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Chapter 10: Q55E (page 470)

Carry out the integration indicated in equation (10.10)

Short Answer

Expert verified

The integration is obtained and proved.

Step by step solution

01

Determine the formulas

Consider the formulato determine the thermal energy as:

E=knT 鈥�.. (1)

Here,kn is Boltzmann constant and the temperature is T.

02

Solve for the integration as:

Solve for the integration as:

Nexcited=鈭�EF+12Egap鈭�1e(E鈭�EF)/knT+1DdE

Considerx=EknT then,dx=dEknT .

Consider xF=EFknTandxgap=EgapknT

Substitute the values in the integration and solve as:

Nexcited=DkBT鈭�xF+12xgap鈭�1exF鈭�xgap+1dx=DkBT鈭�xF+12xgap鈭�exF鈭�xgapexF鈭�xgap+1dx

Solve for the derivative of the function as:

dln(1+exf鈭�xgap)dx=11+exf鈭�xexp鈭�x(鈭�1)

Solve the integration further as:

Nexcited=鈭�DkBTln(1+exF鈭�x)xF+12gap鈭�=DkBTln(1+eEgap2kBT)

Hence, the integration is obtained and proved.

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