Inorganic Chemistry - Final Examination

Provide the energy formula for a one-dimensional particle-in-a-box.

E = (n²h²)/(8m_eL²)

Provide the wavefunction equation for a one-dimensional particle-in-a-box.

|ψ_n(x)⟩ = √(2/L)sin(nπx/L)

Provide the probability equation for a one-dimensional particle-in-a-box.

P_n = |ψ_n(x)⟩² = (2/L)sin²(nπx/L)

What does the radial solution for hydrogen of |1s⟩ look like?

• Entire function in Quadrant I.

• Function begins at y → ∞.

• Function approaches y → 0.

What does the radial solution for hydrogen of |2s⟩ look like?

• Function in both quadrants I and IV.

• Function begins at y → ∞.

• Function possesses a single node (+ → -).

• Function approaches y → 0.

What does the radial solution for hydrogen of |2p⟩ look like?

• Function in Quadrant I.

• Function begins at y = 0.

• Function increases then approaches y → 0.

What does the radial solution for hydrogen of |3s⟩ look like?

• Function in both quadrants I and IV.

• Function begins at y → ∞.

• Function possesses two nodes (+ → -, and - → +).

• Function approaches y → 0.

What does the radial solution for hydrogen of |3p⟩ look like?

• Function in both quadrants I and IV.

• Function begins at y = 0.

• Function possesses one nodes (+ → -).

• Function approaches y → 0.

What does the radial solution for hydrogen of |3d⟩ look like?

• Function in Quadrant I.

• Function begins at y = 0.

• Function increases then approaches y → 0.

For a radial solution of hydrogen, what is the equation to determine the number of nodes?

#nodes = n - ℓ - 1

What is the shape of the s orbital? List any nodes or node planes.

Spherical shape with no nodes.

What is the shape of the p_x orbital? List any nodes or node planes.

Dumbbell shape where p_x projects along the x-axis. There is a node plane along the yz-plane.

What is the shape of the p_y orbital? List any nodes or node planes.

Dumbbell shape where p_y projects along the y-axis. There is a node plane along the xz-plane.

What is the shape of the p_z orbital? List any nodes or node planes.

Dumbbell shape where p_z projects along the z-axis. There is a node plane along the xy-plane.

What is the shape of the d_xy orbital? List any nodes or node planes.

Cloverleaf shape where the node planes lie along the xz and yz planes.

What is the shape of the d_yz orbital? List any nodes or node planes.

Cloverleaf shape where the node planes lie along the xz and xy planes.

What is the shape of the d_xz orbital? List any nodes or node planes.

Cloverleaf shape where the node planes lie along the xy and yz planes.

What is the shape of the d_x2-y2 orbital? List any nodes or node planes.

Cloverleaf shape with the same shape as d_xy but rotated 45 degrees.

What is the shape of the d_z2 orbital? List any nodes or node planes.

Dumbbell shape projected along the z-axis with a torus shape projected around the z-axis at the origin. The node planes are formed by two conical planes projecting along the z-axis converging at the origin.

What are the four quantum numbers, and what values may they possess?

Principal quantum number (n = 1, 2, 3, …)

Angular momentum number (ℓ = 0 to n-1 corresponding to n)

Magnetic quantum number (m_ℓ = -ℓ to 0 to +ℓ)

Spin quantum number (m_s = +1/2, -1/2)

Provide and describe the three rules for electron occupation of orbitals.

1. Aufbau principle

Electrons will fill the lowest energy orbital first to achieve a ground state, with the diagonal rule describing the order.

2. Pauli exclusion principle

Two electrons may not possess the same four quantum numbers in the same atom, visualised by orbital filling diagrams where electron spin is depicted with up or down arrows.

3. Hund’s rule of maximum multiplicity

Electrons will fill the ground state to achieve a maximum total spin, such that electrons will fill orbitals to achieve the maximum number of unpaired electrons before pairing.

What is the formula for calculating multiplicity?

M = 2S + 1

• M is multiplicity

• S is total spin, where S = Σm_s

What is the formula for calculating total pairing energy?

Π = nΠ_c + mΠ_e

• Π is total pairing energy

• Π_c is Coulombic repulsion energy (+)

• n is number of Coulombic repulsions

• Π_e is exchange interaction energy (-)

• m is number of exchange interactions

What value should the total pairing energy be for the ground state of an atom?

