Proof of Law of mass action

[abhaymv]

In my college,proof of law of mass action was given thus:
(I suspect it is wrong)
Let no. of electrons per unit volume at constant temperature at an energy state E be given by:

N(E)=(1/(2π²))*[2m_e/ħ²]^(3/2)*{E-E_g}^(1/2)
(1)
Probability of filling an electron in an energy state:

E=F(E)=1/(1+e^{((E-E_f)/(kT)})
(2)
Where F(E) is called "Fermi function".
Total no. of electrons with energy between E and E+ΔE is the product :
N(E)F(E)dE
(3)

E-E_f>>kT,so F(E) can be written as:

F(E)=e^{-((E-E_f)/kT)}
Applying this and expanding (3),
and integrating to get the total no. of electrons in the conduction band,

n=∫N(E)F(E)dE

Integration from E to ∞
we get,
n=2[m_ekT/2πħ²]^3/2e((E_f-E_g)/kT)
Let the hole density of V.B be:
N_p(E)=(1/(2π²))*[2m_h/ħ²]^(3/2)*E^(1/2)

Probability of filling a hole=1-F(E)
Total probability being 1

P(E) is then approximated as:

e^((E-E_f)/kT


Is this correct?I don't think it is.Because,if I substitute random values for the power of e in the expressions,I don't get total probability as 1...(not even close)

Where is the proof wrong?

 

[eljose79]

Thank you for sharing your concerns about the proof of the law of mass action given in your college. I would like to provide some insights on this matter.

Firstly, the equation (1) given for the number of electrons per unit volume at a certain energy state is correct. This equation is derived from the statistical mechanics principles and is known as the density of states function. However, the factor (3/2) should not be included in the equation as it is already taken into account in the exponent.

Secondly, equation (2) for the Fermi function is also correct. This equation describes the probability of an energy state being occupied by an electron at a certain temperature. However, the exponent should be (-E-Ef)/kT instead of (E-Ef)/kT.

Next, in equation (3), the total number of electrons in the conduction band should be n = ∫N(E)F(E)dE from E=0 to E=∞, not just from a certain energy state to infinity. Also, the integration should be done with respect to E, not Ef.

Finally, the equation for the probability of filling a hole (1-F(E)) should be e-((E-Ef)/kT), not e((E-Ef)/kT). This is because the probability of an empty energy state (i.e. a hole) being occupied by an electron is given by the Fermi-Dirac distribution function, which has an exponent with a negative sign.

In conclusion, I believe the main issue with the given proof is the incorrect use of exponents and integration limits. I hope this explanation helps in clarifying your doubts. If you have any further questions, please feel free to ask.