Atomic structure and properties · AP Chemistry

Chapter 1 · Atomic structure and properties

the mole, the shell, the spectrum

Atoms are far too small to count one by one. Chemistry counts them by weighing, and explains what they do by where their electrons are.

The figure is the shell model of an atom. The nucleus holds the protons (red) and the neutrons. The electrons sit in shells around it, held by their attraction to the nucleus. As the figure adds one proton and one electron at a time, it runs from hydrogen, Z = 1, to calcium, Z = 20.

Electrons in the outermost shell are the valence electrons (red). They are the ones an atom loses, gains or shares, so they decide how an element behaves.

This chapter counts atoms with the mole, reads isotopes from a mass spectrum and works out the composition of compounds and mixtures. Then it builds atoms shell by shell with Coulomb’s law, checks the shells against photoelectron spectra, and explains the periodic trends and the charges of ions.

1.1 · Counting atoms by weighing

moles and molar mass

Each pan receives the same number of atoms, carbon on the left and magnesium on the right. The magnesium pan always weighs 2.02 times as much.

Atoms have tiny masses, so they are measured in atomic mass units (amu): 1 amu = 1.6605 × 10⁻²⁴ g. An average carbon atom has a mass of 12.01 amu, and an average magnesium atom 24.31 amu.

A mole is 6.022 × 10²³ particles, Avogadro’s number N_A. It is chosen so that a mole of atoms has a mass in grams equal to the mass of one atom in amu. That mass is the molar mass: 12.01 g/mol for carbon and 24.31 g/mol for magnesium.

n = \frac{m}{M} \qquad N = n\,N_\text{A}

Here n is the amount in moles, m the mass, M the molar mass and N the number of particles. How many atoms are in 5.00 g of magnesium?

\begin{aligned}n &= \frac{5.00\ \text{g}}{24.31\ \text{g/mol}} = 0.2057\ \text{mol} \\ N &= 0.2057 \times 6.022\times10^{23} = 1.24\times10^{23}\end{aligned}

A compound’s molar mass is the sum of its atoms’ molar masses: H₂O is 2(1.008) + 16.00 = 18.02 g/mol.

1.2 · Isotopes and average mass

the mass spectrum of an element

Atoms of one element all have the same number of protons, but not always the same number of neutrons. These versions are isotopes, and they differ in mass.

A mass spectrum of an element shows one peak for each isotope. A peak’s position is the isotope’s mass; its height is the isotope’s relative abundance, the percentage of the atoms that are that isotope. Magnesium has three:

isotope	²⁴Mg	²⁵Mg	²⁶Mg
mass (amu)	23.985	24.986	25.983
abundance (%)	78.99	10.00	11.01

The average atomic mass on the periodic table is the mean over all the atoms, so each isotope counts by its abundance:

\begin{aligned}&23.985(0.7899) \ &+ 24.986(0.1000) \ &+ 25.983(0.1101) \\ &= 24.31\ \text{amu}\end{aligned}

In the figure the atoms are counted one at a time. Early on the peaks and the red average jump about; as the count grows they settle on the true spectrum and on 24.31 amu.

No single magnesium atom has a mass of 24.31 amu: it is an average. The average lies closest to the most abundant isotope. Chlorine is 75.76% ³⁵Cl and 24.24% ³⁷Cl, so its average, 35.45 amu, sits much nearer 35 than 37.

1.3 · Fixed composition

mass percent and the empirical formula

A pure compound has the same composition in every sample: the same elements in the same proportion by mass. This is the law of definite proportions.

The mass percent of an element in a compound is the mass of that element in one mole of the compound over the molar mass. For ethene, C₂H₄:

\%\,\text{C} = \frac{2(12.01)}{28.05} \times 100 = 85.6\%

The empirical formula gives the simplest whole-number ratio of atoms. Ethene and cyclopropane, C₃H₆, both come apart into CH₂ units, so both have the empirical formula CH₂ and the same mass percents, 85.6% C and 14.4% H. Mass data alone find the empirical formula, not the molecular one.

From mass percents back to a formula: take 100 g, so each percent is grams; convert to moles; divide by the smallest. A compound is 40% C, 6.71% H and 53.3% O:

	grams	moles	÷ smallest
C	40	40 / 12.01 = 3.331	1.00
H	6.71	6.71 / 1.008 = 6.657	2.00
O	53.3	53.3 / 16 = 3.331	1.00

The ratio is 1 : 2 : 1, so the empirical formula is CH₂O. Glucose, C₆H₁₂O₆, has this composition.

1.4 · Mixtures vary

the composition of mixtures

A mixture is two or more substances that are not chemically combined. Unlike a pure compound, its composition can vary from sample to sample.

The figure’s sample is a mixture of sodium chloride and potassium chloride. Each pure salt has its own fixed percent of chlorine by mass:

\begin{aligned}\text{NaCl:}\ \frac{35.45}{58.44} &= 60.66\%\ \text{Cl} \\ \text{KCl:}\ \frac{35.45}{74.55} &= 47.55\%\ \text{Cl}\end{aligned}

As the mix changes, the percent of chlorine slides along a straight line between those two values. If x is the mass fraction of NaCl:

\%\,\text{Cl} = 60.66\,x + 47.55\,(1 - x)

Elemental analysis runs this backwards. A sample is found to be 55.0% Cl by mass. Then

x = \frac{55.0 - 47.55}{60.66 - 47.55} = 0.568

so the sample is 56.8% NaCl by mass. The same reasoning finds the purity of a sample: if an impurity contains none of the element measured, the impurity lowers its mass percent below the pure compound’s value.

