In a certain crystalline material the vacancy concentration at 25 c is one-fourth that at 80

c. at what temperature would the vacancy concentration be 3 times that at 80 c?

At 93 °C, the vacancy concentration will be three times that at 80°C.

The formula for the vacancy concentration in a crystal is a form of the Arrhenius equation.

In logarithmic form, the equation is

ln (N_2/N_1) = (-Q/R) (1/T_2-1/T_1)

where

• Q = the energy required for vacancy formation

• N_2 = the vacancy concentration at T_2

• N_1 = the vacancy concentration at T_1

• R = the gas constant [8.314 J·K^ (-1) mol^ (-1) ]

Let N_80 represent the vacancy concentration at 80 °C.

At 25 °C, ln (N_25/N_80) = ln (0.25N_80/N_80) = ln0.25 = - 1.386

∴ - 1.386 = (-Q/R) (1/298.15 - 1/353.15) = - 1.306 * 10^ (-4) * (Q/R)

Q/R = (-1.386) / [-1.306 * 10^ (-4) ] = 10 620

At T_2, ln (N_T2/N_80) = ln[ (3N_80) / N_80] = ln3 = 1.099

∴ 1.099 = - 10 620 (1/T_2 - 1/353.15) = - 10 620/T_2 + 10 620/353.15

= - 10 620/T_2 + 30.072

10 620/T_2 = 30.072 - 1.099 = 28.97

T_2 = 10 620/28.97 = 366.4 K = 93 °C