AP Chemistry 1.1-1.3: Moles, Mass Spectra, and Composition

How are mass, amount in moles, and molar mass related?

$n = \frac{m}{M}$, where $n$ is amount in moles, $m$ is mass in grams, and $M$ is molar mass in $\mathrm{g/mol}$. Equivalently, $m = nM$.

How many moles are present in a sample with mass $4.70\ \mathrm{g}$ of potassium, given $M_{\mathrm{K}} = 39.10\ \mathrm{g/mol}$?

$n = \frac{4.70\ \mathrm{g}}{39.10\ \mathrm{g/mol}} = 0.120\ \mathrm{mol\ K}$.

What mass corresponds to $2.561\ \mathrm{mol}$ of gold if $M_{\mathrm{Au}} = 196.97\ \mathrm{g/mol}$?

$m = 2.561\ \mathrm{mol} \times 196.97\ \mathrm{g/mol} = 504.5\ \mathrm{g\ Au}$.

How can moles be converted to the number of atoms or molecules?

Multiply the amount in moles by Avogadro's constant: $N = nN_A$. The entity type must be stated, such as atoms, molecules, ions, or formula units.

How can the number of atoms or molecules be converted to moles?

Divide the number of entities by Avogadro's constant: $n = \frac{N}{N_A}$.

How many copper atoms are in $5.00\ \mathrm{g}$ of copper if $M_{\mathrm{Cu}} = 63.55\ \mathrm{g/mol}$?

$5.00\ \mathrm{g} \times \frac{1\ \mathrm{mol}}{63.55\ \mathrm{g}} \times \frac{6.022 \times 10^{23}\ \mathrm{Cu\ atoms}}{1\ \mathrm{mol}} = 4.74 \times 10^{22}\ \mathrm{Cu\ atoms}$.

What is the general factor-label pathway for converting grams of a substance directly to particles?

$\text{grams} \times \frac{1\ \mathrm{mol}}{\text{molar mass in grams}} \times \frac{6.022 \times 10^{23}\ \text{particles}}{1\ \mathrm{mol}}$. Units cancel to leave the requested number of particles.

How many moles of glycine, $\mathrm{C_2H_5NO_2}$, are in $28.35\ \mathrm{g}$ if its molar mass is $75.07\ \mathrm{g/mol}$?

$n = \frac{28.35\ \mathrm{g}}{75.07\ \mathrm{g/mol}} = 0.3776\ \mathrm{mol}$ glycine.

How many molecules are in $0.0400\ \mathrm{g}$ of saccharin, $\mathrm{C_7H_5NO_3S}$, with $M = 183.18\ \mathrm{g/mol}$?

$0.0400\ \mathrm{g} \times \frac{1\ \mathrm{mol}}{183.18\ \mathrm{g}} \times N_A = 1.31 \times 10^{20}$ saccharin molecules.

How can the number of atoms of a particular element in a compound sample be found from the number of molecules?

Multiply the number of molecules by the compound's subscript for that element. For example, each $\mathrm{C_7H_5NO_3S}$ molecule contributes seven carbon atoms.

How many carbon atoms are in $1.31 \times 10^{20}$ molecules of $\mathrm{C_7H_5NO_3S}$?

$1.31 \times 10^{20}\ \text{molecules} \times \frac{7\ \text{C atoms}}{1\ \text{molecule}} = 9.17 \times 10^{20}\ \text{C atoms}$.

How is the percent by mass of an element in a pure compound calculated from its formula?

$\%\text{ element} = \frac{\text{mass contribution of the element in one mole of compound}}{\text{molar mass of the compound}} \times 100\%$. The mass contribution equals the element's atomic mass multiplied by its subscript.

What is the percent by mass of oxygen in $\mathrm{H_2O}$?

$\%\mathrm{O} = \frac{16.00}{18.02} \times 100\% \approx 88.8\%$.

What should the calculated elemental mass percentages of a pure compound add up to?

They should add to approximately $100\%$, with any small discrepancy caused by rounding. A substantial discrepancy indicates an error in the formula or calculation.

How can the mass of a particular element in a compound sample be calculated from its percent composition?

$\text{mass of element} = \text{sample mass} \times \frac{\%\text{ element}}{100}$. This applies to a pure sample whose composition is represented by the compound's formula.

How can an empirical formula be determined from the masses or percentages of elements in a pure compound?