1.5 · Electrons in shells

Coulomb’s law and electron configuration

An atom is a small, dense, positive nucleus with negative electrons around it. The atomic number Z is the number of protons, and in a neutral atom it is also the number of electrons.

What holds an electron is its electrostatic attraction to the nucleus. Coulomb’s law says the force between two charges grows with the charges and falls with the square of the distance between them:

F \propto \frac{q_1 q_2}{r^2}

So an electron is held more tightly, and takes more energy to remove, when the nucleus has more protons or when the electron is closer to it.

Electrons occupy shells (n = 1, 2, 3, …) at increasing distance, and each shell has subshells: s holds 2 electrons, p holds 6. Electrons fill the lowest-energy subshells first (the aufbau principle), in the order 1s, 2s, 2p, 3s, 3p, 4s. The orbital diagram on the right shows each orbital as a box holding at most two electrons of opposite spin. The result is the electron configuration:

N (Z = 7)	1s² 2s² 2p³
Na (Z = 11)	1s² 2s² 2p⁶ 3s¹
Cl (Z = 17)	1s² 2s² 2p⁶ 3s² 3p⁵
Ca (Z = 20)	1s² 2s² 2p⁶ 3s² 3p⁶ 4s²

The inner, filled shells are the core electrons. They lie between the nucleus and the valence electrons and shield them: sodium’s one 3s electron feels its 11 protons partly canceled by 10 core electrons, so it is held loosely.

An ion’s configuration follows from its electron count: Na⁺ has 10 electrons, 1s² 2s² 2p⁶, and Cl⁻ has 18, the configuration of argon.

1.6 · Measuring the shells

photoelectron spectroscopy

Photoelectron spectroscopy (PES) tests the shell model directly. High-energy photons knock electrons out of atoms, and the energy each electron needed to escape is measured.

That energy is the electron’s binding energy. The photon’s energy is known, so the electron’s kinetic energy as it leaves gives it: binding energy = photon energy − kinetic energy.

Electrons in the same subshell have the same binding energy, so each subshell makes one peak. A peak’s position is its binding energy: how strongly that subshell is held. Its height is proportional to the number of electrons in it. Silicon:

subshell	1s	2s	2p	3s	3p
binding energy (MJ/mol)	178	15.1	10.3	1.46	0.79
relative height	2	2	6	2	2

Binding energy is plotted increasing to the left, and in the figure on a log scale so that 178 and 0.79 fit on one axis. Five peaks are five subshells, 1s² 2s² 2p⁶ 3s² 3p²; the heights add to 14, silicon’s atomic number.

The spectrum follows Coulomb’s law. The 1s electrons are closest to the nucleus and unshielded, so their peak is far to the left. The same subshell sits at higher binding energy in an element with more protons: 1s is 104 MJ/mol in sodium and 126 MJ/mol in magnesium.

1.7 · Periodic trends

size, ionization energy and electronegativity

Across a period atoms get smaller and hold their electrons more tightly. Down a group they get larger and hold them less tightly. Coulomb’s law and the shell model explain both.

Across a period each element has one more proton, but the new electron joins the same shell. The core is unchanged, so shielding barely changes, and the valence electrons are pulled in harder. Down a group each element adds a shell: the valence electrons are farther out and shielded by more core electrons.

period 3	Na	Mg	Al	Si	P	S	Cl
radius (pm)	166	141	121	111	107	105	102
IE₁ (kJ/mol)	496	738	578	787	1012	1000	1251

The atomic radius falls across a period and rises down a group. The first ionization energy, the energy to remove one electron from a gaseous atom, does the opposite: it is lowest for the alkali metals and highest for the noble gases. Two small dips, at boron and at oxygen in period 2 (aluminum and sulfur in period 3), come from the subshell structure; the overall pattern is the Coulombic one.

Electronegativity, an atom’s pull on the electrons in a bond it shares, follows the ionization energy: it rises across a period and falls down a group, so fluorine is the most electronegative element. Electron affinity, the energy change when an atom gains an electron, also tends to grow more favorable across a period.

Ions break from their atoms in size: a cation is smaller than its atom, having lost its outer electrons, and an anion is larger, its extra electron repelled by the rest.

1.8 · Valence electrons make ions

charges and formulas of ionic compounds

Sodium has one valence electron and chlorine has seven. When they react, sodium loses its electron, chlorine gains it, and the two ions attract.

For a main-group element the number of valence electrons follows from its group, and the typical charge of its ion follows from that. Metals lose their valence electrons; nonmetals gain enough to fill their valence shell. Either way the ion has the electron configuration of a noble gas.

group	1	2	13	15	16	17
valence electrons	1	2	3	5	6	7
ion	Na⁺	Mg²⁺	Al³⁺	N³⁻	O²⁻	Cl⁻

Sodium stops at +1 because its next electron is a core electron. Its first ionization energy is 496 kJ/mol, but its second is 4562 kJ/mol: that electron is in the second shell, closer to the nucleus and unshielded by the 3s electron that left.

The ions’ attraction is again Coulombic, and an ionic compound has no net charge, so its formula balances the charges: NaCl, MgCl₂, Na₂O, Al₂O₃. Elements in the same group form ions of the same charge and so compounds of the same form: Na₂O and K₂O, MgCl₂ and CaCl₂.

In the solid, ions of opposite charge surround each other in a lattice: in sodium chloride each Na⁺ has six Cl⁻ neighbors, and each Cl⁻ six Na⁺.