Alright, we need to determine the occupation number of particles according to three different statistical distributions at 100 K, energy state of 0.1 eV, and Boltzmann constant [tex]\( k_B = 1.38 \times 10^{-23} \)[/tex] J/K.
### Given:
- Temperature, [tex]\( T = 100 \)[/tex] K
- Energy, [tex]\( E = 0.1 \)[/tex] eV [tex]\( = 0.1 \times 1.60218 \times 10^{-19} \)[/tex] J [tex]\( = 1.60218 \times 10^{-20} \)[/tex] J
- Boltzmann constant, [tex]\( k_B = 1.38 \times 10^{-23} \)[/tex] J/K
First, we calculate the ratio [tex]\(\frac{E}{k_B T}\)[/tex]:
[tex]\[
\frac{E}{k_B T} = \frac{1.60218 \times 10^{-20}}{1.38 \times 10^{-23} \times 100}
\][/tex]
Simplifying the expression within the exponential, we get:
[tex]\[
\frac{E}{k_B T} \approx 1.160997
\][/tex]
#### (a) Fermi-Dirac Statistics
The occupation number according to Fermi-Dirac statistics is given by:
[tex]\[
f_{FD} = \frac{1}{1 + e^{\frac{E}{k_B T}}}
\][/tex]
Substituting the earlier ratio [tex]\(\frac{E}{k_B T} \approx 1.160997\)[/tex]:
[tex]\[
f_{FD} = \frac{1}{1 + e^{1.160997}}
\][/tex]
We evaluated:
[tex]\[
f_{FD} \approx 9.074801286636516 \times 10^{-6}
\][/tex]
#### (b) Classical (Maxwell-Boltzmann) Statistics
The occupation number according to Maxwell-Boltzmann statistics is given by:
[tex]\[
f_{MB} = e^{-\frac{E}{k_B T}}
\][/tex]
Using the ratio [tex]\( \frac{E}{k_B T} \approx 1.160997 \)[/tex]:
[tex]\[
f_{MB} = e^{-1.160997}
\][/tex]
We evaluated:
[tex]\[
f_{MB} \approx 9.074883639402245 \times 10^{-6}
\][/tex]
#### (c) Bose-Einstein Statistics
The occupation number according to Bose-Einstein statistics is given by:
[tex]\[
f_{BE} = \frac{1}{e^{\frac{E}{k_B T}} - 1}
\][/tex]
Again substituting [tex]\( \frac{E}{k_B T} \approx 1.160997 \)[/tex]:
[tex]\[
f_{BE} = \frac{1}{e^{1.160997} - 1}
\][/tex]
We evaluated:
[tex]\[
f_{BE} \approx 9.074965993662668 \times 10^{-6}
\][/tex]
### Summary:
- Fermi-Dirac (F-D) occupation number: [tex]\( f_{FD} \approx 9.074801286636516 \times 10^{-6} \)[/tex]
- Maxwell-Boltzmann (classical) occupation number: [tex]\( f_{MB} \approx 9.074883639402245 \times 10^{-6} \)[/tex]
- Bose-Einstein (B-E) occupation number: [tex]\( f_{BE} \approx 9.074965993662668 \times 10^{-6} \)[/tex]
These results indicate the probability of occupation for each statistical distribution at the given temperature and energy state.
