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For a hydrogen-like orbital the number of radial nodes is given by n − l − 1. For a 3p orbital n = 3 and l = 1, so exactly one radial node is present. The single angular nodal plane of the p orbital is a separate feature and is not counted as a radial node.
First ionisation energy rises across a period as effective nuclear charge increases, but the completely filled 2s subshell makes Be harder to ionise than B. Continuing to carbon, the additional nuclear charge outweighs the extra 2p screening, so the order is Li < B < Be < C.
Xenon in XeF4 carries four bonding pairs and two lone pairs, giving six electron pairs and an octahedral electron-pair arrangement. The two lone pairs occupy trans positions, so the four fluorine atoms define a square plane.
Both internal energy and enthalpy of an ideal gas depend only on temperature, so an isothermal process has ΔU = 0 and ΔH = ΔU + Δ(nRT) = 0. The heat absorbed in a reversible expansion equals the work done, but neither quantity is the enthalpy change.
2,3-Dibromobutane has two stereogenic centres bearing identical substituent sets. The (R,S) and (S,R) arrangements are superimposable and correspond to a single achiral meso form, so the total is the meso compound plus the (R,R) and (S,S) enantiomers.
Nickel(II) is d8 and cyanide is a strong-field ligand, so the eight d electrons pair up completely in a dsp2 hybridisation scheme. Four-coordinate d8 complexes with strong-field ligands adopt square planar geometry and are diamagnetic.
The CH3, CH2 and OH protons each sit in a different magnetic environment, giving three chemically non-equivalent sets of protons. Their different coupling patterns (triplet, quartet and singlet) further confirm the three environments.
Each of the eight corner atoms is shared by eight cells (8 × 1/8 = 1) and each of the six face atoms is shared by two cells (6 × 1/2 = 3). The total is therefore four atoms per unit cell.
The energy levels of a particle in a one-dimensional box are proportional to n², since En = n²h²/8mL². Taking the ratio (2)²/(1)² gives exactly 4.
The integrated rate law for a second-order reaction is 1/[A] = 1/[A]0 + kt, so a plot of 1/[A] versus time is linear with slope k. The other plots are linear only for zero- or first-order kinetics.
At 298 K the relation between standard potential and equilibrium constant is E° = (0.0591/n) log K. With n = 2 and log K = 4, E° = (0.0591 × 4)/2 ≈ 0.118 V.
The Diels–Alder reaction is a [4+2] cycloaddition in which the total number of interacting π electrons is 6, i.e. 4n + 2. For a 4n + 2 system the thermally allowed pathway requires both components to use the same face, giving a suprafacial–suprafacial combination.
A linear triatomic has 3N − 5 = 4 vibrational modes: a symmetric stretch, an asymmetric stretch and a doubly degenerate bend. The symmetric stretch produces no change in dipole moment and is infrared inactive, leaving the asymmetric stretch and the bend as the two observed bands.
Iron in Fe(CO)5 is in the zero oxidation state, so it contributes its eight d electrons. Each of the five carbonyls is a two-electron donor, adding ten electrons to give a total of 18 and a saturated, 18-electron complex.
The aldehydic proton is coupled to the three equivalent methyl protons on the adjacent carbon. Applying the n + 1 rule with n = 3 gives four lines of equal intensity, i.e. a quartet.
Spontaneity requires ΔG° = ΔH° − TΔS° to be negative. Substituting the values gives a change of sign at T = ΔH°/ΔS° = 50 000/100 = 500 K, and because the entropy term is unfavourable the reaction is spontaneous only below this temperature.
Iron(III) is a d5 ion and cyanide is a strong-field ligand, so the electrons occupy a low-spin t2g5 configuration. Three electrons are paired in the t2g set and one remains unpaired.
The cyclopentadienyl anion is planar, cyclic and fully conjugated with six π electrons, satisfying the 4n + 2 rule with n = 1. This closed-shell arrangement confers aromatic stabilisation.
Diborane contains two bridging hydrogen atoms, each held simultaneously by both boron atoms in a B–H–B bridge. Each bridge is a three-centre two-electron bond, giving two such bonds in total; the four terminal B–H linkages are ordinary two-centre two-electron bonds.
After intersystem crossing from the excited singlet to the triplet manifold, emission occurs from the lowest triplet state to the singlet ground state. This T1 → S0 transition is spin-forbidden, which is why phosphorescence is slow and long-lived compared with fluorescence.
Using the Arrhenius expression, ln(k2/k1) = (Ea/R)(1/T1 − 1/T2). Substituting Ea/R = 6.01 × 103 K and the temperature difference gives ln(k2/k1) ≈ 0.65, so the ratio is about 1.9.
The number of microstates for n electrons in a shell of degeneracy N = 4l + 2 is N!/[n!(N − n)!]. For a d shell N = 10 and n = 2, giving 10!/(2!8!) = 45.
The SN1 rate is governed by the stability of the carbocation formed in the rate-determining step. The para-methoxy group donates electron density by resonance and strongly stabilises the benzylic cation, making this substrate the fastest to ionise.
Staggered ethane possesses a principal C3 axis, three C2 axes perpendicular to it, a centre of inversion and an S6 axis. This combination of symmetry elements defines the point group D3d.
Metal-to-ligand back donation populates the CO π* orbital and lowers the C–O stretching frequency. The cationic [Mn(CO)6]+ has the least electron-rich metal centre, so back donation is weakest and the C–O stretch appears at the highest wavenumber.
