Name: significant figures and rounding rules
Uploaded: 2020-09-03T13:38:25.000Z
Duration: 3 min 34 s
Channel: JoVE

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Ohio’s Learning Standards and Model Curriculum

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Chemistry

Structure And Properties Of Matter

Quantifying matter

uncertainty-in-measurement-significant-figures

Uncertainty in Measurement: Significant Figures

Tutti i numeri in una misurazione scientifica sono certi, tranne l'ultima cifra. La certezza della misura dipende da due fattori:il numero di cifre nella misura, e la precisione dello strumento utilizzato. In una quantità misurata, tutte le cifre, inclusa l'ultima cifra incerta, sono definite cifre significative e possono essere determinate utilizzando regole specifiche.

Qualsiasi cifra diversa dallo zero, e tutti gli zeri che si trovano tra due cifre diverse dallo zero, sono significative. Per esempio, 28 ha due cifre significative, mentre 26, 25 ne ha quattro e 208 ne ha tre. Gli zeri iniziali non sono mai significativi, poiché individuano solo il punto decimale.

Per esempio, 0, 00208 ha tre cifre significative. Tali quantità possono essere espresse utilizzando notazioni esponenziali. Pertanto, 0, 00208 può essere scritto come 2, 08 10⁻³.

Gli zeri finali sono significativi solo nei numeri formattati decimali. 2200 ha due zeri finali e due cifre significative, mentre 2200, 0 e 2200, 1 hanno entrambi 5 cifre significative. Per le quantità senza punti decimali, il significato degli zeri finali diventa ambiguo.

Quindi, 2200 può essere scritto come 2, 2 10³ con due cifre significative, o 2, 20 10³ con tre cifre significative. Cifre significative aiutano a raggiungere la certezza anche nelle operazioni matematiche. Nelle operazioni di addizione o sottrazione, il risultato dovrebbe essere arrotondato per avere lo stesso numero di cifre decimali della misurazione con il minor numero di cifre decimali.

L'arrotondamento per difetto dovrebbe essere eseguito quando l'ultima cifra è inferiore a 5 e l'arrotondamento per eccesso quando è 5 o superiore. Altri metodi di arrotondamento vengono talvolta utilizzati quando l'ultima cifra è 5. Per esempio, la somma di 2, 052 e 1, 2 viene arrotondata a 3, 3.

Tuttavia, moltiplicando o dividendo, il risultato dovrebbe essere arrotondato per avere lo stesso numero di cifre significative della misurazione con il minor numero di cifre significative. Pertanto, il prodotto di 2, 052 e 1, 2 viene arrotondato a 2, 5. Gli scienziati spesso ripetono gli esperimenti per ottenere la precisione nelle misurazioni.

La deviazione standard è l'espressione statistica di tale precisione e misura la dispersione dal valore atteso. Se la precisione è alta, la deviazione standard sarà piccola e viceversa. che porta ad una discrepanza consistente nella misurazione.

Per esempio, due gruppi hanno misurato lo spessore di un libro Questi errori tendono ad essere troppo alti o troppo bassi in centimetri. Hanno trovato la stessa media:10, 6 centimetri. rispetto al valore reale.

Tuttavia, le misurazioni del primo gruppo sono più precise, Per esempio, i pesi misurati utilizzando una bilancia calibrata in modo improprio, e quindi hanno una deviazione standard inferiore. Il secondo gruppo ha misurazioni più distribuite ed una deviazione sono prevedibili e per lo più correlati allo strumento. standard più elevata.

nte Tuttavia, a differenza dell'errore casuale, non può essere calcolato come una media con misurazioni ripetute.

incertezza nella misurazione: cifre significative

Tutte le cifre di una misurazione, inclusa l'ultima cifra incerta, sono chiamate cifre significative. Si noti che zero può essere un valore misurato; ad esempio, se una bilancia che mostra il peso alla libbra più vicina indica "140", allora 1 (centinaia), 4 (decine) e 0 (unità) sono tutti valori significativi (misurati).