Convert each element's mass to moles, divide every mole amount by the smallest mole amount, and multiply all ratios by a small integer if needed to obtain whole numbers. Use those whole-number ratios as subscripts.

How can a molecular formula be determined from an empirical formula and molar mass?

Calculate the empirical-formula mass, then divide the compound's molar mass by that value. Multiply every empirical-formula subscript by the resulting whole-number factor.

What particles make up an atom, and where are they located?

Protons and neutrons are concentrated in the nucleus; electrons occupy the surrounding electron cloud. The nucleus contains nearly all of the atom's mass, while electrons occupy most of its volume.

Compare the relative charges and masses of protons, neutrons, and electrons.

A proton has charge $+1$ and mass about $1.0073\ \mathrm{amu}$; a neutron has charge $0$ and mass about $1.0087\ \mathrm{amu}$; an electron has charge $-1$ and mass about $0.00055\ \mathrm{amu}$.

Unified atomic mass unit, $u$ or amu

A mass unit defined as exactly $\frac{1}{12}$ the mass of one carbon-12 atom. It equals approximately $1.6605 \times 10^{-24}\ \mathrm{g}$; the dalton (Da) is an equivalent unit.

Atomic number, $Z$

The number of protons in an atom's nucleus. It uniquely identifies the element; every carbon atom, for example, has $Z = 6$.

Mass number, $A$

The total number of protons and neutrons in one atom's nucleus: $A = Z + N$, where $N$ is the number of neutrons.

How is the number of neutrons in an atom calculated from its mass number and atomic number?

$N = A - Z$. The mass number counts nucleons, while the atomic number counts only protons.

How are the numbers of protons and electrons related in a neutral atom and in an ion?

A neutral atom has equal numbers of protons and electrons. A cation has fewer electrons than protons, while an anion has more electrons than protons.

How is nuclide notation written, and what do its components represent?

It is written as $^{A}_{Z}X$, where $X$ is the element symbol, $A$ is the mass number, and $Z$ is the atomic number. For an ion, the charge is written separately as a superscript at the upper right.

Interpret the nuclide symbol $^{23}_{11}\mathrm{Na}^{+}$.

It represents a sodium species with 11 protons, $23 - 11 = 12$ neutrons, and 10 electrons because the $+1$ charge indicates one fewer electron than protons.

Isotopes

Atoms of the same element that have the same number of protons but different numbers of neutrons. They share the same atomic number $Z$ but have different mass numbers $A$.

Why do isotopes of an element have the same atomic number but different mass numbers?

The element identity is fixed by the number of protons, so isotopes have the same $Z$. They differ in neutron count, changing $A = \text{protons} + \text{neutrons}$.

Average atomic mass

The abundance-weighted average of the masses of an element's naturally occurring isotopes. It is the value listed on the periodic table and is generally not an integer.

How is an element's average atomic mass calculated from isotope data?

$\text{average atomic mass} = \sum (\text{isotope mass} \times \text{fractional abundance})$. Percent abundances must first be converted to decimal fractions.

An element has two isotopes: $10.0\ \mathrm{amu}$ at 20.0% abundance and $11.0\ \mathrm{amu}$ at 80.0% abundance. What is its average atomic mass?

$(10.0)(0.200) + (11.0)(0.800) = 10.8\ \mathrm{amu}$.

How can an unknown isotope abundance be determined when the isotope masses and average atomic mass are known?

Let one isotope's fractional abundance be $x$ and the other's be $1-x$. Set up $\text{average mass} = (\text{mass}_1)x + (\text{mass}_2)(1-x)$ and solve for $x$, then multiply by 100 for percent abundance.

Why is an element's periodic-table atomic mass usually different from the mass number of any single atom?

The periodic-table value is a weighted average of the element's isotopes, whereas a mass number is the integer count of protons plus neutrons for one specific isotope.

What does a mass spectrum of an element show?

It shows the relative abundance of the element's isotopes as a function of their mass-to-charge ratio, commonly plotted with $m/z$ on the horizontal axis and relative intensity or abundance on the vertical axis.

How is the mass-to-charge ratio of an isotope related to its mass number in a typical elemental mass spectrum?

For singly charged isotope ions, $z = 1$, so $m/z$ is approximately equal to the isotope's mass number. The peaks therefore identify isotopes with different numbers of neutrons.

What does the height or relative intensity of a peak in an elemental mass spectrum represent?

It represents the relative abundance of that isotope. A taller peak corresponds to a greater fraction of the element's atoms, after accounting for the spectrum's normalization.

