Chemistry 4A, Fall 2025
Problem Set 7
1. In this problem,you will use the mathematical expressionsofthed orbitals togaina deeper understanding of crystal field theory. up toa constant, the angular part of the wavefunctions are:
( but that isnot relevant to this problem.)
a. show that the above functions depend only on θ and φ,use the conversion formulas from cartesian to spherical coordinates that we learned earlier.
b consider an octahedralcrystalfield with negative chargespositioned at
ri = (1, 0,0); r2 = (-1,0,0) ; r3 = (0, 1,0) ; r4 = (0, 1,0) ; r5 = (0,0, 1) ; rs = (0,0, -1).
Note the bold risavector quantity,not tobe confused with r,which isanumber. Evaluate the square ofthe angular function dx2-y2 at each of the six points, and add up the six numbers. This can be expresse
This number represents the total repulsion from the crystal field experienced by electrons in this orbital, which raises the energy.
c. Repeat the exercise for dz 2 and show that this equals your answer from part b.
Also show that for dxy, dxz, dyz this gives an answer ofzero. This result explains why dz 2 and dx2-y2 are equal in energy and higher than dxy, dxz, dyz.
d. Repeat parts(b)and(c)fora tetrahedral crystalield with negative charges positioned at four corner sofa cube as:
ri = (1, 1, 1); r2 = (-1,-1, 1) ; r3 = (-l, 1,-1) ; r4 = (1,-1,-1)
How arethe d orbitals arranged, andhow does it differ from the octahedral field?
2. Chromium hexacarbonyl Cr(CO)6, iron pentacarbonyl Fe(CO)5 and nickel tetracarbonyl Ni(CO)4 are all stable and neutral coordination complexes.
a. Using the 18-electron rule, explain why these compounds have different numbers of CO ligands.
b. Predict the charge on iron hexacarbonyl Fe(CO)6 assumingthe 18-electron ruleis satisfied.
c. The neutral diiron nonacarbonylcomplex is shown on the right. If each CO ligand donates two electrons, is the 18-electron rule satisfied? Name onepossibility for making up for a shortagein the number of electrons
3. (problem30from chapter8of textbook)For each of the following ions, draw diagrams like the above to show orbital oc cup anc ie sin both weak and strong octahedral fields. Indicate the total number of unpaired electrons in each case.
a. cr2+ b. V3+ c. Ni2+ d. pt4+ e. co2+
4. Assign oxidation numbers to each element in the following compounds:
a. cr2o;2-
b CUSO4
c. (NH4)2MOO4 (Hint:NH4+ has a positive charge.)
d. Formaldehyde,CH2o
Note:Hydrogen has a+1oxidationnumber, except whenbonded to itself(0),or when bonded to an alkali metal(-1).
5. This problem will focus on the kinetic theoryofgases.Hydrogen(H2)is oneofthe lightestgaseswhereas uranium hexafluoride(UF6, shown below)is oneofthe heaviest.
a. calculate the molar mass of uranium hexafluoride.
b. Find the ratio of the molar masses of uranium hexafluoride to hydrogen gas.
c. Using the kinetic theory of gases, find the root mean square velocity of a UF6 molecule and a H2 molecule at standard conditions (25。C, 1 atm) in m/s.
d. Find the ratio ofthe root mean square velocities ofuFsgastoH2 gas.
e. Now take the square ofyour answer from part(d). It shouldmatch the reciprocal ofyour answer from part(b).Thus, the root mean square velocity ofthe gas is inversely proportional to the square root of its molar mass.
f. The Ideal Gas Law predicts the volume that a given amount of gas would occupy at room temperature(use25c, same as part(c)). calculate them as sofa 1.00L sample ofH2 anda 1.00L sample ofuF6 at these conditions. Also estimate the mas sofa 1.00L sample ofliquid water atthe same conditions. Rank the masses of the three samples and write a complete sentence to describe your result.
g. uranium enrichment is a process by which u-235is separated fromu-238by centrifugation ofuF6 gas.calculate the percentage difference in the density ofuF6 gas with u-235vs.u-238atthe same temperature and pressure.(when computing the percentage difference, use the density ofu-238asthe denominator.)
6. (optionalfor10points extra credit)TheLennqrd-Jonespotentialpictured below is a way to model the dispersion interaction in atoms and molecules. In xenon(xe), the parameters for the interaction are given byσ =4.10A, ε = 1.77KJ/mol.
a. calculate the minimum energy distance between two xenon atoms. you will need to take a derivative.
b. calculate the total energy ofa2 × 2 × 2cube of xenon atoms with one atoma teach corner;the cube edge-length is given byyour answer from part(a). provide your answer ink J/mol.
c. Repeat this fora3 × 3 × 3cube. you may need to write a computer program.
d. calculate the total kinetic energy that eight(2 × 2 × 2)xenon atoms would have at room temperature, assumingthat eachatom has kinetic energy equal tothe average kinetic energy from the kinetic theory of gases ·
e. Bycomparing your answers from parts(b)and(d),wouldyouexpectthexenon cube to stay together at room temperature(remaining a solid), or dissociate into individual atoms(turning into a gas)? Is this consistent with the experimental boiling point of165.0k?