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Pushoam
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Homework Statement
Homework Equations
The Attempt at a Solution
The probability of getting a state with energy ## E_v## is ## \frac { N_v } { N_v +N_c } = \frac1{ e^{-(E_v – E_f)/k_BT} +1} ## ………….(1)
Since, ## E_v < E_f, e^{-(E_v – E_f)/k_BT}>>1 ## as ## E_f – E_v>> k_BT ##……….(2)
So, ## \frac { N_v } { N_v +N_c } = \frac1{ e^{-(E_v – E_f)/k_BT} } ## ……….(3)
Similarly, probability of getting a state with energy ## E_c## is ## \frac { N_c} { N_v +N_c } = \frac1{ e^{-(E_c – E_f)/k_BT} +1} ##...(4)
Dividing (1) by (4) gives,
## \frac { N_v } {N_c } =## ## \frac{ e^{-(E_c – E_f)/k_BT} +1}{ e^{-(E_v – E_f)/k_BT} } ## ## = \frac{ e^{-(E_c – E_f)/k_BT} }{ e^{-(E_v – E_f)/k_BT} } ##
## k_BT \ln \frac { N_v } {N_c } = -(E_c – E_f) +(E_v – E_f) =E_v – E_c ##
Is this correct?
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