at constant pressure, \(V \propto T\). Fit a line, calibrate to Celsius/Kelvin, and extrapolate to \(P\to 0\) or \(V\to 0\) to estimate absolute zero.

Inputs support: pi, e, sqrt( ), sin, cos, exp, log. Use *. Temperatures can be entered as °C or K depending on mode.

Thermometer mode: Calibration mode:

Two-point calibration (only if selected)

Reference 1 \(T_1\) (°C): Reference 2 \(T_2\) (°C): Measured \(X_1\) at \(T_1\): Measured \(X_2\) at \(T_2\):

Here \(X\) means \(P\) (const \(V\)) or \(V\) (const \(P\)). If you only have two points, you can still build a scale; if you have more, the tool can do regression on additional rows.

Data points

If “Direct” is selected, each row needs \(T\) (°C or K selectable below) and the measured \(X\). If “Two-point” is selected, rows are optional extra measurements (you may leave table empty).

Units & options

Temperature unit in table: Force line through \(X=0\) at \(T=0\)K: Show regression diagnostics: Real-gas note:

Animation / view

Progress \(\tau\in[0,1]\): Play speed:

The plot reveals the fitted line gradually (from 0% to 100%) using \(\tau\). Drag to pan; wheel to zoom; double-click to reset.

Ready

Steps & Scale

Enter data and click Build scale.

Data plot & fit

\(X\) vs temperature (with linear fit and \(X\to 0\) extrapolation)

Hover to probe \((T,X)\).

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Theory — Gas Thermometer Scale Builder

A gas thermometer is one of the most conceptually direct ways to define temperature: in the ideal-gas limit, the pressure at constant volume (or the volume at constant pressure) is proportional to absolute temperature. Using measured data, we can build a temperature scale and extrapolate to estimate absolute zero.

1) Ideal-gas proportionality

From the ideal gas law \(PV=nRT\):

\[ \text{Constant volume:}\quad P=\frac{nR}{V}\,T \;\Rightarrow\; P\propto T \] \[ \text{Constant pressure:}\quad V=\frac{nR}{P}\,T \;\Rightarrow\; V\propto T \]

In real gases, the best approximation occurs at low pressure (closer to ideal behavior).

2) Linear model and calibration

The calculator fits a line using Kelvin temperature:

\[ X = a\,T_K + b,\quad \text{where }X=P\text{ (const \(V\)) or }X=V\text{ (const \(P\))} \]

Two common calibration approaches:

Two-point calibration (e.g., ice point \(0^\circ\text{C}\) and steam point \(100^\circ\text{C}\))
Direct calibration where each data row includes a measured \(T\) and \(X\)

\[ T_K = T_C + 273.15 \]

3) Extrapolating to absolute zero

If the line is \(X=aT_K+b\), then setting \(X=0\) gives:

\[ 0=aT_0+b \quad\Rightarrow\quad T_0=-\frac{b}{a} \] \[ T_{0,C}=T_0-273.15 \]

If you force the fit through the origin (\(b=0\)), you are imposing the ideal-gas constraint \(X\propto T_K\), so \(X=0\) corresponds exactly to \(T_K=0\) by construction (no estimated intercept).

4) What the plot shows

Data points: measured \\((T_K, X)\\)
Best-fit line (revealed progressively by \\(\tau\\))
Extrapolated intercept at \\(X=0\\) (estimated absolute zero, when applicable)

Extension idea (university): compare multiple low-pressure datasets and show the limit as pressure \(\to 0\), illustrating how real gases approach the ideal-gas temperature scale.

Steps & Scale

Data plot & fit

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