A variable RN should probably be R*N. It seems that your system has no solution in general:

eqs = {N^2  pia1  R == pib1 /2   ,
   1/48  N - 3  R  N^4  pia1 + R  N^4  pia3 == -3  pib1 + pib3 ,  
   (R N^6 pia1)/384 - (R N^6 pia3)/768  == pib1/384 - pib3/768, 
    (R N^10 pia1)/1474560 - (R N^10 pia3)/2211840   == 
    pib1/1474560 - pib3/2211840 ,
   0 == newalb10  (pib1 + omb1  betr) + 
     2  epsb  (pib3  ch3r/chr + omb3 sh3r/chr)  , 
   0 == newala10  (pia1 - oma1  bett) + 
     2  epsa  (pia3  ch3t/cht + omb3 sh3t/cht) ,  
   (R N^10 pia1)/1474560 - (R N^10 pia3)/2211840 == 
    pib1/1474560 - pib3/2211840 , -((R  N^12  pia1)/176947200) + (
     R N^12 pia3)/247726080  == -(pib1/176947200) + pib3/247726080};
symbs = Cases[eqs, _Symbol, All] // Union
vars = {pia1, pib1, oma1, omb1, pia3, oma3, pib3, oma3}
params = Complement[symbs, vars]
eqs /. Thread[params -> 1]
Reduce[eqs /. Thread[params -> 1], vars]

Fixing the RN error, and replacing N with n because N[] is a built-in function, Solve alone still produces no general solution, but adding MaxExtraConditions->All, after a pretty long time (minutes), produces a whole list. Which suggests you may get your answer if you can add some assumptions to your input.

Solve[{
  n^2 pia1 R == pib1/2,
  1/48 n - 3 R n^4 pia1 + R n^4 pia3 == -3 pib1 + pib3,
  (R n^6 pia1)/384 - (R n^6 pia3)/768 == pib1/384 - pib3/768,
  (R n^10 pia1)/1474560 - (R n^10 pia3)/2211840 == 
   pib1/1474560 - pib3/2211840,
  0 == newalb10 (pib1 + omb1 betr) + 
    2 epsb (pib3 ch3r/chr + omb3 sh3r/chr),
  0 == newala10 (pia1 - oma1 bett) + 
    2 epsa (pia3 ch3t/cht + omb3 sh3t/cht),
  (R n^10 pia1)/1474560 - (R n^10 pia3)/2211840 == 
   pib1/1474560 - pib3/2211840,
  -((R n^12 pia1)/176947200) + (R n^12 pia3)/
    247726080 == -(pib1/176947200) + pib3/247726080},
 {pia1, pib1, oma1, omb1, pia3, oma3, pib3}, 
 MaxExtraConditions -> All]

Maybe that is the error. Buona fortuna!

Please keep the conversation in English. This may be helpful for future visitors.

Thank you very much GianLuca. Yes, I will check that. On a first glimpse, it seems that omb3 is lacking. Hope this is all the problem

It is a cumbersome system of equations and it is very easy not to count the number of the equations correctly.

Take care, I am very grateful,

Gustavo