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Chapter 17: Problem 29

Consider the following energy levels, each capable of holding two particles: $$ \begin{array}{l} E=2 \mathrm{~kJ}\\\ \boldsymbol{E} \quad E=1 \mathrm{~kJ}\\\ E=0 \quad \quad X X \end{array} $$ Draw all the possible arrangements of the two identical particles (represented by \(X\) ) in the three energy levels. What total energy is most likely, that is, occurs the greatest number of times? Assume that the particles are indistinguishable from each other.

Short Answer

Expert verified
The possible arrangements of the two indistinguishable particles in the three energy levels are: (XX, 0, 0), (0, XX, 0), (0, 0, XX), (X, X, 0), (X, 0, X), and (0, X, X). Their respective total energies are: 0 kJ, 2 kJ, 4 kJ, 1 kJ, 2 kJ, and 3 kJ. Since the total energy of 2 kJ occurs twice, it is the most likely total energy.

Step by step solution

01

Create every possible arrangement of particles

We place the particles either in the same energy level or in different energy levels. Here are all the possible arrangements: 1. Both particles in A (E=0 kJ) (XX, 0, 0) 2. Both particles in B (E=1 kJ) (0, XX, 0) 3. Both particles in C (E=2 kJ) (0, 0, XX) 4. One particle in A and one in B (X, X, 0) 5. One particle in A and one in C (X, 0, X) 6. One particle in B and one in C (0, X, X) Step 2: Calculate total energy for each arrangement
02

Calculate the total energy of each configuration

For each arrangement, sum the energy levels of the particles: 1. (XX, 0, 0) - Total energy = 0 kJ + 0 kJ = 0 kJ 2. (0, XX, 0) - Total energy = 1 kJ + 1 kJ = 2 kJ 3. (0, 0, XX) - Total energy = 2 kJ + 2 kJ = 4 kJ 4. (X, X, 0) - Total energy = 0 kJ + 1 kJ = 1 kJ 5. (X, 0, X) - Total energy = 0 kJ + 2 kJ = 2 kJ 6. (0, X, X) - Total energy = 1 kJ + 2 kJ = 3 kJ Step 3: Find the most frequently occurring energy
03

Determine the most common total energy

We count the occurrences of each energy value: - 0 kJ appears once - 1 kJ appears once - 2 kJ appears twice - 3 kJ appears once - 4 kJ appears once The total energy of 2 kJ occurs most frequently, with two possible arrangements (configurations 2 and 5). Thus, the most likely total energy is 2 kJ.

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