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<h2 class="hd hd-2 unit-title">Practice 1.1</h2>
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Concentration of Vacancy
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Helium is unique among the elements in that it remains a liquid at 0K at atmospheric pressure. At pressures above about 20 atm, solid BCC helium forms, but the small mass of the helium atoms and the weak bonding between atoms gives rise to defects in the solid even at 0K. This quantum mechanical result is due to the presence of &#8220;zero-point energy&#8221; which is sufficient to produce vacancies in the crystal. The classical formula for the equilibrium number of vacancies in a crystal, however, requires that at T = 0K, the concentration of vacancies is zero:
\[
N_v = N exp(-\frac{Q_v}{kT})
\]
The concentration of defects in solid helium can be described by using an equivalent classical solid at a temperature T that corresponds to its melting point.
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<p>If the zero-point energy of the helium crystal is 1.3 meV and the melting point of the equivalent classical solid is 1.5 K, calculate a lower bound on the concentration of vacancies in solid helium at 0K.</p>
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<h2 class="hd hd-2 unit-title">Practice 1.2</h2>
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Concentration of Defects
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>The energy of vacancy formation for copper at 1000&#176;C is 0.9 eV/atom. By what factor does the concentration of defects increase from room temperature (23&#176;C) to 1000&#176;C?</p>
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<h2 class="hd hd-2 unit-title">Practice 1.3</h2>
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Forging Steel
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>During the forging process, a piece of high strength steel is exposed to a 900&#176;C environment containing gaseous hydrogen. If the surface of the steel has a steady hydrogen concentration of 0.01 wt% (100 ppm), what will be the concentration 1 mm from the surface after 1 hour? Assume the diffusion coefficient for hydrogen in steel at 900&#176;C is \(0.9\times10^{-7}m^{2}/s\).
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<h2 class="hd hd-2 unit-title">Practice 1.4</h2>
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Hydrogen Concentration
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Hydrogen concentrations of 1 ppm and above in steel may lead to crack formation and catastrophic failure (hydrogen embrittlement). During the forging process, a piece of high strength steel is exposed to a 900&#176;C environment containing gaseous hydrogen introduced by the presence of water vapor.</p>
<p>What is the maximum concentration of hydrogen on the surface of the steel that will prevent hydrogen embrittlement if the steel is to be subjected to the environment for 0.1 hour and if just over 10 mm of material from the surface is to be removed by a surface treatment? Assume the diffusion coefficient for hydrogen in steel at 900&#176;C is \(0.9\times10^{-7}m^{2}/s\).</p>
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<h2 class="hd hd-2 unit-title">Practice 1.5</h2>
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Practice 1.5
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>A membrane is to be manufactured to the following specifications. At \(700 ^{\circ} C\) the leak rate of hydrogen at steady state is not to exceed \(10^{3} mol \cdot cm^{-2} hr^{-1}\) when the concentrations of hydrogen are maintained at \(C_s^{high}\) on one side of the membrane and \(C_s^{low}\) = effectively zero on the other side. What is the minimum thickness, \(d\), of iron that will meet these requirements? The diffusion coefficient of atomic hydrogen in iron, \(D_H\) at \(700^{\circ}\) is \(3.091 \times 10^{-4} cm^2s^{-1}\). Express your answer in units of \(cm\).</p>
<p>This problem is taken from MIT Open Courseware, 3.091SC Introduction to Solid State Chemistry. CC BY-SA-NC 4.0</p>
<p>\(d\) (in \(cm\)):</p>
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Practice 1.6A
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<p>Determine the vacancy formation energy for each of the solids (one mole of each) with Arrhenius plots below. Provide your answer in eV/atom.</p>
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Practice 1.6B
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<p>Referring to the plot below, calculate the approximate number of vacancies in one mole of nickel, aluminum, and cadmium at 727&#176;C (1000 K). What percentage of the atomic sites are vacancies?</p>
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