Why is \frac{dN_i}{N}\neq dX_i when using the mole fraction concept?

[roldy]

It looks like my first post on this did not make it on this forum some how.

I came across this statement.
Even though $$\frac{N_i}{N}=X_i$$, $$\frac{dN_i}{N}\neq dX_i$$

How does this work? The book offered no help as well as searches on the internet.

 

[Borek]

My guess is that as N is sum of all N_k, if N_i changes, denominator changes as well.

 

[roldy]

I guess that makes sense. Seems logical to me.

 

[DrStupid]

My guess is that as N is sum of all N_k, if N_i changes, denominator changes as well.

Yep, that's it.

$dx_i = \frac{{N \cdot dN_i - N_i \cdot dN}}{{N^2 }}$

and

$dN = dN_i$

gives

$dx_i = \frac{{N - N_i }}{{N^2 }} \cdot dN_i$

But for small $x_i$

$dx_i = \frac{{dN_i }}{N}$

is a good approximation.