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Gas Thermometers and Absolute Temperature

Level 1 - Physics topic page in Temperature and Heat.

Principle

A constant-volume gas thermometer defines an absolute temperature scale by using gas pressure as the measurable property.

For a fixed amount of dilute gas held at fixed volume, pressure is proportional to Kelvin temperature.

Notation

\(p\)
absolute pressure of the gas
\(\mathrm{Pa}\)
\(T\)
absolute temperature
\(\mathrm{K}\)
\(T_C\)
Celsius temperature
\(\mathrm{deg\,C}\)
\(p_1,p_2\)
two gas pressures at fixed volume
\(\mathrm{Pa}\)
\(T_1,T_2\)
two absolute temperatures
\(\mathrm{K}\)

Method

The gas thermometer is useful because a dilute gas at fixed volume gives a nearly linear pressure-temperature relation. Extrapolating that line identifies the zero of the absolute scale.

Fixed gas sample
\[p\propto T\]
The amount of gas and volume are held constant.
Pressure ratio
\[\frac{p_1}{p_2}=\frac{T_1}{T_2}\]
Temperature from reference
\[T=T_{\mathrm{ref}}\frac{p}{p_{\mathrm{ref}}}\]
Celsius offset
\[T=T_C+273.15\]

The Kelvin scale is absolute because \(T=0\\,\\mathrm\{K\}\) corresponds to the extrapolated zero-pressure intercept of this gas-thermometer model. The gas would not remain an ordinary gas all the way to that point; the intercept defines the scale, not a practical gas state.

Rules

These are the compact gas-thermometer and absolute-temperature relations.

Kelvin temperature
\[T=T_C+273.15\]
Celsius temperature
\[T_C=T-273.15\]
Fixed-volume ratio
\[\frac{T_1}{T_2}=\frac{p_1}{p_2}\]
Reference calibration
\[T=T_{\mathrm{ref}}\frac{p}{p_{\mathrm{ref}}}\]

Examples

Question
A constant-volume gas thermometer reads
\[1.20\times10^5\,\mathrm{Pa}\]
at
\[300\,\mathrm{K}\]
What temperature corresponds to pressure
\[1.60\times10^5\,\mathrm{Pa}\]
?
Answer
\[T=T_{\mathrm{ref}}\frac{p}{p_{\mathrm{ref}}}=300\frac{1.60\times10^5}{1.20\times10^5}=400\,\mathrm{K}\]

Checks

  • Gas thermometer pressures must be absolute pressures.
  • Kelvin temperatures are not written with degree symbols.
  • Ratios such as \(T_1/T_2\) require absolute temperature, not Celsius temperature.
  • The zero-pressure intercept defines the scale; it is not a claim that an ordinary gas can be cooled to \(0\\,\\mathrm\{K\}\).

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Temperature Measurement Scales

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Thermal Expansion