Question

We have 6 atoms in a box and we place a divider in it at a random time. What are the possible macrostates of the atoms? How many microstates exist at each macrostate? Determine the entropy for each macrostate.

Answer 1

Okay so we put a divider inside a box , so we have so many possibilies,

let's say the box is diveded into two sides - (left , right)

possible macrostates are (6 , 0) , (5, 1) , (4,2) , (3,3) , (2,4) , (1,5) , (0,6) . Out of these, (3,3) is the most probable macrostate.

Now, How many microstates at each macrostate,

Only one microstate for (6,0) and (0,6)

Number of microstates can be calculates as N! / n₁! * n₂! ( N is total number of particles, n₁ and n₂ are particles on eah side of divider)

so, For (1,5) and (5,1) macrostate , Numbr of microstate = 6! / 1! 5! = 6

for (4,2) and (2,4) , Number of micorsstates = 6! / 2! 4! = 15

for (3,3), Number of microstates = 6! / 3! 31 = 20

Entropy can be calculated as

S = k* ln (number of microstates) where k = 1.38e-23 (Boltzmann constant)

S = 1.38e-23* ln (20) = 4.13e-23 (for (3,3) macostate)