Grand Canonical Ensemble Calculator
Grand Canonical Ensemble Calculator computes ⟨N⟩, ⟨E⟩ for ideal gas at fixed T, V, μ, using relations like (∂T/∂V)_S.
Grand Canonical Ensemble Overview
The grand canonical ensemble describes systems at fixed \\(T\\), \\(V\\), \\(\mu\\):
\\[
\Xi = \sum_{N=0}^\infty e^{\beta \mu N} Z_N, \quad \beta = \frac{1}{k T}
\\]
Key properties:
- Average particle number: \\(\langle N \rangle = \frac{1}{\beta} \frac{\partial \ln \Xi}{\partial \mu}\\)
- Average energy: \\(\langle E \rangle = -\frac{\partial \ln \Xi}{\partial \beta} + \mu \langle N \rangle\\)
Maxwell relation for internal energy (\\(U\\)):
\\[ \left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V \\]For an ideal gas: \\( P = \frac{\langle N \rangle k T}{V} \\)
Example Calculation
Example: Ideal Gas with \\(T = 298 \, \text{K}, V = 22.4 \, \text{L}, \mu = -0.1 \, \text{kJ/mol}\\)
Differential: \\( dU = T dS – P dV \\)
Maxwell Relation: \\(\left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V\\)
Pressure: \\( P = \frac{\langle N \rangle k T}{V} \\)