the van der Waals equation
/ vahn der VAHLS /
A repaired version of the ideal gas law that admits real molecules take up space and gently attract one another. It bends P V = n R T so that it fits real gases at higher pressure and lower temperature, exactly the regime where the ideal model fails.
Precisely: (P + a n^2 / V^2)(V - n b) = n R T. The term a n^2/V^2 corrects for the attraction between molecules, which slightly lowers the pressure the gas actually exerts, while the b term subtracts the volume the molecules themselves occupy, since they are not truly points. Here a and b are constants fitted to each particular gas.
Why it matters: it was the first equation to capture the condensation of a gas into a liquid and the critical point, an honest step from idealization toward reality. It is still only approximate, and more accurate equations of state exist, but it captures the right physics in a simple form. Van der Waals won the 1910 Nobel Prize for this work.
Near its condensation point carbon dioxide departs noticeably from ideal-gas behaviour, and the van der Waals equation describes it much better than P V = n R T alone.
The gas equation corrected for molecular volume and attraction.
When a approaches 0 and b approaches 0, meaning a dilute gas at high temperature, the van der Waals equation reduces back to the ideal gas law.