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Entropy of mixing - Ideal gas. What is x?

  1. Apr 26, 2016 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations
    ds= integral (nR/v) dv

    3. 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
     
  2. jcsd
  3. Apr 26, 2016 #2
    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.
     
  4. Apr 26, 2016 #3
    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 .
     
  5. Apr 26, 2016 #4
    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 cant mix any more. im guessing the entropy change at the end is 0 because its just a homogenous mix now and theyve reached equilibrium now entropy is maximum
     
  6. Apr 26, 2016 #5
    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.
     
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