Worksheet: Calculating Entropy by Counting Microstates

In this worksheet, we will practice using the Boltzmann equation to calculate the entropy of particles from the logarithm of the number of microstates.

Q1:

A system contains six particles distributed between two boxes.

Calculate the fractional probability of all particles occupying one of the two boxes.

  • A 1 1 2 8
  • B 1 6 4
  • C 1 1 6
  • D 1 8
  • E 1 3 2

The system originally contains an equal number of particles in each box. The particles are redistributed so that one box contains only one particle. Calculate to 4 significant figures the entropy change Δ𝑆 for this process.

  • A 1 . 3 9 2 × 1 0 J/K
  • B 1 . 5 1 7 × 1 0 J/K
  • C 2 . 4 3 5 × 1 0 J/K
  • D 1 . 6 6 2 × 1 0 J/K
  • E 2 . 8 7 1 × 1 0 J/K

Calculate to 4 significant figures the maximum value of Δ𝑆 when one distribution converts to another in this system.

  • A 6 . 6 9 8 × 1 0 J/K
  • B 3 . 7 3 9 × 1 0 J/K
  • C 4 . 1 3 6 × 1 0 J/K
  • D 4 . 9 0 9 × 1 0 J/K
  • E 3 . 1 7 9 × 1 0 J/K

Q2:

A system contains four particles evenly distributed between two boxes. Before heat transfer, two units of energy are distributed between the particles in one box and the particles in the other box have zero energy. After heat transfer, the two units of energy are evenly distributed between the two boxes. Calculate to 4 significant figures the entropy change Δ𝑆 for this process.

  • A 3 . 9 7 2 × 1 0 J/K
  • B 1 . 7 2 5 × 1 0 J/K
  • C 1 . 9 1 4 × 1 0 J/K
  • D 9 . 5 7 0 × 1 0 J/K
  • E 3 . 4 3 1 × 1 0 J/K

Q3:

Consider two systems that are identical in all respects, except that in one the molecules are distinguishable, whereas in the other the molecules are indistinguishable. What would be the difference between the molar entropies of these two systems?

Q4:

What is the total number of microstates in a 5D term of the ground state iron atom?

Q5:

There are sixteen microstates associated with placing four particles in two boxes. The microstates may be collected into five distributions, (a)–(e), involving the same numbers of particles in each box.

Calculate the entropy change when distribution (a) converts to distribution (d).

  • A 8 . 3 1 2 × 1 0 J/K
  • B 1 . 7 7 3 × 1 0 J/K
  • C 1 . 9 1 4 × 1 0 J/K
  • D 7 . 5 3 8 × 1 0 J/K
  • E 8 . 2 8 4 × 1 0 J/K

Calculate the entropy change when distribution (c) converts to distribution (b).

  • A 1 . 4 3 9 × 1 0 J/K
  • B 1 . 7 8 4 × 1 0 J/K
  • C 2 . 4 3 1 × 1 0 J/K
  • D 5 . 5 9 8 × 1 0 J/K
  • E 1 . 9 0 6 × 1 0 J/K

Calculate the maximum value of Δ𝑆 when one distribution converts to another in this system.

  • A 8 . 2 8 4 × 1 0 J/K
  • B 5 . 5 7 0 × 1 0 J/K
  • C 1 . 9 1 4 × 1 0 J/K
  • D 2 . 4 7 4 × 1 0 J/K
  • E 1 . 6 3 1 × 1 0 J/K

Q6:

Which of the following substances has the highest molar entropy?

  • A C H O H ( ) 4 9 l
  • B C H O H ( ) 2 5 l
  • C C H O H ( ) 2 5 g
  • D C H O H ( ) 3 7 g
  • E C H O H ( ) 3 7 l

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