Question

. Given that the activation energy for vacancy formation in Copper is 0.90 eV/atom, calculate the temperature (in Kelvin) at which 1 of every 10,000 Copper lattice sites is vacant.

Answer 1

Answer-

Let the total number of vacant site = P_i

And total number lattice site = P_j

Now, the probability of a site being vacant is = P_i / P_j

This ratio is called the Boltzmann factor and given by the relation -

  P_i / P_j = e^{-(E_j - E_i )/kT}

Here, E_i = energy of the state i

and E_j = energy of the state j

Now, E_j - E_i = energy required to remove the electrons from the lattice site = activation energy = 0.90 ev / atom

1 ev = 1.60218 x 10^-19 joule

0.90 ev = (0.90) x ( 1.60218 x 10^-19 ) = 1.442 x 10^-19 joule

So, activation energy = E_i - E_j = 1.442 x 10^-19 joule

Now we have to find the temperature at which 1 out of every 10000 copper lattice sites are vacant

So, P_i = 1

and P_j = 10000

Now put all these values in equation 1)-

1 / 10000 = e^-(1.442 x 10^-19 joule / k_BT)

k_B = 1.380649×10⁻²³ J⋅K⁻¹

1 / 10000 = e^{-( 1.442 x 10^-19 joule / 1.380649 x 10^-23 J/K x T)}

Take ln both side-

-9.2 = -(1.442 x 10^-19 joule) / (1.380649×10⁻²³ J⋅K⁻¹) x (T)

T = (1.442 x 10^-19 joule) / 9.2 x ( 1.380649×10⁻²³ J⋅K⁻¹)

T = 0.11352 x 10⁴ kelvin

T = 1135.2 Kelvin

So the required temperature is 1135.2 Kelvin