Question

Calculate the concentration of vacancies in copper at room temperature (25oC). What temperature will be needed to heat treat copper such that the concentration of vacancies produced will be 1000 times more than the equilibrium concentration of vacancies at room temperature? Assume that 20,000 cal are required to produce a mole of vacancies in copper.

Answer 1

Answer:

1) 1.808 *10^8 vacancies per cm^3

2) 101,64ºC

Explanation:

You can use the following equation relating the number of vacancies with the energy required to produce them and the temperature:

$n_{v} =Ne^{\frac{-E}{K_{b}*T} }$

Where N is the number atoms per unit of volume, you can calculate this number with the density of copper:

$8.96 \frac{g}{cm^{3} } *\frac{1 mole copper}{63.54 g } *\frac{6.023*10^{23} atoms }{1 mole} = 8.47*10^{22}atoms$

And you can substitute all values in the equation:

$n_{v} =8.47*10^{22}atoms*e^{\frac{-20 000 cal}{6.023*10^{23}mol^{-1}*3.297*10^{-24}cal*k^{-1}*298.15K} }=1.808*10^{8}atoms$

Now you solve for temperature and and use n as 1000 times the value of before:

$T=\frac{-E}{k_{b}ln(n/N)} =\frac{-(2*10^{4}cal)}{(3.297*10^{24}calk^{-1})*ln(1.8083*10^{11}/8.47*10^{22})} =374.79 K = 101.64 C$