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 <em>Arrhenius equation</em>.
In logarithmic form, the equation is
ln(<em>N</em>_2/<em>N</em>_1) = (-<em>Q</em>/<em>R</em>)(1/<em>T</em>_2-1/<em>T_</em>1)
where
• Q = the energy required for vacancy formation
• <em>N</em>_2 = the vacancy concentration at <em>T</em>_2
• <em>N</em>_1 = the vacancy concentration at <em>T</em>_1
• <em>R</em> = the gas constant [8.314 J·K^(-1)mol^(-1)]
Let <em>N</em>_80 represent the vacancy concentration at 80 °C.
At 25 °C, ln(<em>N</em>_25/<em>N</em>_80) = ln(0.25<em>N</em>_80/<em>N</em>_80) = ln0.25 = -1.386
∴ -1.386 =(-<em>Q</em>/<em>R</em>)(1/298.15 – 1/353.15) = -1.306 × 10^(-4) × (<em>Q</em>/<em>R</em>)
<em>Q</em>/<em>R</em> = (-1.386)/[-1.306 × 10^(-4)] = 10 620
At <em>T</em>_2, ln(<em>N</em>_<em>T</em>2/<em>N</em>_80) = ln[(3<em>N</em>_80)/<em>N</em>_80] = ln3 = 1.099
∴ 1.099 = -10 620(1/<em>T</em>_2 – 1/353.15) = -10 620/<em>T</em>_2 + 10 620/353.15
= -10 620/<em>T</em>_2 + 30.072
10 620/<em>T</em>_2 = 30.072 – 1.099 = 28.97
<em>T</em>_2 = 10 620/28.97 = 366.4 K = 93 °C
