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Chapter 4: Problem 3

(a) Calculate the fraction of atom sites that are vacant for copper \((\mathrm{Cu})\) at its melting temperature of \(1084^{\circ} \mathrm{C}(1357 \mathrm{~K})\). Assume an energy for vacancy formation of \(0.90 \mathrm{eV} /\) atom. (b) Repeat this calculation at room temperature \((298 \mathrm{~K})\) (c) What is the ratio of \(N_{v} / N(1357 \mathrm{~K})\) and \(N_{v} / N\) \((298 \mathrm{~K}) ?\)

Short Answer

Expert verified
Based on the given information and the calculation, the ratio of the fractions of vacancies at 1357 K and 298 K for copper is approximately \(e^{27.38}\).

Step by step solution

01

Convert the energy for vacancy formation into Joules

The energy for vacancy formation is given in electron volts. Let's convert it to Joules using the conversion factor \(1.6 \times 10^{-19} \text{ J/eV}\): \(E_{v} = 0.90 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = 1.44 \times 10^{-19} \text{ J}\)
02

Calculate the fraction of vacant sites at 1357 K

Using the formula for N_v/N with the given values, we can calculate the fraction of vacant sites at 1357 K: \(N_{v} / N(1357 \text{K}) = Ce^{\left(-\frac{1.44 \times 10^{-19} \mathrm{J}}{1.38 \times 10^{-23} \mathrm{J/K} \times 1357 \mathrm{K}}\right)} = Ce^{-\frac{1.44 \times 10^{-19}}{1.873 \times 10^{-21}}} = Ce^{-7.68}\)
03

Calculate the fraction of vacant sites at 298 K

Similarly, we can calculate the fraction of vacant sites at 298 K: \(N_{v} / N(298 \text{K}) = Ce^{\left(-\frac{1.44 \times 10^{-19} \mathrm{J}}{1.38 \times 10^{-23} \text{J/K} \times 298 \mathrm{~K}}\right)} = Ce^{-\frac{1.44 \times 10^{-19}}{4.114 \times 10^{-21}}} = Ce^{-35.06}\)
04

Calculate the ratio of the fractions at 1357 K and 298 K

Finally, we are asked to find the ratio between these two fractions. The constant \(C\) will cancel out when finding the ratio: \(\frac{N_{v} / N(1357 \text{K})}{N_{v} / N(298 \text{K})} = \frac{Ce^{-7.68}}{Ce^{-35.06}} = e^{35.06 - 7.68} = e^{27.38}\) The ratio of the fractions of vacancies at 1357 K and 298 K is approximately \(e^{27.38}\).

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