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Structure.questions_VIII_answer -...

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MTLE-2100 Structure of Engineering Materials , Questions VIII--Answers 1. What is the ratio of V Al ’’’ and V O •• of the Schottky defect in Al 2 O 3 ? To maintain the charge neutrality, 2 Al vacancy, 2 V Al ’’’ and three oxygen vacancy, 3V O •• , will be created. 2. Take the derivative of the following equation with respect to n and set it to equal to zero to obtain n/ NN’ = exp (-g/2kT). G = H - T S = ng - T kln{N!/[(N-n)!n!]}{N’!/[(N’-n)!n!]} By using the Stirling’s approximation, G= ng -kT{NlnN-N -(N-n)ln(N-n) +(N-n) -nln n +n
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Unformatted text preview: +N’lnN’ -N’ -(N’-n)ln(N’-n) +(N’-n) -nln n+n} ∂∆ G/ ∂ n = g -kT{ln(N-n) +1 -1-ln n -1+1 ln(N’-n) +1 -1 -ln n -1 +1} = g -kTln{(N-n)(N’-n)/n 2 } =0 (N-n)(N’-n)/n 2 = exp (g/kT) Since N, N’ >>n NN’/n 2 = exp (g/kT) n/ √ NN’ = exp (-g/2kT) 3. Consider TiO 1.95 . Or what fraction of oxygen is vacant? What fraction of titanium is in Ti 3+ state? Oxygen vacancy 0.05/2 = 0.025 Assume x fraction of Ti is Ti 3+ , 1-x fraction is Ti 4+ . 3x + 4(1-x) = 2 times1.95 4 -x = 3.90 x= 0.1...
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