Dalton's law of partial pressures equivalent to volume ratio?

[Syrus]

Homework Statement

I had a general inquiry about a relation in my physical chemistry textbook. It stated that when determining the partial pressure of an ideal gas component in a mixture/solution, instead of using the standard Dalton's law: P_i = x_iP_tot, one can simply use the relation P_i = (volume i : volume mixture or solution)P_tot.

I am having a hard time convincing myself of this and have not been able to find a derivation of this in either my book or online. Can someone please help me get started?

Homework Equations

The Attempt at a Solution

 

[Borek]

PV=nRT is all you need. How does n depend on V?

Well, perhaps adding obvious

$$n_{total} = \sum n_i$$

will help.

 

[Syrus]

Assuming we are dealing with a binary solution, that gets me here...(still stuck)

x_i = n_i/n_tot

n_i = P_iV_i/RT

n_tot = P_iV_i/RT + P_jV_j/RT

n_i/n_tot = P_iV_i/(P_iV_i + P_jV_j)

...?


Obviously we want x_i to somehow equal V_i/ V_tot

 

[Borek]

You don't have separate P_iV_i pairs. If you sum partial pressures P_i it is equivalent to assuming all gases occupy the same volume V_total, if you sum partial volumes V_i it is equivalent to assuming all gases are under the same pressure P_total.

 

[Syrus]

Ah, i see what you are saying now.

If we write n_i = V_iP_tot/RT this leads us to the conclusion.

I am curious now why n_i = V_iP_tot/RT = V_totP_i/RT in general