Calculating chemical potenntial

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In summary, the question is asking for the chemical potential, \mu, at T=0 K for 44 ^{6}Li atoms in a 3D harmonic potential well. The equation for the energy levels is given as \epsilon_{i}=(i+3/2)\hbar\omega with degeneracies g_{i}= 1/2(i+1)(i+2). The solution involves using Fermi-Dirac statistics for spin 1/2 particles and setting \mu equal to the Fermi energy at T=0 K. To obtain \mu, the value of n_{i} is needed, which represents the number of particles in each degenerate energy level g_{i}.
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Mattofix
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Homework Statement



What is the chemical potential, [tex]\mu[/tex] (in units of [tex]\hbar\omega[/tex]) at T = 0 K of 44 [tex]^{6}Li[/tex] atoms with spin 1/2 contained in a 3D harmonic potential well (energy levels, [tex]\epsilon_{i}=(i+3/2)\hbar\omega[/tex] with degeneracies [tex]g_{i}= 1/2(i+1)(i+2)[/tex] )?

Homework Equations



a74431b34814575af95738dd8de59715.png


The Attempt at a Solution



spin = 1/2 [tex]\Rightarrow[/tex] Fermoins

t=0 [tex]\Rightarrow[/tex] [tex]\mu=\epsilon_{f}[/tex] (fermi energy)

therefore:
a74431b34814575af95738dd8de59715.png
= 1/2

So i need 'i' to then use in [tex]\epsilon_{i}=(i+3/2)\hbar\omega[/tex] to obtain [tex]\mu[/tex].

I know that [tex]\frac{n_{i}}{g{i}} = 1/2[/tex] and that [tex]g_{i}= 1/2(i+1)(i+2)[/tex] so all i need is the value of [tex]n_{i}[/tex], which is the number of particles in each [tex]g_{i}[/tex].

Is this all correct? If how do i obatin [tex]n_{i}[/tex]?
 
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Anyone got any ideas?
 

Related to Calculating chemical potenntial

1. What is chemical potential?

The chemical potential is a measure of the change in the free energy of a system when the number of particles in the system is changed by one.

2. How is chemical potential calculated?

Chemical potential can be calculated using the equation μ = ∂G/∂n, where μ is the chemical potential, G is the free energy, and n is the number of particles in the system.

3. What is the importance of calculating chemical potential?

Calculating chemical potential allows scientists to understand how a system will behave when the number of particles changes, and can help predict the direction of chemical reactions.

4. What factors can affect the calculation of chemical potential?

The temperature, pressure, and composition of a system can all affect the calculation of chemical potential, as well as any external factors such as electric or magnetic fields.

5. How is chemical potential used in practical applications?

Chemical potential is used in a variety of practical applications, such as in the design of chemical processes and in the study of phase transitions. It is also used in the field of electrochemistry to predict and understand the behavior of redox reactions.

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