Microstates & Entropy of two subsystems of a 4 particle system? (ensembles?)

S is the entropy.In summary, a four-particle system composed of 2 two-particle subsystems is initially isolated before being brought into thermal contact with each other. The maximum internal energy of subsystem I is 4{E} and subsystem II has an internal energy of 0. The total number of microstates of the combined system is the product of the numbers of microstates of the 2 subsystems. The entropy of the system can be calculated using the Boltzmann constant and the number of microstates. However, the process of finding the number of microstates is not provided in the given data and equations.
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Homework Statement



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A four-particle system is composed of 2 two–particle subsystems. Subsystem I has par-
ticles A and B, which can have a maximum internal energy U(AB) = 4{E}. Subsystem II
has particles C and D, in which the internal energy U(CD) = 0.

The subsystems are initially isolated from each other, before being brought into thermal
contact (but still isolated from the rest of the universe). By calculating and comparing the
number of microstates of the combined system before and after being brought into thermal
contact, state if and how the entropy of the combined system changes.
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Homework Equations



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Let W = number of microstates
Let kb = Boltzmann constant

W = 2^N

S = (kb)lnW
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The Attempt at a Solution



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So, I've theoretically laid out:


subsystem I, at any time t, it is likely to exist as W1(U1)
subsystem II, at ant time t is likely to be in U2(U2) microstate


Thus, the combined subsystem is likely to be in W(U1, U2) microstate
where N(total) = N(I) + N(II)
and U(total) = U(I) + U(II)


The total number of microstates of the composite system is the product of the numbers of microstates of the 2 subsystems.

W(U(total), N(total)) = W1(U1, N1)W2(U1,N1) = W1(U1,N1)W2(U(total) - U1, N(total) - N1)

Thus, entropy S is
S = S1(U1, N1) + S2(U(total) - U1, N(total) - N1)
****************************************************************

However, I have no clue as to how to actually calculate W from the data and formulae given. I also do not know how to calculate S, since I do not know how to find W.

Any help would be appreciated!
Much love. =)

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PS. N is the number of particles
 

Related to Microstates & Entropy of two subsystems of a 4 particle system? (ensembles?)

1. What is a microstate in the context of a 4 particle system?

A microstate in this context refers to a specific arrangement or configuration of the 4 particles in the system. This can include their positions, velocities, and other properties that define their state.

2. How is entropy related to the microstates of a 4 particle system?

Entropy is a measure of the disorder or randomness of a system. In the case of a 4 particle system, the number of microstates that the particles can occupy directly affects the entropy of the system. More microstates result in higher entropy.

3. What are the possible ensembles that can be used to describe the 4 particle system?

The most commonly used ensembles for describing a 4 particle system are the microcanonical ensemble, canonical ensemble, and grand canonical ensemble. These ensembles differ in their constraints and variables, but all aim to describe the system in equilibrium.

4. How does the entropy of a 4 particle system change when it is divided into two subsystems?

When a 4 particle system is divided into two subsystems, the total entropy of the system remains the same. However, the entropy of each subsystem may change depending on the constraints and interactions between the two subsystems.

5. Can the microstates of a 4 particle system be used to predict its behavior?

No, the behavior of a 4 particle system cannot be predicted solely based on its microstates. Other factors such as external influences and interactions between particles must also be considered in order to accurately predict the behavior of the system.

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