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**Unformatted text preview: **CHAPTER 4 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS 4.1 In order to compute the fraction of atom sites that are vacant in lead at 600 K, we must employ Equation (4.1). As stated in the problem, Q v = 0.55 eV/atom. Thus, N V N = exp − Q V kT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = exp − 0.55 eV /atom 8.62 x 10 − 5 eV /atom -K ( ) (600 K) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 2.41 x 10-5 4.2 Determination of the number of vacancies per cubic meter in gold at 900 ° C (1173 K) requires the utilization of Equations (4.1) and (4.2) as follows: N V = N exp − Q V kT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = N A ρ Au A Au exp − Q V kT ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 6.023 x 10 23 atoms/mol ( ) 19.32 g/ cm 3 ( ) 196.9 g/mol exp − 0.98 eV / atom 8.62 x 10 − 5 eV /atom − K ( ) (1173 K) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 3.65 x 10 18 cm-3 = 3.65 x 10 24 m-3 4.3 This problem calls for the computation of the energy for vacancy formation in silver. Upon examination of Equation (4.1), all parameters besides Q v are given except N , the total number of atomic sites. However, N is related to the density, ( ρ ), Avogadro's number ( N A ), and the atomic weight ( A ) according to Equation (4.2) as N = N A ρ Ag A Ag = 6.023 x 10 23 atoms /mol ( ) 10.49 g/cm 3 ( ) 107.87 g/mol 1 = 5.86 x 10 22 atoms/cm 3 = 5.86 x 10 28 atoms/m 3 Now, taking natural logarithms of both sides of Equation (4.1), and, after some algebraic manipulation Q V = − RT ln N V N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − 8.62 x 10-5 eV/atom - K ( ) (1073 K) ln 3.60 x 10 23 m − 3 5.86 x 10 28 m − 3 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = 1.11 eV/atom 4.4 In this problem we are asked to cite which of the elements listed form with Cu the three possible solid solution types. For complete substitutional solubility the following criteria must be met: 1) the difference in atomic radii between Cu and the other element ( ∆ R% ) must be less than ±15%, 2) the crystal structures must be the same, 3) the electronegativities must be similar, and 4) the valences should be the same, or nearly the same. Below are tabulated, for the various elements, these criteria. Crystal ∆ Electro- Element ∆ R% Structure negativity Valence Cu FCC 2+ C -44 H -64 O -53 Ag +13 FCC 0 1+ Al +12 FCC -0.4 3+ Co -2 HCP -0.1 2+ Cr -2 BCC -0.3 3+ Fe -3 BCC -0.1 2+ Ni -3 FCC -0.1 2+ Pd +8 FCC +0.3 2+ Pt +9 FCC +0.3 2+ Zn +4 HCP -0.3 2+ 2 (a) Ni, Pd, and Pt meet all of the criteria and thus form substitutional solid solutions having complete solubility. (b) Ag, Al, Co, Cr, Fe, and Zn form substitutional solid solutions of incomplete solubility. All these metals have either BCC or HCP crystal structures, and/or the difference between their atomic radii and that for Cu are greater than ±15%, and/or have a valence different than 2+. ...

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