For an n-th order reaction the rate constant has units mol^(1-n) L^(n-1) s^-1. A first-order reaction has n = 1, so the concentration terms cancel and the unit is s^-1.
Carbocation stability increases with the number of alkyl groups attached to the positively charged carbon because of inductive and hyperconjugative electron release. The tert-butyl cation has three such groups and is therefore the most stable of those listed.
Ethene contains only C-C and C-H sigma bonds plus one pi bond; the HOMO is the pi orbital and the LUMO is the pi* orbital. The lowest-energy absorption therefore corresponds to the pi to pi* transition.
The complex cation contains five neutral NH3 ligands and one chloride ligand, and the two outside chlorides are counterions. Charge balance requires cobalt to be in the +3 oxidation state, so the correct name is pentaamminechloridocobalt(III) chloride.
Mn2+ is a d5 ion. In a high-spin octahedral field all five d electrons are unpaired, giving S = 5/2 and a spin-only moment of sqrt(4S(S+1)) = sqrt(35), approximately 5.92 BM.
Peroxides initiate a radical chain reaction in which the bromine radical adds to the less substituted carbon of the double bond, generating the more stable secondary radical. Hydrogen abstraction then gives the anti-Markovnikov product, 1-bromopropane.
For an ideal gas the internal energy depends only on temperature, so an isothermal process has Delta U = 0. The gas does work and absorbs an equal amount of heat, so w and q are nonzero, and Delta S is positive for expansion.
For a weak acid, [H+] is approximately sqrt(Ka x C) when dissociation is small. Here sqrt(1.0 x 10^-6 x 0.01) = 1.0 x 10^-4 M, giving pH = 4.
Ligands are arranged in the spectrochemical series according to the magnitude of Delta-o they produce. Cyanide is a strong-field ligand that lies far to the strong-field end, well above water and the halides.
2,3-Dibromobutane has two stereogenic centres with identical substituent sets, so the maximum of 2^2 = 4 stereoisomers is reduced to three by the presence of a meso form. The three are the (R,R), (S,S) pair and the achiral meso compound.
For a first-order reaction, t1/2 = ln 2 / k, which contains no concentration term, so the half-life is independent of initial concentration. Zero- and second-order half-lives depend on C0.
Bromine consists of 79Br and 81Br in nearly equal natural abundance. A molecule with one bromine atom therefore gives two molecular-ion peaks of similar intensity separated by two mass units.
n-Butane has two types of hydrogen: those on C1/C4 and those on C2/C3. Substitution at C1 gives one achiral product, while substitution at C2 gives 2-chlorobutane, which is chiral and therefore exists as two enantiomers, giving three products in total.
For an isothermal reversible expansion of an ideal gas, Delta S = nR ln(V2/V1). With n = 1 and V2/V1 = 2, this gives Delta S = R ln 2, a positive value because the gas becomes more dispersed.
Jahn-Teller distortion is strong when the octahedral d-electron configuration is degenerate in the eg set, as for d9. Cu2+ is d9 (t2g6 eg3), giving an unevenly occupied eg set and a strong tetragonal distortion.
The Moho is the seismic discontinuity across which compressional-wave velocity jumps from roughly 7 km/s in the crust to about 8 km/s in the uppermost mantle. It is identified by refraction and reflection of seismic waves and represents a change in rock composition from crustal silicates to peridotitic mantle.
The horizontal component of planetary vorticity is f = 2Ω sin φ, obtained by projecting the Earth's rotation vector onto the local vertical. It vanishes at the equator and reaches its maximum of 2Ω at the poles, which is why geostrophic adjustment is strongest at high latitudes.
Nitrogen makes up about 78% of the atmosphere by volume, with oxygen at roughly 21%. Argon (about 0.93%) and carbon dioxide (about 0.04%) are comparatively minor constituents despite their importance for other processes.
Diamond is assigned hardness 10 on the Mohs scale, the highest value, because of its rigid three-dimensional covalent bonding of carbon atoms. Corundum is 9, topaz 8 and quartz 7, so each of those can be scratched by diamond but not vice versa.
The Pacific Ocean covers roughly one-third of the Earth's surface and about half of the total ocean area, making it by far the largest basin. It also contains the greatest average depth and the deepest known trench.
At the Mid-Atlantic Ridge, two plates move apart and new oceanic lithosphere is created by sea-floor spreading, so the boundary is divergent. The ridge axis is offset by transform faults, but the spreading segments themselves are constructive margins.
Seismology shows that S waves propagate through the inner core, which requires it to be solid; its density and the Earth's bulk composition indicate an iron–nickel alloy. The solid state is maintained despite high temperature because pressure raises the melting point.
P waves are compressional and involve a bulk modulus as well as a shear modulus, so their velocity is always greater than that of shear waves in the same material. This is why P waves arrive first at a seismograph and why the S–P time difference increases with distance from the source.
The discontinuous branch of Bowen's series begins with olivine at the highest temperature and proceeds through pyroxene, amphibole and biotite. Quartz, muscovite and potassium feldspar all appear at the low-temperature end of the series and are typical of felsic, late-stage melts.
Geostrophic balance assumes frictionless, straight-line flow above the boundary layer, where the horizontal pressure-gradient force is exactly opposed by the Coriolis force. The resulting wind blows parallel to the isobars with low pressure to the left in the Northern Hemisphere.