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Crystal defects and diffusion

The diffusion coefficient of an element was 1.0×10⁻¹⁴ m²/s at 800 K and 4.0×10⁻¹³ m²/s at 1000 K. Assuming the Arrhenius form D = D₀exp(−Q/RT), which value is closest to the activation energy Q? Take R = 8.314 J/(mol·K).

  1. AAbout 53 kJ/mol
  2. BAbout 123 kJ/mol
  3. CAbout 245 kJ/mol
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Answer: B. About 123 kJ/mol

Taking the ratio of the expressions at the two temperatures eliminates D₀ and gives ln(D₂/D₁) = (Q/R)(1/T₁ − 1/T₂). D₂/D₁ = 40, ln 40 ≈ 3.689 and 1/800 − 1/1000 = 2.5×10⁻⁴ K⁻¹, so Q = 8.314 × 3.689 / (2.5×10⁻⁴) ≈ 1.23×10⁵ J/mol, about 123 kJ/mol. About 53 kJ/mol uses the common logarithm (log₁₀ 40 ≈ 1.602) instead of the natural logarithm, and about 245 kJ/mol is double the value, from confusing the expression with the form exp(−Q/2RT). In practice, ln D is plotted against 1/T and Q is obtained from the slope −Q/R of the straight line.

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