Question

Consider copper from 0 degrees Celsius to 1000 degrees Celsius. The density for copper at 0 degrees is 8.92 g/cm^3. Assume the vacancy formation energy is 0.91 eV/atom and that the volume coefficient of thermal expansion is 3 times the the linear coefficient of thermal expansion. 1) Determine the density change between these two temperatures due to the change in the number of vacancies.

Thermal expansion coefficient of copper is 9.3 (10-6 in/(in R))*) OR 16.6 (10-6 m/(m K))

Answer 1

We know that density changes with temperature as follows

.........(1)

= density at 0 degree = 8.92 gm.cm^-3

= density at t degree

= coefficient of volume expansion = 3*16.6*10^-6 m/mK

C or K

Putting these values in equation (1) and simplifying,

  ........(2)

Now, we know the formula .........(3) , where N = atomic sites/volume , A = molar mass , =density

is Avogadro's number.

For copper, A=63.5 g/mol

N_A = 6.02*10²³

Putting these values in eqn (3),

N = = 8.5*10²² atomic sites/cm³

Now, number of vacancies per unit volume is given by the equation

........(4)

Q_v = vacancy formation energy , here Q_v = 0.91 eV = 0.91*1.6*10^-19 J

k = Boltzmann's constant , T = temperature in Kelvin scale , here T= 1000 + 273 K = 1273 K

Putting these values in equation (4) and simplifying,

N_v = 2.85*10¹⁹ / cm³

We now go back to equation (3), to calculate density of copper at 1000 degree Celsius.

Now, you can again calculate dnsity from quation (3) and compare it with the same in equation (2) .

Finally, you can calculate the change in these two dnsities.