Entropy of mixing - Ideal gas. What is x?

[navm1]

Homework Statement

A bottle with volume v containing 1 mole of argon is next to a bottle of volume v with 1 mole of xenon. both are connected with a pipe and tap and are same temp and pressure. the tap is opened and they are allowed to mix. What is the total entropy change of the system? Once the gases have fully mixed, the tap is shut and the gases are no longer free. what is the entropy change with this process?

Homework Equations

ds= integral (nR/v) dv

The Attempt at a Solution

imagining it as a reversible isotherm. I used
deltaS_mix = deltaS_1+deltaS_2

deltaS_1 = n_1*R*ln(V_1+V_2)/V_1

deltaS_2 = n_2*R*ln(V_1+V_2)/V_2

then adding them together and cancelling down from 2V/V etc i ended up with
R(ln2+ln2) = Rln4.

My Questions are:

Why does it ask for two separate entropy change calculations in the question?
In my textbook it uses xV and (x-1)V for the respective volumes and it ends up as

deltaS = -NK_b(x*lnx+(1-x)ln(1-x))

What does x represent here?

Thanks

 

[drvrm]

Why does it ask for two separate entropy change calculations in the question?

as the two gases have expanded there will be entropy change for each one of them i.e. under expansion from a state of order to more disorder.

 

[drvrm]

In my textbook it uses xV and (x-1)V for the respective volumes and it ends up as

well it might be taking V as the total volume of the system. and if x fraction of V is being occupied by one then 1-x times V must be the volume of the other one .

 

[navm1]

got it. so i used x as 1/2 and got Rln(2). makes sense to me and seems to match my course material.

I still don't quite understand why it asks for an entropy change of the system when they mix and when theyve mixed and the tap is shut so they can't mix any more. I am guessing the entropy change at the end is 0 because its just a homogenous mix now and theyve reached equilibrium now entropy is maximum

 

[drvrm]

I still don't quite understand why it asks for an entropy change of the system when they mix and when theyve mixed and the tap is shut so they can't mix any more. I am guessing the entropy change at the end is 0 because its just a homogenous mix now and theyve reached equilibrium now entropy is maximum

Entropy is a property reflected in the ways in which a system of N particles can get described.

the more ordered a system is- it gets to less entropy- and the opposite is also true.
as one opens the tap-
the two gases are free to diffuse throughout the volume of two containers. For an ideal gas, the energy is not a function of volume,

and, for each gas, there is no change in temperature. The entropy change of each gas is affected as for a reversible isothermal expansion from the initial volume to a final volume
In terms of the overall spatial distribution of the molecules of the two gases , one can say that
final state was more random, more mixed, than the initial state in which the two types of gas molecules were confined to specific regionsof space in the bottles..

Another way to say this is in terms of ``disorder;'' there is more disorder in the final state than in the initial state.
the perspective/background of entropy is thus that increases in entropy are connected with increases in randomness or disorder.no doubt in the final state they can not take any path of more randomness or disorder.