Un risultato di misurazione viene riportato correttamente quando le sue cifre significative rappresentano accuratamente la certezza del processo di misurazione. Di seguito sono riportate una serie di regole per determinare il numero di cifre significative in una misurazione:

<ol>
	<li>Tutte le cifre diverse da zero sono significative. Iniziando dalla prima cifra diversa da zero a sinistra, conta questa cifra e tutte le altre rimanenti a destra. Questo è il numero di cifre significative nella misurazione. Ad esempio, 843 ha tre cifre significative; 843.12 ha 5 cifre significative.</li>
	<li>Gli zeri vincolati, che sono zeri tra due cifre diverse da zero, sono significativi. Ad esempio, 808.101 ha due zeri vincolati e 6 cifre significative.</li>
	<li>Gli zeri iniziali sono zeri a sinistra della prima cifra diversa da zero. Le cifre iniziali non sono mai significative; rappresentano semplicemente la posizione del punto decimale. Ad esempio, gli zeri iniziali in 0,008081 non sono significativi. Questo numero può essere espresso utilizzando la notazione esponenziale come 8.081 × 10^−3, quindi il numero 8.081 contiene tutte le cifre significative e 10^-3 individua il punto decimale.</li>
	<li>Il significato degli zeri finali, che sono zeri alla fine di un numero, dipende dalla loro posizione. Gli zeri finali prima (ma dopo una cifra diversa da zero) e dopo il punto decimale sono significativi. Tuttavia, per i numeri che non hanno punti decimali, gli zeri finali possono essere significativi o meno. Questa ambiguità può essere risolta con l'uso della notazione esponenziale. Ad esempio, la misura 1300 può essere scritta come 1,3 ×10^3 (due cifre significative), 1,30 ×10^3 (tre cifre significative, se è stata misurata la cifra delle decine), oppure 1.300 × 10^3 (quattro cifre significative, se è stata misurata anche la cifra delle unità).</li>
</ol>

Cifre significative nei calcoli

L'incertezza nelle misurazioni può essere evitata riportando i risultati del calcolo con il corretto numero di cifre significative. Ciò può essere determinato dalle seguenti regole per l'arrotondamento dei numeri:

<ol>
	<li>Quando si sommano o sottraggono numeri, bisogna arrotondare il risultato allo stesso numero di cifre decimali del numero con il minor numero di cifre decimali.</li>
	<li>Quando si moltiplicano o si dividono numeri, si arrotonda il risultato allo stesso numero di cifre del numero con il minor numero di cifre significative.</li>
	<li>Se la cifra da eliminare (quella immediatamente a destra della cifra da conservare) è inferiore a 5, “arrotondare per difetto” e lasciare invariata la cifra mantenuta.</li>
	<li>Se la cifra da eliminare (quella immediatamente a destra della cifra da conservare) è 5 o superiore, "arrotondare per eccesso" e aumentare la cifra conservata di 1. È possibile utilizzare anche metodi di arrotondamento alternativi se la cifra eliminata è 5 . La cifra mantenuta viene arrotondata per eccesso o per difetto, a seconda di quale dei due restituisce un valore pari.</li>
</ol>

Una nota importante è che l'arrotondamento delle cifre significative dovrebbe essere effettuato preferibilmente alla fine di un calcolo multifase per evitare l'accumulo di errori in ogni fase a causa dell'arrotondamento. Pertanto, le cifre significative e gli arrotondamenti facilitano la corretta rappresentazione della certezza dei valori misurati riportati.

Questo testo è adattato da <a href="https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision">Openstax, Chemistry 2e, Section 1.5: Measurement Uncertainty, Accuracy, and Precision. </a>

Incertezza nella misurazione: cifre significative

Tutte le cifre di una misurazione, inclusa l'ultima cifra incerta, sono chiamate cifre significative o cifre significative. Si noti che zero può essere un ...

Education

Didattica

JoVE Core

Chimica

Introduction: Matter and Measurement

Introduzione: Materia e Misurazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

significant figures and rounding rules

Significant figures and rounding rules help students report measurements with the right level of certainty. A measurement includes every digit that was actually measured, including the last uncertain digit. That is why a value can look simple but still carry useful information about the tool used to measure it.

Zero can also be significant when it is part of a measured value. For example, if a scale reads 140 pounds to the nearest pound, the 1, 4, and 0 are all significant. Reporting measurements this way shows how certain the result really is.

To count significant figures, all nonzero digits are significant. Start with the first nonzero digit on the left and count that digit and every digit to the right. By this rule, 843 has three significant figures, and 843.12 has five.

Zeros can change the count in different ways. Captive zeros, which are zeros between two nonzero digits, are significant, as in 808.101, which has six significant figures. Leading zeros are not significant because they only show where the decimal point is. For example, 0.008081 has four significant figures, and it can also be written as 8.081 × 10^-3.

Trailing zeros are significant only in some cases. Zeros at the end of a number are significant if they come after a decimal point, or if they are before a decimal point but still follow a nonzero digit. If a number has no decimal point, trailing zeros may or may not be significant. Exponential notation can remove that uncertainty. For example, 1300 could mean 1.3 × 10^3, 1.30 × 10^3, or 1.300 × 10^3, depending on how many digits were measured.

