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.

 

[LudusRex]

Hello,

I am happy to help clarify the concept of mole fraction for you. Mole fraction is a way to express the concentration of a particular component in a mixture. It is defined as the ratio of the number of moles of a specific component (N_i) to the total number of moles in the mixture (N). This can also be written as X_i, where X represents mole fraction and the subscript i indicates the specific component.

Now, to address the statement you came across. The equation \frac{N_i}{N}=X_i is correct and represents the definition of mole fraction. However, the statement \frac{dN_i}{N}\neq dX_i is also true. This is because the partial derivative of the number of moles of a specific component with respect to the total number of moles in the mixture is not equal to the partial derivative of mole fraction with respect to the total number of moles. In simpler terms, the change in the number of moles of a component is not equal to the change in its mole fraction when the total number of moles in the mixture changes.

I hope this helps clarify the concept of mole fraction for you. If you have any further questions, please don't hesitate to ask. Good luck with your studies!