Proof that volume mol for ideal gases

Consider a gas which contains ideal gas A and ideal gas B. Volume% A and volume% B in the gas are defined as  

Consider a moles of ideal gas A in a freely expanding/contracting box at constant temperature T° and constant pressure P°. The volume occupied by A is described by the ideal gas law  

Remove a wall between the two boxes and join them. Allow the gases to mix. For ideal gases, there is no change in gas volume when the gases mix so that  

The left side of Eq. (E.IO) is volume% A (Eq. E.l), whereas the right side is mole% A (Eq. E.3) so that the equation becomes  

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Appendix E Proof that volume% = mol% (for ideal gases)... 

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