Question

What is the entropy of a closed system in which 25 distinguishable grains of sand are distributed among 1000 distinguishable equalsized compartments with only one grain per compartment?

Answer 1

The entropy of the closed system is

S = (1.58 × 10⁻²¹) J/K

Step-by-step explanation:

Entropy of a system is proportional to the number of different microstates. The constant of proportionality is the Boltzmann's constant.

S = K In W

K = Boltzmann's constant = (1.381 × 10⁻²³) J/K.

Number of different microstates possible for distributing 25 distinguishable grains of sand among 1000 distinguishable equal sized compartments with only one grain per compartment = ¹⁰⁰C₂₅ = (4.7642 × 10⁴⁹)

S = (1.381 × 10⁻²³) In (4.7642 × 10⁴⁹)

S = 1.381 × 10⁻²³ × 114.4

S = (1.58 × 10⁻²¹) J/K

Hope this Helps!!!

Answer 2

S = 172.69 J/K

Step-by-step explanation:

For a closed system, the entropy is given as the natural logarithm of microstates.

The number of particles (grains of sand); N = 25

The number of boxes (compartments); M = 1000

Entropy is the logarithm of number of microstates; Thus,

S = In ψ

However, the number of microstates is given by the formula;

ψ = Mⁿ

Thus, S = In Mⁿ

Plugging in the relevant values to obtain;

S = In (1000)^(25)

S = 172.69 J/K