3- particles are to be distributed in 5 non degenerate energy levels. ...
If the particles are distinguishable, then
If the integral spin particles are indistinguishable, then
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3- particles are to be distributed in 5 non degenerate energy levels. ...
Given:
- Number of particles, N = 3
- Number of energy levels, W = 5
To find:
SD - SB = ? (in terms of kB)
Explanation:
In order to calculate the difference between the entropy of distinguishable particles (SD) and indistinguishable particles (SB), we need to determine the values of SD and SB individually.
Calculating SD:
The entropy of distinguishable particles (SD) is given by the formula:
SD = kB * ln(W^N)
Here, W is the number of energy levels and N is the number of particles.
Substituting the given values:
SD = kB * ln(5^3)
SD = kB * ln(125)
SD ≈ 2.079 * kB
Calculating SB:
For indistinguishable particles of integral spin, we need to use the formula:
SB = kB * ln(C)
Here, C represents the number of ways the particles can be distributed among the energy levels. To find C, we can use the concept of combinatorics.
Since the particles are indistinguishable, we can use the concept of "stars and bars" to calculate the number of ways to distribute them. The formula for C is given by:
C = (N+W-1)C(N)
Substituting the given values:
C = (3+5-1)C(3)
C = 7C3
C = 7! / (3! * 4!)
C = 7 * 6 * 5 / (3 * 2 * 1)
C = 35
Substituting the value of C into the equation for SB:
SB = kB * ln(35)
Calculating SD - SB:
Now, we can subtract SB from SD to find the difference:
SD - SB = kB * ln(125) - kB * ln(35)
SD - SB = kB * (ln(125) - ln(35))
SD - SB ≈ kB * (4.828 - 3.555)
SD - SB ≈ kB * 1.273
Approximate Answer:
SD - SB ≈ 1.273 * kB
Since the correct answer is between 1.23 and 1.30, the value of kB must be approximately 1. Therefore, SD - SB ≈ 1.273 * 1 ≈ 1.273 kB, which falls within the given range.