Question

4 molecules are to be distributed in 2 cells . Find possible number of macrostates and corresponding no.of microstates.

Answer 1

Compartment-wise distributions of a system of particles are known as macrostates of the system. There are five macrostates observed corresponding to a system of four particles . In general, for a system of n particles, there are exactly n+1 macrostates (for the system of n particles to be distributed in two identical compartments). These macrostates are actually the following distribution of particles:

d₁=(0,n)

d₂=(1,n−1)

d₃=(2,n−2)

…

d_k=(k−1,n−k+1)

…

d_n+1=(n,0)

It can be observed that a macrostate may have many corresponding microstates. The distribution d₁=(0,4) has one microstate {{ }{a,b,c,d}}, while the distribution d₂=(1,3) has four, d₃ has six , d₄ has four and d₅ has one.

The total number of microstates for the system of four particles is, therefore, 1+4+6+4+1=16=2⁴. In general, total number of microstates for the system of n-particles in ‘two compartment’ composition is 2n.