Theory — Constant-Volume/Pressure Gas Thermometer
A gas thermometer uses the near-linear relationship between a gas property (pressure or volume) and temperature.
By extrapolating that linear trend to where the property would vanish, we obtain an experimental estimate of
absolute zero and motivate the Kelvin temperature scale.
1) Ideal-gas basis
- \(T_K\) is temperature in kelvins (absolute temperature).
- \(R\) is the ideal gas constant, \(n\) is amount of gas, and \(P,V\) are pressure and volume.
2) Constant-volume gas thermometer
If \(V\) is held constant (and \(n\) fixed), then:
In practice, we often record \(P\) at several Celsius temperatures \(T\) (°C) and fit a line
\(\,P=aT+b\). If the gas behaves ideally over the measured range, then extending the line to \(P=0\)
gives the Celsius value of absolute zero.
If \(T_0\approx -273.15^\circ\text{C}\), then \(T_K \approx T+273.15\).
3) Constant-pressure gas thermometer
If \(P\) is held constant (and \(n\) fixed), then:
Similarly, fitting \(V=aT+b\) and extrapolating to \(V=0\) gives \(T_0=-b/a\) and again \(T_K=T-T_0\).
4) Why do we use a linear fit?
Over moderate temperature ranges and at low gas densities, many gases behave close to ideally, so \(P\) (or \(V\))
changes nearly linearly with \(T\). A least-squares line fit averages small measurement errors and gives a stable estimate of the slope and intercept.
5) Real gas deviations (advanced note)
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At very low temperatures or higher pressures, gases deviate from the ideal law, and the \(P\)-\(T\) or \(V\)-\(T\) relation may curve.
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A classic approach is to use helium at low density and extrapolate to zero pressure to reduce interaction effects.
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Your calculator reports a best-fit line and \(R^2\); a noticeably lower \(R^2\) suggests nonlinearity or noisy data.
6) Worked example (constant volume)
Suppose \(P(0^\circ\text{C})=1\,\text{atm}\) and \(P(100^\circ\text{C})=1.366\,\text{atm}\).
The line through these two points has slope
