Pressure–temperature relationship at constant volume
A graph of Gay-Lussac’s law (pressure–temperature law) describes a fixed amount of gas in a rigid container, so volume and moles remain constant while temperature changes. Under ideal-gas behavior, pressure increases linearly with absolute temperature.
\[ \frac{P}{T}=\text{constant}\qquad (n,\ V\ \text{constant}) \]
\[ \frac{P_1}{T_1}=\frac{P_2}{T_2} \]
\[ P=\left(\frac{nR}{V}\right)T \]
Shape of the graph in Kelvin
With temperature on the x-axis in Kelvin and pressure on the y-axis, the graph is a straight line with positive slope. The line passes through the origin because \(P=0\) corresponds to \(T=0\ \mathrm{K}\) in the idealized extrapolation.
The slope equals \(\dfrac{nR}{V}\), so larger amounts of gas (larger \(n\)) or smaller container volume (smaller \(V\)) produce a steeper line.
Celsius graph and the absolute-zero intercept
With temperature on the x-axis in degrees Celsius, the graph remains a straight line, but it does not pass through the origin because Celsius is offset from Kelvin: \[ T(\mathrm{K}) = T(^\circ\mathrm{C}) + 273.15 \] The extrapolated line crosses the temperature axis near \(-273.15^\circ\mathrm{C}\), reflecting the absolute-zero reference for the Kelvin scale.
Graph interpretation summary
| Temperature scale on x-axis | Graph shape | Intercept behavior | Meaning of slope |
|---|---|---|---|
| Kelvin (\(\mathrm{K}\)) | Straight line, increasing | Through the origin in ideal extrapolation | \(\dfrac{nR}{V}\) (pressure rise per kelvin) |
| Celsius (\(^\circ\mathrm{C}\)) | Straight line, increasing | Crosses x-axis near \(-273.15^\circ\mathrm{C}\) | Same proportionality, shifted by the temperature offset |
Numerical proportionality example
A rigid container with a fixed gas sample that has \(P_1=1.00\ \mathrm{atm}\) at \(T_1=300\ \mathrm{K}\) has pressure at \(T_2=450\ \mathrm{K}\) given by
\[ P_2 = P_1\frac{T_2}{T_1} = (1.00\ \mathrm{atm})\frac{450}{300} = 1.50\ \mathrm{atm}. \]
The linear increase visible in the graph corresponds exactly to this constant ratio \(P/T\).
Common pitfalls
- Temperature units: Kelvin is required in \(\dfrac{P_1}{T_1}=\dfrac{P_2}{T_2}\) and in \(P=\left(\dfrac{nR}{V}\right)T\).
- Container condition: rigid volume; flexible containers change volume and do not follow a pure pressure–temperature line.
- Non-ideal behavior: high pressures or very low temperatures create curvature relative to the ideal straight line.