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The Pressure Law for an Ideal Gas

Macroscopically, at a constant volume, the pressure  
\[p\]
  exerted by a gas is proportional to its temperature  
\[T\]
  in Kelvins. We can write this as  
\[p=kT\]
  where  
\[k\]
  or  
\[\frac{p}{T}=k\]
  where  
\[k\]
  is a constant.
Microscopically, this can be explained in the following way.
  • If the temperature of a gas increases, the molecules of the gas have more kinetic energy (
    \[kinetic \: energy_{average} = \frac{3}{2}kT\]
    ) and move faster (because
    \[kinetic  \: energy=\frac{1}{2}mv^2\]
    ).
  • Faster moving molecules will have a larger change in momentum when they hit the walls of the container, and will hit the walls of the container more often, since they cover the distance between the walls in a shorter time.
  • Force is equal to change in momentum each second, so grater change in momentum means a greater force.
  • Pressure equals force divided by area, so increasing temperature implies increasing change in momentum for gas molecules implies increasing force then increasing area.

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