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Whole number rule

Whole number rule is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Whole number rule rather than just read about it. In short: In chemistry, the whole number rule states that the masses of the isotopes are whole number multiples of the mass of the hydrogen atom. The rule is a modified version of Prout's hypothesis proposed in 1815, to the effect that atomic weights are multiples of the weight of the hydrogen atom.

Whole number rule — main illustration
Whole number rule — illustration

Key takeaways

  • Whole number rule belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Whole number rule to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Whole number rule from memory before moving on to harder problems.

Reference excerpt

In chemistry, the whole number rule states that the masses of the isotopes are whole number multiples of the mass of the hydrogen atom. The rule is a modified version of Prout's hypothesis proposed in 1815, to the effect that atomic weights are multiples of the weight of the hydrogen atom. It is also known as the Aston whole number rule after Francis W. Aston who was awarded the Nobel Prize in Chemistry in 1922 "for his discovery, by means of his mass spectrograph, of isotopes, in a large number of non-radioactive elements, and for his enunciation of the whole-number rule".

Law of definite proportions

The law of definite proportions was formulated by Joseph Proust around 1800 and states that all samples of a chemical compound will have the same elemental composition by mass. The atomic theory of John Dalton expanded this concept and explained matter as consisting of discrete atoms with one kind of atom for each element combined in fixed proportions to form compounds.

Prout's hypothesis In 1815, William Prout reported on his observation that the atomic masses of the elements were whole multiples of the atomic mass of hydrogen. He then hypothesized that the hydrogen atom was the fundamental object and that the other elements were a combination of different numbers of hydrogen atoms.

Aston's discovery of isotopes In 1920, Francis W. Aston demonstrated through the use of a mass spectrometer that apparent deviations from Prout's hypothesis are predominantly due to the existence of isotopes. For example, Aston discovered that neon has two isotopes with masses very close to 20 and 22 as per the whole number rule, and proposed that the non-integer value 20.2 for the atomic weight of neon is due to the fact that natural neon is a mixture of about 90% neon-20 and 10% neon-22). A secondary cause of deviations is the binding energy or mass defect of the individual isotopes.

Discovery of the neutron

During the 1920s, it was thought that the atomic nucleus was made of protons and electrons, which would account for the disparity between the atomic number of an atom and its atomic mass. In 1932, James Chadwick discovered an uncharged particle of approximately the mass as the proton, which he called the neutron. The fact that the atomic nucleus is composed of protons and neutrons was rapidly accepted and Chadwick was awarded the Nobel Prize in Physics in 1935 for his discovery. The modern form of the whole number rule is that the atomic mass of a given elemental isotope is approximately the mass number (number of protons plus neutrons) times a dalton (approximate mass of a proton, neutron, or hydrogen-1 atom). This rule predicts the atomic mass of nuclides and isotopes with an error of at most 1%, with most of the error explained by the mass deficit caused by nuclear binding energy.

References

Further reading Harkins WD (1925). "The Separation of Chlorine into Isotopes (Isotopic Elements) and the Whole Number Rule for Atomic Weights". Proc. Natl. Acad. Sci. U.S.A. 11 (10): 624–8. Bibcode:1925PNAS...11..624H. doi:10.1073/pnas.11.10.624. PMC 1086175. PMID 16587053.

External links 1922 Nobel Prize Presentation Speech

Illustrations

Whole number rule: Francis W. Aston received the 1922 Nobel Prize in Chemistry for his enunciation of the whole-number rule.
Francis W. Aston received the 1922 Nobel Prize in Chemistry for his enunciation of the whole-number rule.
Whole number rule: John Dalton's list of atomic weights and symbols
John Dalton's list of atomic weights and symbols
Whole number rule: James Chadwick, discoverer of the neutron, with General Leslie Groves, director of the Manhattan Project.
James Chadwick, discoverer of the neutron, with General Leslie Groves, director of the Manhattan Project.

Worked examples

Example 1 — a first encounter with Whole number rule

Start with the simplest possible case. Write down what Whole number rule claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Whole number rule before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Whole number rule ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Whole number rule

In research
Whole number rule appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Whole number rule in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Whole number rule is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mass spectrometry, Periodic table, so understanding it makes those chapters shorter.
In everyday life
Look for Whole number rule outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Whole number rule in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Whole number rule means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Whole number rule out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Whole number rule in simple terms?

In chemistry, the whole number rule states that the masses of the isotopes are whole number multiples of the mass of the hydrogen atom. The rule is a modified version of Prout's hypothesis proposed in 1815, to the effect that atomic weights are multiples of the weight of the hydrogen atom.

Why does Whole number rule matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Whole number rule?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Whole number rule.

Tags

  • Mass spectrometry
  • Periodic table

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