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Answer on Solid State Physics Question for JOEL LOBO

Question #2151
The fraction of vacancy sites in a metal is 1X10-10 at 500C. What will be the
fraction of vacancy sites at double the temperature?
Expert's answer
The fraction of vacant sites is given by the following expression:

<img src="/cgi-bin/mimetex.cgi?%5Cfrac%7BN_%7B%5Cnu%7D%7D%7BN%7D%20=%20e%5E%7B%5Cleft%20%28%20-%5Cfrac%7BQ_%7B%5Cnu%7D%7D%7BkT%7D%20%5Cright%20%29%7D" title="\frac{N_{\nu}}{N} = e^{\left ( -\frac{Q_{\nu}}{kT} \right )}">

At 500C we can find the activation energy :
<img src="http://latex.codecogs.com/gif.latex?Q_%7B%5Cnu%7D%20=%20%5Cln%20%7B%5Cfrac%7BN_%7B%5Cnu%7D%7D%7BN%7D%7DkT%20=%20-%5Cln%7B1%20%5Ctimes%2010%5E%7B-10%7D%7D%5Ccdot%208.31%5Ccdot%28500+273%29%20=%20147909.5%20%5C%20J/mol" title="Q_{\nu} = \ln {\frac{N_{\nu}}{N}}kT = -\ln{1 \times 10^{-10}}\cdot 8.31\cdot(500+273) = 147909.5 \ J/mol">

At double temperature 1000C = 1000+273 = 1273K the fraction fo vacant sites is:
<img src="http://latex.codecogs.com/gif.latex?%5Cfrac%7BN_%7B%5Cnu%7D%7D%7BN%7D%20=%20e%5E%7B%5Cleft%20%28%20-%5Cfrac%7B147909.5%7D%7B8.31%5Ccdot1273%7D%20%5Cright%20%29%7D%20=%200.85%20%5Ctimes%2010%5E%7B-8%7D" title="\frac{N_{\nu}}{N} = e^{\left ( -\frac{147909.5}{8.31\cdot1273} \right )} = 0.85 \times 10^{-8}">

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