Question

what is the entropy of a closed system in which 20 distinguishable grains of sand are distributed among 1000 distinguishable equal-sized compartments? express your answer using three significant digits.

Answer 1

The entropy of a closed system with 20 distinguishable grains of sand in 1000 compartments is 1.90 x 10^-21 J/K, calculated using the Boltzmann entropy formula and taking into account the number of possible microstates.

Step-by-step explanation:

Calculating Entropy in a Closed System

For a closed system with 20 distinguishable grains of sand distributed among 1000 distinguishable equal-sized compartments, the entropy can be calculated using the formula for the entropy (S) in terms of the number of microstates (W):

S = k * ln(W)

where k is the Boltzmann constant (k = 1.38 x 10-23 J/K) and ln represents the natural logarithm. In this scenario, the number of microstates (W) is equal to 100020, since each of the 20 grains can be in any of the 1000 compartments independently. Therefore:

W = 100020 microstates

The entropy of the system is then:

S = k * ln(100020)
= (1.38 x 10-23 J/K) * ln(100020)
= 1.38 x 10-23 J/K * 20 * ln(1000)
= 1.38 x 10-23 J/K * 20 * 6.907
= 1.90 x 10-21 J/K

Therefore, the entropy (to three significant digits) is 1.90 x 10-21 J/K.