Calculation of No. of micro states (in equilibrium)

In summary, the conversation discusses determining the total number of micro states of a system in equilibrium within a certain value, 1/σ. It is suggested to use the factorials and Stirling's approximation to calculate lnΩ as a function of Q_A.
  • #1
Guffie
23
0

Homework Statement



Hello,

I am required to determine the total number of micro states of a system in equilibrium within a certain value, 1/σ.

The number of micro states for this system is given by,

[itex] \Omega =\frac{({{Q}_{A}}+{{N}_{A}}-1)!}{{{Q}_{A}}!({{N}_{A}}-1)!}\frac{({{Q}_{B}}+{{N}_{B}}-1)!}{{{Q}_{B}}!({{N}_{B}}-1)!}[/itex]

In the equilibrium state Q_a,e = 5000 and Q_b,e = 10000
Also Na = 5000 and Nb = 10000 and E = e(Qa+Qb)

So I need to find the total number of micro states in this system at equilibrium within 1/σ.
In this case σ=sqrt(5000).

Could anyone help me figure out how I would do this?

Homework Equations


The Attempt at a Solution



Is the correct way to do this by summing all the micro states together from

Q_a,e + 1/σ to Q_a,e + 1/σ,

if so how is that possible? the typical factorial function applies to integers.

1/σ is pretty small as well 0.014 so should i just ignore this?

The other part of the question asks to calculate hte total number of micro states outside of this interval.

So that means calculating every possible combination of Qa and Qb and adding them all together right?

Is there a way I can do this quickly in mathematica or something?
 
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  • #2
Guffie said:
[itex] \Omega =\frac{({{Q}_{A}}+{{N}_{A}}-1)!}{{{Q}_{A}}!({{N}_{A}}-1)!}\frac{({{Q}_{B}}+{{N}_{B}}-1)!}{{{Q}_{B}}!({{N}_{B}}-1)!}[/itex]

In the equilibrium state Q_a,e = 5000 and Q_b,e = 10000
Also Na = 5000 and Nb = 10000 and E = e(Qa+Qb)
These factorials will result in very large numbers. It will be easier to work with [itex]\ln \Omega[/itex], and take the exponential at the end to convert back to a multiplicity.

Can you convert write this [itex]\ln \Omega[/itex] as a function of [itex]Q_A[/itex]? (Hint: You will need Stirling's approximation.)
 

1. How do you calculate the number of microstates in equilibrium?

The number of microstates in equilibrium can be calculated using the equation W = N!, where N is the number of particles in the system. This equation assumes that the particles are distinguishable and that all microstates are equally probable.

2. What is the significance of calculating the number of microstates in equilibrium?

Calculating the number of microstates in equilibrium allows us to understand the distribution of energy and particles in a system. It is a key concept in statistical mechanics and helps us to predict the behavior of large systems.

3. How does the number of microstates change with an increase in temperature?

As temperature increases, the number of microstates in equilibrium also increases. This is because at higher temperatures, there is more thermal energy available for the particles to occupy different energy levels and configurations.

4. Can the number of microstates in equilibrium ever decrease?

No, the number of microstates in equilibrium can never decrease. This is because the universe tends towards increasing entropy, which is related to the number of microstates in a system. Therefore, the number of microstates will either remain constant or increase.

5. How does the number of microstates relate to entropy?

The number of microstates in equilibrium is directly proportional to the entropy of a system. As the number of microstates increases, so does the entropy. This means that systems with a higher number of microstates have a higher degree of disorder and randomness.

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