A Understanding an Approximation in Statistical Physics

[Arman777]

In a book that I am reading it says

$$(V - aw)(V - (N-a)w) \approx (V - Nw/2)^2$$

Where $V$ is the volume of the box, $N$ is the number of the particles and $w$ is the radius of the particle, where each particle is thought as hard spheres.

for $a = [1, N-1]$
But I don't understand how this can be possible ? Any ideas

 

[etotheipi]

\begin{align*}
(V - w)(V - (N-1)w) &= V^2 - wV - (N-1)wV + (N-1)w^2 \\

&= V^2 - NwV + O(w^2)
\end{align*}it is the same to first order in $w$

 

[Arman777]

\begin{align*}
(V - w)(V - (N-1)w) &= V^2 - wV - (N-1)wV + (N-1)w^2 \\

&= V^2 - NwV + O(w^2)
\end{align*}it is the same to first order in $w$

Thanks for your answer, however it seems that I made a mistake while typing. Can you also look my answer and maybe change yours as well ?

 

[etotheipi]

the factor $a$ you inserted in the edit doesn't change anything, they are still the same to order $w$

 

[Arman777]

the factor $a$ you inserted in the edit doesn't change anything, they are still the same to order $w$

I see but I still cannot understand where that $1/2$ comes I mean its strange. It seems like its just invented from 'nothing'

 

[etotheipi]

I see but I still cannot understand where that $1/2$ comes I mean its strange

$(V - \frac{Nw}{2})^2 = V^2 - NwV + \frac{N^2 w^2}{4}$
multiplying out bracket
and $\frac{N^2 w^2}{4} = O(w^2)$

 

[Arman777]

I guess I kind of understand..