text: The Joule–Thomson coefficient, , is of practical importance because the two end states of a throttling process () lie on a constant enthalpy curve. Although ideal gases, for which , do not change temperature in such a process, real gases do, and it is important in applications to know whether they heat up or cool down.
  This coefficient can be found in terms of the previously described derivatives as,
  so when  is positive the gas temperature decreases when it passes through a throttle, and if it is negative the temperature increases. Therefore the condition  defines a curve that separates the region of the  plane where  from the region where it is less than zero. This curve is called the inversion curve, and its equation is . Using the expression for  derived previously for the van der Waals equation this is
  Note that for  there will be cooling for  or in terms of the critical temperature . As Sommerfeld noted, "This is the case with air and with most other gases. Air can be cooled at will by repeated expansion and can finally be liquified."
  The inversion curve can be found by solving its equation for , and substituting into the vdW  equation. This produces , where, for simplicity,  have been replaced by .
  The maximum of this, quadratic, curve occurs, with , for
  which gives , or , and the corresponding . The zeros of the inversion curve , are, making use of the quadratic formula, , or  and  ( and ). In terms of the dimensionless variables,  the zeros are at  and , while the maximum is , and occurs at . Note from Fig. 5 that there is an overlap between the saturation curve and the inversion curve plotted there. 
  This region is shown enlarged in the right hand graph of the figure. Therefore a van der Waals gas can be liquified by passing it through a throttle under the proper conditions; real gases are liquified in this way.