Unveiling the Mystery of O in F = F0 + kBT log c + ½ cb1 + O (c2)

[KALEB5]

Please,

This equation is based on the free energy of perfect gaz :

F = F0 + kBT log c + ½ cb1 + O (c2)

½ cb1 is the excluded volume term (c is the concentration) but what is O ?

 

[diazona]

Did you mean this?
$$F = F_0 + k_B T \log c + \frac{1}{2} c b_1 + \mathcal{O}(c^2)$$
If so, the O(c²) at the end refers to "terms of order c² and higher." This formula appears to be a power series in c, which means that if written out in full, it would take the form
$$F = F_0 + k_B T \log c + F_1 c + F_2 c^2 + F_3 c^3 + F_4 c^4 + \cdots + F_n c^n + \cdots$$
But if c is a small number, all the terms with large powers of c will be so much smaller that they can safely be ignored. In that situation, it's common to only write the first few terms and omit all the rest because they are so small. You write O(c²) to indicate that the terms with c² and higher powers of c have been omitted; that way people will know that the formula is only valid for small values of c.