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?

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="https://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="https://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}">

<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="https://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="https://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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