the Equation of State. 199 



through a distance dx the increase of potential energy is 

 — (m — b'p)(</ + *j--)d.v or— (m — b'p)(gdx+cidp). Whence, 



introducing the assumption which relates the temperature 

 the potential energy, and concentration of the molecules, we 

 obtain 



Q + dp ( m-Vp)(gdx+adp-) 



V 



dp _ (m — b'p)(gdx-\-adp) 

 p kt 



Substituting dp for pgdx and solving the equation, 

 we find 



kt , m — J/p a 



If in this equation we put /? = — , v being the volume 

 occupied by a molecule, the equation reduces to 



kt 1 v — V a* 



p=-v lo z— — p • • • • • (2) 



V 



2 



, urn 



where a= -=-. 



If in the above equation (2) P is substituted for » + — , 



v 2 

 the total internal pressure, the relation expressed by the 

 equation 



I?f 

 kt^f- = ?(v-b f ) ...-..- (3) 



e kt -l 



can easily be shown to hold. 



The similarity between the left-hand side of equation (3) 

 and the expression deduced by Planck for the mean energy 

 of a resonator at temperature t will be immediately re- 

 cognized, and the similarity suggests that there may be a 

 close relation between the product VU and the quantum hv. 



* If a and b are assumed to be constant the above equation has the 

 defects of Van der Waals' equation. For example, it leads to the relation 

 li.t c 



^=2-6. 



Ve°c 