The minimum (most negative) energy value.

What is the formula for the effective nuclear charge?

Z_eff = Z - S

• Z_eff is effective nuclear charge

• Z is nuclear charge

• S is shielding constant

What are Slater’s four rules (method) to calculate the shielding constant?

1. Reorder and group orbitals:

(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)(5d)(5f)(6s,6p)(6d)(6f)

2. Electrons above in order will not shield those below.

3. If concerned with an s or p orbital:

Each electron in the same n grouping contributes 0.35 (except 1s, which contributes 0.30). Each electron in the n-1 groupings contributes 0.85. Each electron in the n-2 and below groups contributes 1.00.

4. If concerned with a d or f orbital:

Each electron in the same n grouping contributes 0.35. Each electron in the n-1 and below groups contributes 1.00.

What is ionisation energy?

The minimum energy required to eject the most loosely bound valence electron.

What is electron affinity?

The energy change (usually released) when a neutral gaseous state atom accepts an electron to form an anion.

What is the general trend of ionisation energy across the periodic table, and what is the main exception?

In general, ionisation energy increases across the period (→) and up the group (↑).

The main exception is nitrogen and oxygen, where IE_N > IE_O because oxygen has destabilising Π_c interactions.

What is the general trend of electron affinity across the periodic table, and what are the three main exceptions?

In general, electron affinity increases in magnitude across the period (→) and up the group (↑), and is usually negative in value.

The three exceptions are:

• N (0) compared to C (-122) because addition of an electron to nitrogen increases coulombic repulsion forces.

• Noble gases because addition of an electron results in their entry to a new, highly shielded shell.

• F (-328) and Cl (-349) because the 2p orbital of F is small, so there is high electron repulsion from high density.

What is the formula to calculate the formal charge of an atom in a molecule?

FC = (valence electrons) - (lone pair electrons) - 0.5(electrons engaging in covalent bonding)

Differentiate between the D_∞h and C_∞v point groups.

D_∞h are linear molecules that are purely symmetrical (e.g., CO₂)

C_∞v are linear molecules that are not symmetrical (e.g., NO₂)

What is the five step checklist for determining the point group of a molecule?

1. Determine if linear.

If linear and symmetrical, D_∞h. If linear and nonsymmetrical, C_∞v. If not linear, neither.

1. Determine if high, low, or intermediate symmetry.

If high symmetry, determine if perfectly tetrahedral (T_d), square bipyramidal (O_h), or icosahedral (I_h).

If low symmetry, determine if no symmetry operations (C₁), only reflection (C_s), or only inversion (C_i).

If intermediate, none of these.

1. Locate subsidiary axes, if any.

If possessing subsidiary axes, belongs in the D group.

If not possessing subsidiary axes, belongs in the C group.

1. Locate planes of reflection, if any.

If D, determine if no symmetry planes (D_n), horizontal reflection (D_nh), or dihedral reflection (D_nd).

If C, determine if no symmetry planes (C_n), horizontal reflection (C_2h), or vertical reflection (C_2v).

Which three symmetry operations preclude chirality?

Reflection, inversion, and improper rotation.

Which two symmetry operations preclude polarity?

C₂’ and horizontal symmetry.

What are the four rules of determining the value of a basis set vector superimposed onto an atom?

1. If the atom moves, x = 0

2. If the vector retains direction, x = +1

3. If the vector inverses direction, x = -1

4. If the vector arrows swap directions simultaneously, x = 0

When utilising the Greater Orthogonality Theorem for determining the irreducible representations, what four values are involved in determining the coefficients of the parts of the final irreducible representation?

Coefficient = Σ(ABC)/h

• A is the column coefficient, found from the character table.

• B is the column internal coordinate value, found from performing the symmetry operations with the basis set.

• C is the irreducible character value, found from the character table.

• h is the order of the point group, which is the number of symmetry operations times their respective coefficients.

How do you determine the vibrational representations of a molecule from its irreducible representation?

Γ_vib = Γ_irr - Γ_trans - Γ_rot

• Γ_irr is the total modes of movement, determined with the Greater Orthogonality Theorem.

• Γ_trans is the translational modes of movement, determined with the linear relations column (x, y, z).

• Γ_rot is the rotational modes of movement, determined with the quadratic relations column (xy, yz, zx).