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Chapter 4: Problem 1

Calculate the fraction of atom sites that are vacant for lead at its melting temperature of \(327^{\circ} \mathrm{C}(600 \mathrm{~K})\). Assume an energy for vacancy formation of \(0.55 \mathrm{eV} /\) atom.

Short Answer

Expert verified
Answer: The fraction of atom sites that are vacant for lead at its melting temperature of 600 K is approximately \(2.47 \times 10^{-5}\).

Step by step solution

01

Write down the Arrhenius equation for vacancy concentration

The Arrhenius equation for the equilibrium vacancy concentration is given by: \(n_v = N \cdot \exp\left(\dfrac{-E_{\text{v}}}{k_B T}\right)\) where \(n_v\) is the number of vacancies, \(N\) is the total number of atomic sites, \(E_{\text{v}}\) is the vacancy formation energy (0.55 eV/atom in our case), \(k_B\) is the Boltzmann constant (\(8.617 \times 10^{-5} \text{ eV/K}\)), and \(T\) is the temperature in Kelvin (600 K for our case).
02

Convert the energy of vacancy formation to eV

Since the energy of vacancy formation is already given in eV, we don't need to make any conversions: \(E_{\text{v}} = 0.55 \text{ eV/atom}\)
03

Calculate the exponential term in the Arrhenius equation

We can now evaluate the exponential term in the Arrhenius equation: \(\exp\left(\dfrac{-E_{\text{v}}}{k_B T}\right) = \exp\left(\dfrac{-0.55}{8.617 \times 10^{-5} \cdot 600}\right) = \exp(-10.61) \approx 2.47 \times 10^{-5}\)
04

Calculate the fraction of vacancies

Finally, we can express the fraction of vacancies by dividing the number of vacancies by the total number of atomic sites: \(\text{Vacancy fraction} = \dfrac{n_v}{N} = 2.47 \times 10^{-5}\) Hence, the fraction of atom sites that are vacant for lead at its melting temperature of 600 K is approximately \(2.47 \times 10^{-5}\).

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Determine the approximate density of a highleaded brass that has a composition of \(64.5\) \(\mathrm{wt} \% \mathrm{Cu}, 33.5 \mathrm{wt} \% \mathrm{Zn}\), and \(2 \mathrm{wt} \% \mathrm{~Pb}\).

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