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XXXIV. Fundamental Properties of the Equation of State. 

 By R. D. Kleeman, D.Sc, B.A.* 



A General Form for the Equation of State. 



THE equation of state of a substance must conform to 

 the laws of thermodynamics. It should, therefore, be 

 possible to deduce a fundamental form for the equation of 

 state which takes these laws into account. Such a form for 

 the equation can be easily obtained. It is shown in treatises 



on thermodynamics that (t^L^I"^) > where dQ denotes 



the amount of heat absorbed when the volume of the sub- 

 stance, whose mass will be taken as one gram, increases by 

 dv at a constant temperature T, and dp denotes the change 

 in pressure when the temperature changes by dT keeping 



the volume constant. Now Q= — (U +u) +p .dv f, where 



ni a 



U is the work done per molecule against the molecular 

 attraction on separating the molecules by an infinite distance 

 from one another, u is the internal energy of a molecule 

 and m a its absolute mass, and p . dv the external work done 

 when the volume of the substance changes by dv. Substi- 

 tuting this expression for Q in the thermodynamical equation 

 and dividing it by T 2 , and integrating we obtain 



_ T C(dJJ\ ST T r/du\ ST T^ 

 P ~ ro« J \ dv ) T T* + m a ) \dvh ^ + ™* 



(1) 



where Z is a function of v only. This is a fundamental form 

 for the equation of state. It contains the three functions 

 U, u, and Z, and its form is therefore completely obtained if 

 the form of each of these functions is known. On considera- 

 tion it will be evident that the state of a quantity of matter 

 must be regulated by five things, viz., (1) the attraction of 

 the molecules upon one another which gives rise to a nega- 

 tive potential energy represented per molecule by U, (2) the 

 internal energy of a molecule represented by u, (3) the 

 kinetic energy of a molecule, (4) the mass of a molecule, 

 (5) the actual volume of a molecule. Since the equation of 

 state must contain a function which represents the effect 

 of the kinetic energy and actual volume of the molecules of 

 a substance on its state, Z cannot be equal to zero. A mole- 

 cule may have no actual volume in the strict sense of the 



* Communicated by tiie Author. 



t Proc. Camb. PJjjl. Soc. vol. xvi. pt. vi. 



2 D 2 



