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- The wavefunction of one of the d orbitals is proportional to sin2θ sin 2ϕ. At what angles does it have nodal planes?(a) The nitrogen atom has one electron in each of the 2px,2py and 2pz orbitals. By using the form of the angularwave functions, show that the total electron density,c2(2px) +c2(2py) +c2(2pz), is spherically symmetric(that is, it is independent of the angles u and f). Theneon atom, which has two electrons in each 2porbital, is also spherically symmetric.(b) The same result as in part (a) applies to d orbitals,thus a filled or half-filled subshell of d orbitals isspherically symmetric. Identify the spherically symmetric atoms or ions among the following: F=, Na, Si,S2-, Ar+, Ni, Cu, Mo, Rh, Sb, W, Au.Consider the hydrogen atom in the ℓ = 3 state. Calculate the magnitude of L→, the allowed values of Lz, and the corresponding angles θ that L→ makes with the z axis.
- When β-carotene is oxidized in vivo, it breaks in half and forms two molecules of retinal (vitamin A), which is a precursor to the pigment in the retina responsible for vision. The conjugated system of retinal consists of 11 C atoms and 1 O atom. In the ground state of retinal, each level up to n = 6 is occupied by two electrons. Assuming an average internuclear distance of 140 pm, calculate (a) the separation in energy between the ground and first exciteted state in which one electron occupies the state with n = 7, and (b) the frequency of radiation required to produce a transition between these two states. From your results, correct the following sentence (from the options in brackets). The absorption spectrum of a linear polyene shifts to (higher/lower) frequency as the number of conjugated atoms (increases/decreases).P7E.10 Show that !, and l both commute with the hamiltonian for a hydrogen atom. What is the significance of this result? Begin by noting that -;+1;+1? . Then show that [1,1] = [!,!,]1,+ 4,LL ] and then use the angular momentum commutation relations in eqn 7F.14.Sketch the boundary surface of a dx2 −y2 and a py orbital. Be sure to show and label the axes.
- 9A.2 Write the valence bond wavefunction of the o bond in a C-H group of a molecule.3- Consider the atom having the electron configuration 1s 2s2 2p 3s 3p. Assume that the z components of both the orbital and spin angular momenta of the electron in the 3p subshell are positive. What are the quantum numbers that describe the state of this electron? On=3,1= 1, m = 1, s = 1/2 n = 3,1 = 2, m = 1, s = 1/2 On= 3,1= 2, m 2, s =-1/2 On=3,13D 1, m = 2, s = 1/2 On=3,1 1, m = -1, s = 1/2 %3D8C.4 (a) the moment of inertia of a CH4 molecule is 5.27 x 10^-47 kg m^2. What is the minimum energy needed to start it rotating? 8C.5 (a) use the data in 8C.4 (a) to calculate the energy needed excite a CH4 molecule from a state with l=1 to a state with l=2