The number of vacancies in some hypothetical metal increases by a factor of 3 when the temperature is increased from 1020 ˚c to 1290 ˚c. calculate the energy for vacancy formation (in j/mol) assuming that the density of the metal remains the same over this temperature range.

first step here is to substitute the 3 of your two equations into the second;

3 Ne^(-Q_v/k(1293)) = Ne^(-Q_v/k(1566))

Since 'N' is a constant, we can remove it from both sides.

We also want to combine our two Q_v values, so we can solve for Q_v, so we should put them both on the same side:

3 = e^(-Q_v/k(1293)) / e^(-Q_v/k(1566))

3 = e^(-Q_v/k(1293) + Q_v/k(1566) ) (index laws)

ln (3) = -Q_v/k(1293) + Q_v/k(1566) (log laws)

ln (3) = -0.13Q_v / k(1566) (addition of fractions)

Q_v = ln (3)* k * 1566 / -0.13 (rearranging the equation)

Now, as long as you know Boltzmann's constant it's just a matter of substituting it for k and plugging everything into a calculator.