Possibility of separating Isotopes. 525- 



It is clear that p cannot be identical over a wide range of T, 

 i. e. that two substances must be separable by fractionation 

 unless v m is identical for both as well as A, and C p and 



. All these constants may change with the atomic volume, 



K 



i. e. they may and probably do depend upon the total pres- 

 sure. Therefore if the theorem that the isotopes are not 

 separable is mathematically true, at any rate \ and C v must 

 be identical over a finite range of atomic volumes. It has 

 been shown by Soddy, at any rate in the case of lead, that 

 the distances between the centres of the atoms in the solid 

 state are identical. If this is so, it seems impossible for A, 

 and v m to be identical. 



Consider an atom at the surface of a plane of the isotope 

 defined by ~ = 0. Let the forces acting upon it have a com- 

 ponent in the Z axis (/> (z). The latent heat at the absolute 



zero A is then proportional to I cf>(z)dz. Since r is identical 



for two isotopes the condition that X is identical means that 



j <f>(z).dz is identical. Further, they must be identical 



although r may be varied over a finite range. 



Now imagine another plane of the same material placed 

 in contact with the first. The force on the atom will be 

 <f>(z) — cj)(z), so that the quasi-elastic restoring force is by 

 Taylor's theorem 2Az(fr'(z). The frequency v m is therefore 



1 r/UU) 



2tt V M 



If this is to be identical in two isotopes 4>'{z) must be 

 proportional to the atomic weight M. 



Therefore if two isotopes are to be inseparable by frac- 

 tionation, cp(z) must be a function such that I <f>(z)dz is 



J r 



identical in both cases, whereas <$>'(z) must be proportional 

 to the atomic weight. 



If </>(-) may be represented as a power series, sav 

 a n z n + a n -iZ n ~ x -f .... in the case of one isotope and 

 b n z n + b n _iZ n ~ l -{- .... for the other, one has 



cj> 1 '(z) = na n z»- 1 +{n-l)a n _ 1 z»- 2 + 



M M 



= ^<#» 2 ^)=»M"- 1 +(«-i)i»- 2 -'"- a + ...)g?- 



