Atomic Structure and Isotopes: From Nuclide Symbols to Atomic Weight

Protons, neutrons and electrons in measured units

The proton carries a charge of +1 in units of the elementary charge, e = 1.602 × 10⁻¹⁹ C, and the electron carries −1. The neutron has no charge. Masses are usually given in unified atomic mass units (u, also called daltons, Da), where 1 u is one twelfth of the mass of a carbon-12 atom, 1.6605 × 10⁻²⁷ kg.

From the 2022 CODATA values published by NIST, the proton is 1.007276 u, the neutron is 1.008665 u and the electron is 0.000549 u. The proton is about 1836 times heavier than the electron, so almost all of an atom's mass sits in the nucleus. The nucleus is also tiny: its radius is of order 10⁻¹⁵ m, while the atom's radius is of order 10⁻¹⁰ m. The electrons therefore set an atom's size and its chemistry, and the nucleus sets its mass.

Reading nuclide symbols and counting particles in ions

The atomic number Z is the number of protons. It defines the element: every atom with 17 protons is chlorine. The mass number A is the total number of protons and neutrons, so the neutron count is A − Z. A nuclide symbol writes A as a left superscript and Z as a left subscript on the element symbol, as in ³⁵₁₇Cl. Because the symbol already fixes Z, the subscript is often dropped, giving ³⁵Cl or chlorine-35.

Electrons equal protons only in a neutral atom. An ion with charge q has Z − q electrons, so a positive charge means fewer electrons and a negative charge means more. The charge never changes the nucleus.

Worked example 1. For ⁵⁶Fe³⁺: Z = 26, so 26 protons; 56 − 26 = 30 neutrons; 26 − 3 = 23 electrons. For ¹²⁷I⁻: 53 protons, 127 − 53 = 74 neutrons and 53 + 1 = 54 electrons. For ³⁴S²⁻: 16 protons, 18 neutrons and 18 electrons. Notice that ³⁴S²⁻ has the same electron count as an argon atom; matching electron counts are what make ions isoelectronic, not matching mass numbers.

Why isotopes react alike but differ in mass

Isotopes are atoms with the same Z but different numbers of neutrons. Chlorine-35 and chlorine-37 both have 17 protons and, as neutral atoms, 17 electrons. Chemical behavior is controlled by electrons, so the two isotopes form the same compounds in the same ratios.

Properties that depend on mass do differ slightly. Lighter molecules move faster at a given temperature, and bonds to heavier isotopes react a little more slowly. These effects are small for heavy elements. Uranium enrichment shows how small. By Graham's law, the effusion rates of ²³⁵UF₆ and ²³⁸UF₆ are in the ratio √(M₂/M₁). Using NIST isotope masses, M(²³⁵UF₆) = 349.034 g/mol and M(²³⁸UF₆) = 352.041 g/mol, so the ratio is √(352.041/349.034) = 1.0043. One pass through a barrier enriches the lighter isotope by less than half a percent, which is why enrichment plants need many stages in series.

Weighted averages: from isotope masses to atomic weight

The atomic weight on a periodic table is not the mass of any one atom. It is the average mass of the element's atoms in a normal sample, weighted by how common each isotope is. Multiply each isotope's mass by its fractional abundance and add the products. The fractions must sum to 1.

Worked example 2. Magnesium has three stable isotopes. From the NIST Atomic Weights and Isotopic Compositions tables: ²⁴Mg, 23.985042 u, abundance 0.7899; ²⁵Mg, 24.985837 u, 0.1000; ²⁶Mg, 25.982593 u, 0.1101. The products are 18.94578, 2.49858 and 2.86068 u, and their sum is 24.30505 u. Strict significant-figure rules applied to the four-digit abundances would round this to 24.31 u. Kept to three decimal places, it is 24.305 u, the conventional value printed on most periodic tables.

Run the same calculation for chlorine, using ³⁵Cl at 34.968853 u with abundance 0.7576 and ³⁷Cl at 36.965903 u with 0.2424: 26.4924 + 8.9605 = 35.453 u. A quick estimate catches errors: the average must lie between the isotope masses and closer to the more abundant one.

Working backward to find an isotope abundance

If an element has two isotopes and you know the atomic weight, you can solve for the abundances. Call the fraction of the first isotope x; the second is then 1 − x. Write the weighted average as an equation and solve for x.

Worked example 3. Copper has a standard atomic weight of 63.546 and two stable isotopes, ⁶³Cu at 62.929598 u and ⁶⁵Cu at 64.927790 u (NIST). Then 62.929598x + 64.927790(1 − x) = 63.546. Rearranging gives x = (64.927790 − 63.546)/(64.927790 − 62.929598) = 1.381790/1.998192 = 0.6915. So copper is 69.15% ⁶³Cu and 30.85% ⁶⁵Cu, which reproduces the NIST abundances of 0.6915 and 0.3085.

This method works only for two isotopes. With three, one average gives one equation in two unknowns, and you need another measurement, such as a mass spectrum.

Isotope patterns in a mass spectrum

A mass spectrometer ionizes a sample, separates the ions by mass-to-charge ratio and records how many arrive at each value. For a single element, the peak heights give the isotope abundances directly. For molecules, isotopes combine and produce recognizable patterns.

Worked example 4. Predict the molecular-ion pattern of Cl₂. Let p = 0.7576 for ³⁵Cl and q = 0.2424 for ³⁷Cl. A molecule is ³⁵Cl–³⁵Cl (mass 70) with probability p² = 0.5740, ³⁵Cl–³⁷Cl (mass 72) with probability 2pq = 0.3673, and ³⁷Cl–³⁷Cl (mass 74) with probability q² = 0.0588. The factor of 2 counts the two ways to get one of each. Scaled so the tallest peak is 100, the pattern is 100 : 64.0 : 10.2.

Two shortcuts follow. A compound with one chlorine atom shows peaks two mass units apart in a ratio of about 3 : 1, because p/q = 3.13. A compound with one bromine atom shows two peaks of nearly equal height, because NIST lists ⁷⁹Br at 0.5069 and ⁸¹Br at 0.4931.

Atomic-structure errors that cost marks

Most mistakes come from mixing up the three counts or from treating the atomic weight as a property of a single atom. Write Z, A and the charge before you count anything.

• Changing the proton count to make an ion. Charge changes electrons only; protons fix the element.
• Adding electrons for a positive ion. A 3+ ion has three fewer electrons than the atom.
• Reading the atomic weight as a mass number. No chlorine atom has a mass of 35.45 u; it is an average of atoms at about 35 u and 37 u.
• Using percentages as fractions without dividing by 100, which gives an average 100 times too large.
• Forgetting the factor of 2 for mixed isotope pairs, such as ³⁵Cl–³⁷Cl, when predicting a mass spectrum.
• Calling two nuclides isotopes because they have the same mass number. ¹⁴C and ¹⁴N share A but are different elements.

Mass defect and interval atomic weights

Adding six protons, six neutrons and six electrons with the CODATA masses gives 12.0989 u, yet a carbon-12 atom is exactly 12 u. The missing 0.0989 u, about 0.82% of the total, corresponds to the energy released when the nucleus formed. This mass defect is why isotope masses are not whole numbers, and it is the starting point for nuclear binding energy.

For some elements, the isotope mix varies enough between natural sources that IUPAC's Commission on Isotopic Abundances and Atomic Weights reports an interval instead of a single number. Chlorine is listed as [35.446, 35.457] and magnesium as [24.304, 24.307]. For ordinary calculations, use the conventional single values, 35.45 and 24.305, unless your course says otherwise.