Four particles have free access to distribute into four different energy levels 0,5,10, and 20J. If these four particles are to create two macrostates having total energies 10J and 40J, which macrostate has more entropy?

To determine which macrostate has more entropy, we need to calculate the number of microstates corresponding to each macrostate. The entropy of a system is directly proportional to the logarithm of the number of microstates.

In this case, we have four particles distributed among four different energy levels: 0J, 5J, 10J, and 20J. We want to find the number of microstates for two macrostates, one with a total energy of 10J and the other with a total energy of 40J.

Let's start with the macrostate of total energy 10J. We need to count the number of different ways the four particles can distribute themselves to achieve a total energy of 10J.

Since the particles are distinguishable, we can use the concept of combinations to calculate the number of microstates. We need to distribute 10J among four levels, and the number of possibilities for each particle at each energy level is:

- For 0J: 1 possibility (all particles at 0J)
- For 5J: 2 possibilities (particles at 5J and particles at 0J)
- For 10J: 3 possibilities (particles at 10J, particles at 5J, and particles at 0J)
- For 20J: 4 possibilities (particles at 20J, particles at 10J, particles at 5J, and particles at 0J)

Now, we can calculate the number of microstates for the macrostate with a total energy of 10J by multiplying the possibilities for each energy level:

Number of microstates for macrostate with 10J = 1 * 2 * 3 * 4 = 24

Next, let's repeat the process for the macrostate with a total energy of 40J.

- For 0J: 4 possibilities (each particle can be at 0J)
- For 5J: 4 possibilities (each particle can be at 5J)
- For 10J: 4 possibilities (each particle can be at 10J)
- For 20J: 4 possibilities (each particle can be at 20J)

Number of microstates for macrostate with 40J = 4 * 4 * 4 * 4 = 256

Now, we can compare the entropy of the two macrostates. The entropy is given by the logarithm of the number of microstates:

Entropy of macrostate with 10J = log2(24) ≈ 4.58 bits
Entropy of macrostate with 40J = log2(256) = 8 bits

Therefore, the macrostate with a total energy of 40J has more entropy (8 bits) compared to the macrostate with a total energy of 10J (4.58 bits).