Concept 01 / 07States of Matter
Gas Laws: Boyle, Charles, Gay-Lussac and Avogadro
The empirical gas laws relating pressure, volume, temperature and amount, the absolute temperature scale from Charles' law, graphical forms (isotherms, isobars, isochores), the combined gas law and typical applications such as bubbles rising in a lake.
The gaseous state is the simplest state of matter: molecules are far apart, intermolecular forces are weak, and gases fill any container. Four measurable quantities describe a gas: pressure , volume , temperature and amount . The gas laws, found experimentally, relate them two at a time.
Boyle's law (1662). At constant and , , that is constant and . A plot of against is a rectangular hyperbola (an isotherm); higher temperatures give isotherms farther from the axes. Plots of against are straight lines through the origin, and against is a horizontal line. Since density , at constant density is proportional to pressure: air is denser at sea level than on a mountain.
Charles' law (1787). At constant and , where is absolute temperature: . In Celsius, : volume changes by of its value per degree. Every isobar, extrapolated, reaches zero volume at . This defines absolute zero and the Kelvin scale . (Real gases liquefy long before this.)
Gay-Lussac's law. At constant and , : . A pressure cooker or a tyre heated in summer shows it. The graphs of against are isochores.
Avogadro's law (1811). At constant and , : equal volumes of all gases contain equal numbers of molecules. The molar volume at STP (, ) is ; at , it is .
Combined gas law. For a fixed amount, . Always use kelvin; pressure and volume units only need to match on both sides.
Units. ; .
Key Takeaways:
- Boyle const (, fixed); Charles const (, fixed); Gay-Lussac const (, fixed); Avogadro const (, fixed).
- Temperatures in gas laws must be in kelvin.
- Isotherm: hyperbola in -; isobar: line through in -.
- Molar volume: at , ; at , .
- Pitfall: doubling Celsius temperature does not double volume; convert to kelvin first.
Brain-Teaser 1: An air bubble at the bottom of a lake, where the pressure is and the temperature , rises to the surface, where the pressure is and the temperature . By what factor does its volume change?
Solution: . The volume grows about times: mostly from the drop in pressure, a little from the warming.
Brain-Teaser 2: An open flask contains air at . To what temperature must it be heated so that one-third of the air escapes?
Solution: Pressure and volume are fixed (open flask), so constant. If one-third escapes, and .
Brain-Teaser 3: A gas occupies at . To what pressure must it be compressed at constant temperature to occupy , and what happens to its density?
Solution: . Mass is unchanged and volume falls by the factor , so density rises times, in proportion to pressure.
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