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Statistical Physics: Cubic lattice of two molecules

  1. Apr 7, 2015 #1
    1. The problem statement, all variables and given/known data
    A mixture of two substances exists on a cubic lattice of N sites, each of which is occupied by either an A molecule or a B molecule. The number of A molecules is NA and the number of B molecules is NB, such that NA + NB = N. The energy of interaction is [itex]k_BT\chi_{AA}[/itex] between two nearest neighbour A molecules, [itex]k_BT\chi_{BB} [/itex] between two nearest neighbour B molecules and [itex]k_BT\chi_{AB}[/itex] between a neighbouring pair of an A and a B molecule. When the interaction energies are written in this form, [itex]\chi_{ij}[/itex] is dimensionless.

    1) State,
    a. what the order parameter defined as [itex]\phi_A = NA/N[/itex] physically represents.
    b. the range of values that [itex]\phi_A[/itex] can take.
    c. the relation between [itex]\phi_A[/itex] and [itex]\phi_B[/itex] ([itex]\phi_B = NB/N[/itex]).

    2) By determining the number of A-A, A-B and B-B nearest neighbour contacts in terms of [itex]\phi_A[/itex], show that, if [itex]\chi_{AA} = \chi_{BB}[/itex], the total energy can be written as;

    $$ E = E_0 + 3Nk_BT \chi \phi_A (1−\phi_A )$$
    Where [itex]E_o[/itex] and [itex]\chi[/itex] are constants.

    2. Relevant equations


    3. The attempt at a solution
    1a) The percentage of molecule A in lattice sites N.
    1b) [itex]\phi_A=0...1/2[/itex] (due to max entropy being when there is 50% of molecule A and 50% of molecule B)
    1c) [itex]\phi_B=1-\phi_A[/itex]

    2) so this is where im stuck, i found;
    [itex]N_{AA}=6N\phi_A^2[/itex] (each site has 6 nearest neighbours, so number of A molecules x no of NN x P(NN being a A molecule)
    [itex]N_{BB}=6N(1-\phi_A)^2[/itex]
    [itex]N_{AB}=12N\phi_A(1-\phi_A)[/itex]

    and then using;
    $$E=N_{AA}V_{AA}+N_{BB}V_{BB}+N_{AB}V_{AB}$$
    I get that
    $$E=6VN+6VN\phi_A(\phi_A-1)$$
    Where [itex]V=V_{AA}=V_{BB}=V_{AB}[/itex] as [itex]\chi_{AA} = \chi_{BB}[/itex].

    I'm thinking i might've worked out the number of A-A, B-B and A-B interactions wrong. could someone give me a nudge in the right direction?
     
    Last edited: Apr 7, 2015
  2. jcsd
  3. Apr 8, 2015 #2
    Figured out what i was doing wrong now! all is fine :D
     
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