Relation between mole, volume and pressure fraction

[subhradeep mahata]

Homework Statement

If mole fraction, pressure fraction and volume fraction are denoted by X_mol , X_p, X_V respectively, of a gaseous component, then what is the relation between them?

Homework Equations

mole fraction = mole of component / total moles
pressure fraction = pressure of component / total pressure
volume fraction = volume of component / total volume

The Attempt at a Solution

Well, to be honest, I simply don't know where to start from. But still, have a look at my method (even if it is stupid) :
Let's assume there are 3 gaseous components.
mole fraction of 1st component (at const. temp.) = n1 / n1+n2+n3
= P1V1 /( P1V1+P2V2+P3V3) (cancelling the RT terms, T being constant) (equation 1)
=P1 / (P1+P2+P3) ...pressure fraction assuming constant volume (from eq.1)
=V1 / (V1+V2+V3) ...volume fraction assuming constant pressure (from eq. 1 )
So, my answer is that all three are same, which happens to be the correct answer according to my book. But i still need some clarifications, is it justified to assume volume to be constant for calculating pressure fraction? please help me out. If you find my answer to be incorrect, please give me the correct proof.

 

[Gene Naden]

I think that the relation "pressure fraction = pressure of component / total pressure" assumes that all three components are contained in a single vessel of volume $v$. So you have $X_{p}=\frac{p1\:v}{p1\:v+p2\: v+p3\:v}$

 

[mjc123]

There is some inconsistency in the way people talk about gas mixtures. Sometimes each gas is assumed to occupy the total volume, with partial pressures P₁, P₂ and P₃, and total pressure P = P₁ + P₂ + P₃. Sometimes they are thought of as all being at a constant pressure P, and occupying fractions of the total volume equal to their mole fractions (assuming ideality). The first is more physically realistic - the concept of volume fraction is not so meaningful as it is in a liquid mixture, where each component excludes the others from the volume it occupies, whereas in the gas the molecules are zooming round in empty space and not excluding each other from their "fraction" of the volume. Nevertheless, the second approach is sometimes useful to work with, particularly in the case of chemical reactions that produce or consume gas. For example, suppose you have a solid under an atmosphere of air of volume V at a given pressure P. If the solid decomposes producing a gas, and at constant pressure the total gas volume increases by ΔV, you can say that "a volume ΔV of gas at pressure P was produced by the reaction" and calculate the number of moles. In reality of course the gas will mix with the air and not occupy a separate volume ΔV - but not instantaneously.

 

[Chestermiller]

When people talk about volume fraction, what they really mean is mole fraction. This has never made sense to me either, but that's the way it's done.

Of course, by Dalton's law, pressure fraction and mole fraction are the same.

 

[subhradeep mahata]

@Gene Naden yes that's what i have assumed.

 

[subhradeep mahata]

@Chestermiller
We know, partial pressure=mole fraction *total pressure
or, partial pressure/ total pressure = mole fraction
or, pressure fraction = mole fraction
Is it correct?
And is it correct to use mole fraction and volume fraction interchangeably ?