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Due: Thursday, Feb 06

Chemistry 150 Problem Set 1

1) a) Consider a face centered cubic array of chloride ions with sodium atoms occupying

the octahedral holes (i.e. the NaCl structure type). The diagram below shows a planar {100} slice of the NaCl structure with four chloride ions in contact with each other as well as the central sodium ion. Using rx to denote the radius of the chloride ions and rm to denote the sodium ion radius, derive the necessary radius ratio (rm/rx) for all nearest neighbor atoms to remain in contact.

b) Would a structure be more likely to adopt the NaCl structure type if its rm/rx radius

ratio were larger or smaller than the one you calculated above? Explain briefly what a smaller or larger ratio means physically.

c) Derive the minimum radius ratio (rm/rx) possible for a cation in contact with eight

anions in the CsCl structure type.

d) Which is more likely to follow the radius ratio rules, AuCl or KCl? Why? 2) Powder x-ray diffraction is an invaluable tool for analysis of solid-state structure and

purity. In this technique, a polycrystalline sample is exposed to a beam of x-rays. Planes of atoms within the sample will diffract the beam when the Bragg equation is satisfied (2dsinθ = nλ) where d is the distance between lattice planes, θ is the angle between the incoming x-ray beam and the diffraction plane, λ is the wavelength of the radiation source, and n is the order of the diffraction.

Plutonium has been called “the most complex element on the periodic table”1 because six different crystalline phases have been reported at various temperatures. For instance, the δ phase of plutonium metal is a face centered cubic crystalline solid that is stable between ~600 to 700 °C. Shown below is the powder x-ray diffraction pattern of δ phase plutonium.

1 Söderlind, Per; Sadigh, Babak. Phys. Chem. Rev. (2004) 92.

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Powder X-ray diffraction Pattern of Plutonium (delta phase)

0 10 20 30 40 50 60 70 80 90 100

2θ (deg)

Inte

nsity

ty

2θ = 38.80°

2θ = 83.28°

2θ = ??°

a) Draw a unit cell for δ-Pu and sketch in a (111), (100), and (110) lattice plane.

b) Fill in the missing data in the following table where 2θ is the full diffraction angle, d (Ǻ) is the distance between planes and h, k, and l are Miller indices. [NOTE: all peaks are the result of first order diffraction (n=1), and the x-ray source is CuKα1 (λ = 1.540598 Å)].

2θ (deg) d (Ǻ) h k l

1 1 1 38.80 2 0 0 83.28 0 0

c) What is the volume of the cell? d) The (200) peak is the second largest peak for δ-Pu. Where are the peaks for the (020), (002), (02 0), and (002 ) sets of lattice planes? e) Based on the atomic arrangement of the fcc cell, make a qualitative argument as to

why the (111) peak is more intense than the (200) peak.

f) Calculate the crystallographic density of δ-Pu in units of g/cm3. 3) A tube furnace with a platinum heating element is heated to very high temperatures for

long periods of time under an atmosphere of O2. Eventually platinum crystals begin to form near the outer edges of the furnace. Write a chemical equation and brief explanation of why the crystals are forming. Your explanation should include the van’t Hoff equation.

Pt(s) + O2(g) PtO2(g)

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4) Consider the following (unbalanced) solid state reaction:

a) In the product, oxygen atoms form an fcc lattice with Mg occupying one eighth of the tetrahedral sites and Fe occupying half of the octahedral sites. Given this information, what is the correct formula for the product compound?

b) Despite a strong thermodynamic favorability, this reaction does not occur under

standard conditions. Why is this, and what synthetic methods could be employed to make the reaction occur?

c) The diagram below shows the interface between two reactant crystals before, and

after, the reaction has begun. Write balanced equations for the reactions at each interface such that they add to form the overall reaction. Be sure to account for the ions being transported through the product lattice to the other interface by subtracting them from your equations.

d) The rates of product growth at the interfaces are not equal; what ratio would you

predict between the rates of growth at interface I and interface II?

5) Consider an infinite one-dimensional chain of equally spaced hydrogen atoms. The band diagram for such a structure is shown below. (Note: the unit cell has been chosen to include 2 hydrogen atoms).

MgO(s) + Fe2O3(s) MgxFeyO4 (s)

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a) Write the form of χn for for the bands labeled i) and ii) using the Hückel approximation (hint: these are just the molecular orbitals of H2).

b) What is the range of k in the band diagram?

c) Using a representative three unit cells for each, draw in the phases of the 1s orbitals at locations i, ii, and iii on the band diagram

d) In reality, hydrogen does not form infinite one dimensional chains. Draw a new band diagram showing the effect of shortening one H-H bond and lengthening the other to form separate H2 molecules.

e) How is this distortion similar to what happens to copper (II) in an octahedral environment?

6) Consider a linear chain of eclipsed PtH4

2- units as depicted below.

a) The diagram to the left depicts the bands of

some of the orbitals of the platinum metal. Label the orbitals which compose each of the bands shown. (Hint: only bands corresponding to the platinum d orbitals and the platinum p(z) orbital are shown).

b) Why do some bands run up from k = 0 to k = π/a and some run down?

c) Which d orbital band has the most dispersion

(largest bandwidth)? Explain this in terms of its bonding.

d) Using four repeat units of the structure, draw

the p(z) band at k = 0 and k = π/a.

e) Where is the Fermi level for this one dimensional structure? Would you predict it to be a conductor or an insulator?

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f) Say the platinum is partially oxidized (~0.3 less electrons per Pt on average). How would this change the conductive properties? Would you predict the Pt – Pt distance to increase, decrease, or remain the same?

7) Consider an infinite two dimensional sheet of hydrogen atoms. a) On separate 4 x 4 grids draw pictorial representations of the 1s orbital phases at Г, Х,

and M.

b) Order Х, Г, and M from lowest to highest energy.

c) Label them as bonding, non-bonding, or antibonding. 8) Decide whether or not the following systems are likely to be p-type or n-type semiconductors.

a) As-doped Ge

b) Ga-doped Ge c) Si-doped Ge