How can isotope abundances be determined from a mass spectrum?

Divide each isotope peak's intensity by the sum of all isotope peak intensities, then multiply by 100 to obtain its percent abundance. If the intensities are already normalized to 100, each intensity directly gives the percent abundance.

A mass spectrum has isotope peaks with relative intensities 75 and 25. What are the approximate isotope abundances?

The total intensity is $100$, so the isotopes have abundances of $75\%$ and $25\%$, respectively.

How can an element's average atomic mass be calculated from a mass spectrum?

Identify each isotope's mass from its $m/z$ peak, convert its relative intensity to a fractional abundance, and calculate $\overline{m} = \sum (m_i f_i)$.

Formula mass

The sum of the average atomic masses of all atoms represented in a chemical formula. It is reported in atomic mass units (amu or u) for a single formula entity and numerically corresponds to the molar mass in g/mol.

How is the molecular mass of a covalent compound calculated?

Multiply each element's average atomic mass by the number of that element's atoms in one molecule, then sum the contributions: $\text{molecular mass} = \sum (\text{subscript} \times \text{average atomic mass})$.

Calculate the molecular mass of $\mathrm{CHCl_3}$ using $\mathrm{C} = 12.01$, $\mathrm{H} = 1.008$, and $\mathrm{Cl} = 35.45\ \mathrm{amu}$.

$12.01 + 1.008 + 3(35.45) = 119.37\ \mathrm{amu}$.

How does the formula of an ionic compound differ conceptually from the formula of a covalent compound?

A covalent formula represents the atoms in one discrete molecule. An ionic formula represents the simplest whole-number ratio of cations and anions in an electrically neutral lattice, not a discrete molecule.

Formula unit

The lowest whole-number ratio of ions represented by an ionic compound's formula. For example, one formula unit of $\mathrm{NaCl}$ corresponds to one $\mathrm{Na}^{+}$ ion and one $\mathrm{Cl}^{-}$ ion in a 1:1 ratio.

How is the formula mass of an ionic compound calculated?

Sum the average atomic masses of all atoms indicated by the ionic formula, including atoms inside parentheses multiplied by the outside subscript. The result is called formula mass rather than molecular mass.

Calculate the formula mass of $\mathrm{Al_2(SO_4)_3}$ in terms of the atoms represented.

Rewrite the composition as $\mathrm{Al_2S_3O_{12}}$ and add $2$ Al, $3$ S, and $12$ O atomic masses. Using common atomic masses gives approximately $342.15\ \mathrm{amu}$.

Why may neutral-atom atomic masses be used when calculating the formula mass of an ionic compound?

The electron masses gained or lost when ions form are negligible compared with nuclear masses. In an ionic compound, using neutral atomic masses gives the correct mass to the significant digits normally reported.

Mole

The SI amount unit for counting microscopic entities. One mole contains exactly $6.02214076 \times 10^{23}$ specified entities, such as atoms, molecules, ions, or formula units.

Avogadro constant, $N_A$

The number of entities per mole: $N_A = 6.02214076 \times 10^{23}\ \mathrm{mol^{-1}}$, commonly rounded to $6.022 \times 10^{23}\ \mathrm{mol^{-1}}$.

What is the difference between one mole of carbon atoms and one mole of lead atoms?

Each contains the same number of atoms, $N_A$. Their masses differ because carbon and lead atoms have different average atomic masses.

Molar mass

The mass of exactly one mole of a substance, expressed in $\mathrm{g/mol}$. For an element or compound, its numerical value equals the atomic mass or formula mass in amu.

Why are atomic mass in amu and molar mass in g/mol numerically equal but physically different?

Atomic mass describes the mass of one atom or formula entity on an atomic scale, whereas molar mass describes the mass of $6.022 \times 10^{23}$ entities. Their numerical equality results from the definition of the atomic mass unit relative to carbon-12.

How is the molar mass of a compound determined from its chemical formula?

Multiply each element's atomic mass by the number of atoms in one formula unit or molecule, then add all contributions. The numerical result is expressed in $\mathrm{g/mol}$.

What is the molar mass of $\mathrm{H_2O}$ if $\mathrm{H} = 1.008$ and $\mathrm{O} = 16.00\ \mathrm{g/mol}$?

$M(\mathrm{H_2O}) = 2(1.008) + 16.00 = 18.016\ \mathrm{g/mol}$, typically reported as $18.02\ \mathrm{g/mol}$.