Examveda

Consider a system of two non-interacting classical particles which can occupy any of the three energy levels with energy values E = 0, ε and 2ε having degeneracies g(E) = 1, 2 and 4 respectively, The mean energy of the system is

A. $$\varepsilon \left[ {\frac{{4\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 8\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}{{1 + 2\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 4\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}} \right]$$

B. $$\varepsilon \left[ {\frac{{2\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 8\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}{{1 + 2\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 4\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}} \right]$$

C. $$\varepsilon {\left[ {\frac{{2\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 4\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}{{1 + 2\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 4\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}} \right]^2}$$

D. $$\varepsilon \left[ {\frac{{\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 2\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}{{1 + \exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + \exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}} \right]$$

Answer: Option D


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Related Questions on Thermodynamics and Statistical Physics

A system has two energy levels with energies E and 2E. The lower level is four-fold degenerate while the upper level is doubly degenerate. If there are N non-interacting classical particles in the system, which is in thermodynamic equilibrium at temperature T, the fraction of particles in the upper level is

A. $$\frac{1}{{1 + {e^{ - \varepsilon /{k_B}T}}}}$$

B. $$\frac{1}{{1 + 2{e^{\varepsilon /{k_B}T}}}}$$

C. $$\frac{1}{{2{e^{\varepsilon /{k_B}T}} + 4{e^{2\varepsilon /{k_B}T}}}}$$

D. $$\frac{1}{{2{e^{\varepsilon /{k_B}T}} - 4{e^{2\varepsilon /{k_B}T}}}}$$