How Does Adding a Zero-Energy Particle Affect System Entropy?

In summary, the conversation discussed the solution to a problem involving two distinguishable particles and nondegenerate energy levels. The total energy is U = 2e and a third distinguishable particle with zero energy is added to the system. The entropy of the assembly is increased by a factor of 1.63, which corresponds to 6 possible final configurations. The interpretation of the wording "a distinguishable particle with zero energy is added to the system" was also discussed, with differing opinions on whether the third particle can interact with the other particles or not.
  • #1
the keck
22
0
[SOLVED] Basic Statistical Thermodynamics

Homework Statement



Two distinguishable particles are to be distributed among nondegenerate energy levels 0,e,2e,3e... such that the total energy is U = 2e

If a distinguishable particle with zero energy is added to the system show that the entropy of the assembly is increased by a factor of 1.63 when compared to the entropy of the original assembly.

Homework Equations


S=k*ln(W), where k is Boltzmann's Constant and W is thermodynamic probability


The Attempt at a Solution



I know that without the particle added, W is three. And so a factor of 1.63 means that W must equal to 6 i.e. Solving ln(x)/ln(3) = 1.63 => x = 6

However, I do not know how adding an extra particle would produce three more states.

Regards,
The Keck
 
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  • #2
the keck said:
I know that without the particle added, W is three

That's not what I get. Could you write down your three configurations?
 
  • #3
George Jones said:
That's not what I get. Could you write down your three configurations?

george, I do agree with his/her result. Which number do you not agree with? # initial configurations or 6 final?
 
  • #4
nrqed said:
george, I do agree with his/her result. Which number do you not agree with? # initial configurations or 6 final?

With the initial configuration of 3 possible states.

Since the particles are distinguishable, aren't there two separate states where each particle has energy e?
 
  • #5
George Jones said:
With the initial configuration of 3 possible states.

Since the particles are distinguishable, aren't there two separate states where each particle has energy e?

I think that this is a single state. There is no way to distinguish "particle 1 having energy e and particle 2 having energy e" from the reverse situation even if the aprticles are distinguishable. I think this is a single state.


to the OP: I do get 6 states for the case of 3 particles.
 
  • #6
nrqed said:
I think that this is a single state. There is no way to distinguish "particle 1 having energy e and particle 2 having energy e" from the reverse situation even if the aprticles are distinguishable. I think this is a single state.

to the OP: I do get 6 states for the case of 3 particles.

I agree. That's what I get as well.
 
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  • #7
nrqed said:
I think that this is a single state. There is no way to distinguish "particle 1 having energy e and particle 2 having energy e" from the reverse situation even if the aprticles are distinguishable. I think this is a single state.

It seems that I had talked myself into thinking that order matters. Thanks.

Writing things a different way, suppose one particle is green and one particle is yellow. Green with energy e and yellow with energy e is that same state as yellow with energy e and green with energy e.

I also see the six states for the configuration with three distinguishable particles, but I have a bit of a problem with the wording "a distinguishable particle with zero energy is added to the system".

Start out with a 2-particle system consisting of one green particle and one yellow particle. Add a third particle. Since the particles are distinguishable, we know which particle was added, say a blue particle. Under this interpretation, the blue particle has to have energy zero, and there still are only three states.

My interpretation of the wording probably is non-standard, or maybe even, as before, outright wrong.
 
  • #8
George Jones said:
It seems that I had talked myself into thinking that order matters. Thanks.

Writing things a different way, suppose one particle is green and one particle is yellow. Green with energy e and yellow with energy e is that same state as yellow with energy e and green with energy e.
Right. If I close my eyes and you switch them, I won't be able to tell if they ahve been switched after I open my eyes.
I also see the six states for the configuration with three distinguishable particles, but I have a bit of a problem with the wording "a distinguishable particle with zero energy is added to the system".

Start out with a 2-particle system consisting of one green particle and one yellow particle. Add a third particle. Since the particles are distinguishable, we know which particle was added, say a blue particle. Under this interpretation, the blue particle has to have energy zero, and there still are only three states.

My interpretation of the wording probably is non-standard, or maybe even, as before, outright wrong.

I see what you are saying. Good point. The question is a bit ambiguous because of that. It depends if the particles interact or not. One other interpretation is the following: the third particle has no energy when it is added (so the total energy remains 2E) but since it may interact with the others (which is the assumption I made and which is not clear from the question, I grant you that!), once it is added teh energy may be redistributed among all three particles. If there is no energy exchanged, I agree that the number of states remains 3. If energy may be redistributed among all three particles, the number of states increases to 6.
 
  • #9
Thanks for your help, George and nrqed. I also took the assumption that the particles cannot interact and so the third particle only has energy zero, leading to only three states.

I admit the question is definitely ambiguous, but that seems to be one of the those things which seem to occur more often in Statistical Physics questions than in other types of physics questions.

Thanks again

Regards,
The Keck
 

FAQ: How Does Adding a Zero-Energy Particle Affect System Entropy?

What is the definition of Basic Statistical Thermodynamics?

Basic Statistical Thermodynamics is a branch of thermodynamics that uses statistical methods and concepts to explain and analyze the behavior of large systems of particles. It is based on the idea that the macroscopic properties of a system can be described by the statistical behavior of its microscopic constituents.

What are the key concepts in Basic Statistical Thermodynamics?

The key concepts in Basic Statistical Thermodynamics include entropy, temperature, energy, and probability. Entropy is a measure of disorder or randomness in a system, while temperature is a measure of the average kinetic energy of particles in a system. Energy is the ability to do work, and probability is used to describe the likelihood of different states of a system.

How is Basic Statistical Thermodynamics different from Classical Thermodynamics?

Classical Thermodynamics is based on macroscopic observations and does not take into account the individual behavior of particles. On the other hand, Basic Statistical Thermodynamics uses statistical methods to explain the macroscopic behavior of a system based on the microscopic behavior of its particles. It provides a more detailed and accurate description of thermodynamic systems.

What is the importance of Basic Statistical Thermodynamics in science and engineering?

Basic Statistical Thermodynamics is essential in understanding and predicting the behavior of complex systems, such as chemical reactions, phase transitions, and material properties. It is also used in the design and optimization of industrial processes and in the development of new materials and technologies.

Can Basic Statistical Thermodynamics be applied to real-world systems?

Yes, Basic Statistical Thermodynamics is widely applied in various fields, including physics, chemistry, biology, and engineering, to analyze and predict the behavior of real-world systems. It has been successfully used to explain and predict phenomena at different scales, from the behavior of individual molecules to the properties of large-scale systems.

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