Boltzmann Distribution Plotter
Boltzmann Distribution Plotter plots probabilities for energy levels in thermal equilibrium using temperature and degeneracies.
Formulas Used in Boltzmann Distribution Plotter
The plotter computes the probability of a system occupying each energy level in thermal equilibrium:
Probability:
\\[ P_i = \frac{g_i e^{-E_i / (k_B T)}}{Z} \\]Partition Function:
\\[ Z = \sum_i g_i e^{-E_i / (k_B T)} \\]Where:
- \\( P_i \\): Probability of the system being in state \\(i\\)
- \\( E_i \\): Energy level \\(i\\) (J)
- \\( g_i \\): Degeneracy of energy level \\(i\\)
- \\( T \\): Temperature (K)
- \\( k_B \\): Boltzmann constant (\\( 1.380649 \times 10^{-23} \, \text{J/K} \\))
- \\( Z \\): Canonical partition function
Example Calculations
Example: System with \\( T = 300 \, \text{K}, E_i = [0, 1 \times 10^{-20}, 2 \times 10^{-20}] \, \text{J}, g_i = [1, 2, 2] \\)
\\[
k_B = 1.380649 \times 10^{-23} \, \text{J/K}
\\]
\\[
k_B T = 1.380649 \times 10^{-23} \cdot 300 \approx 4.141947 \times 10^{-21} \, \text{J}
\\]
\\[
Z = 1 \cdot e^{-0 / (4.141947 \times 10^{-21})} + 2 \cdot e^{-1 \times 10^{-20} / (4.141947 \times 10^{-21})} + 2 \cdot e^{-2 \times 10^{-20} / (4.141947 \times 10^{-21})}
\\]
\\[
Z \approx 1 \cdot 1 + 2 \cdot e^{-2.414} + 2 \cdot e^{-4.828} \approx 1 + 0.1792 + 0.0160 \approx 1.1952
\\]
\\[
P_1 = \frac{1 \cdot e^{-0 / (4.141947 \times 10^{-21})}}{1.1952} \approx \frac{1}{1.1952} \approx 0.8367
\\]
\\[
P_2 = \frac{2 \cdot e^{-1 \times 10^{-20} / (4.141947 \times 10^{-21})}}{1.1952} \approx \frac{2 \cdot 0.0896}{1.1952} \approx 0.1499
\\]
\\[
P_3 = \frac{2 \cdot e^{-2 \times 10^{-20} / (4.141947 \times 10^{-21})}}{1.1952} \approx \frac{2 \cdot 0.0080}{1.1952} \approx 0.0134
\\]
Result: \\( Z \approx 1.1952, P_i \approx [0.8367, 0.1499, 0.0134] \\)