3- particles are to be distributed in 5 non degenerate energy levels. Let 'SD' be the entropy of such a system if the particles are distinguishable and 'SB' be the entropy if they are indistinguishable particles of integral spin. Then SD-SB =_____ kB (upto 2 decimal places)Correct answer is between '508.00,511.00'. Can you explain this answer?

Question

Answer

To calculate the difference in entropy between distinguishable and indistinguishable particles, we need to consider the number of ways the particles can be distributed among the energy levels.

1. Distinguishable particles:
When the particles are distinguishable, each particle can occupy any of the energy levels independently. Therefore, the number of ways to distribute the particles among the energy levels is given by the product of the number of particles in each level. In this case, the number of particles in each level can range from 0 to the total number of particles.

2. Indistinguishable particles of integral spin:
For indistinguishable particles of integral spin, we need to consider the Pauli exclusion principle, which states that no two identical fermions (particles with half-integer spin) can occupy the same quantum state. Since the particles have integral spin, they are bosons and do not follow this exclusion principle. Therefore, the number of ways to distribute the particles among the energy levels is given by the number of ways to distribute identical objects into distinct boxes, which is given by the partition function.

The partition function for bosons is given by the formula:
Z = (n + k - 1)! / (n!(k - 1)!)

Where n is the total number of particles and k is the number of energy levels.

3. Calculating the entropy difference:
The entropy is related to the partition function by the formula:
S = k ln(Z)

Therefore, the difference in entropy between distinguishable and indistinguishable particles is given by:
SD - SB = k ln(ZD) - k ln(ZB) = k ln(ZD/ZB)

Now, let's calculate the value of SD - SB using the given information:
n = 3 (total number of particles)
k = 5 (number of energy levels)

ZD = 5^3 = 125 (since each particle can occupy any of the 5 levels)
ZB = (3 + 5 - 1)! / (3!(5 - 1)!) = 35 (using the partition function formula)

SD - SB = k ln(125/35) = k ln(25/7) ≈ 510.83 kB

Therefore, the correct answer is between 508.00 and 511.00 kB.

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