The same idea applies to calculations. When adding or subtracting, round the answer to the same number of decimal places as the value with the fewest decimal places. When multiplying or dividing, round the answer to the same number of significant figures as the value with the fewest significant figures. If the digit to be dropped is less than 5, round down. If it is 5 or greater, round up.

An alternative method may be used when the digit being dropped is 5. In that case, the retained digit is rounded to the even value. It is also best to round only at the end of a multistep calculation. That helps avoid extra error from rounding too early. Significant figures and rounding together help students show measurement certainty correctly.

Significant Figures and Rounding Rules

scientific laws and theories

How Science Explains Laws and Theories

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the scientific method

How Scientists Test a Hypothesis

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physical and chemical properties of matter

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Distinguishing Physical and Chemical Changes

Matter can be identified by its properties, which are the features that help us tell one substance from another. These properties fall into two main groups: physical properties and chemical properties. Understanding the difference helps explain how matter changes. A physical property is a feature of matter that is not linked to a change in chemical composition. Examples include density, color, hardness, melting point, boiling point, and electrical conductivity. Some physical properties can be...

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Energy Forms and Conservation

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measurement: derived units

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uncertainty in measurement: accuracy and precision

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Measurement Error: Reading Results Correctly

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uncertainty in measurement: reading instruments

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Measurement Uncertainty and Significant Figures

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uncertainty in measurement: significant figures

Significant figures and rounding rules help students report measurements with the right level of certainty. A measurement includes every digit that was actually measured, including the last uncertain digit. That is why a value can look simple but still carry useful information about the tool used to measure it. Zero can also be significant when it is part of a measured value. For example, if a scale reads 140 pounds to the nearest pound, the 1, 4, and 0 are all significant. Reporting...

key terms and concepts

Learn how to count significant figures, identify leading and trailing zeros, and apply rounding rules in measurement calculations for high school chemistry.

Significant Figures and Rounding Rules | JoVE K12

chemistry

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Stoichiometry Problem-Solving

The Atomic Theory of Matter

The earliest recorded discussion of the basic structure of matter comes from ancient Greek philosophers. Leucippus and Democritus argued that all matter ...

The Bohr Model

Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded ...

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of ...

Subatomic Particles

Dalton was only partially correct about the particles that make up matter. All matter is composed of atoms, and atoms are composed of three smaller ...

Quantum Numbers

It is said that the energy of an electron in an atom is quantized; that is, it can be equal only to certain specific values and can jump from one energy ...

Atomic Orbitals

An atomic orbital represents the three-dimensional regions in an atom where an electron has the highest probability to reside. The radial distribution ...

The Pauli Exclusion Principle

The arrangement of electrons in the orbitals of an atom is called its electron configuration. We describe an electron configuration with a symbol that ...

The Energies of Atomic Orbitals

In an atom, the negatively charged electrons are attracted to the positively charged nucleus. In a multielectron atom, electron-electron repulsions are ...

The Aufbau Principle and Hund's Rule

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Electron Configuration of Multielectron Atoms

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Thomson's e/m Experiment

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Atomic structure
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Content elaboration: Physical Science included properties and locations of protons, neutrons and electrons, atomic number, mass number, cations and anions, isotopes and the strong nuclear force which holds the nucleus together. In this course, the historical development of the atomic model and the positions of electrons are explored in greater detail. Atomic models are constructed to explain experimental evidence and make predictions. The changes in the atomic model over time exemplify how scientific knowledge changes as new evidence emerges and how technological advancements like electricity extend the boundaries of scientific knowledge. Thompson's study of electrical discharges in cathode-ray tubes led to the discovery of the electron and the development of the plum pudding model of the atom. Rutherford's experiment, in which he bombarded gold foil with a-particles, led to the discovery that most of the atom consists of empty space with a relatively small, positively charged nucleus. Bohr used data from atomic spectra to propose a planetary model of the atom in which electrons orbit the nucleus, like planets around the sun. Later, Schrödinger used the idea that electrons travel in waves to develop a model in which electrons travel randomly in regions of space called orbitals (quantum mechanical model). Based on the quantum mechanical model, it is not possible to predict exactly where electrons are located but there is a region of space surrounding the nucleus in which there is a high probability of finding an electron (electron cloud or orbital). Data from atomic spectra (emission and absorption) gives evidence that electrons can only exist at certain discrete energy levels and not at energies between these levels. Atoms are usually in the ground state where the electrons occupy orbitals with the lowest available energy. However, the atom can become excited when the electrons absorb a photon with the precise amount of energy (indicated by the frequency of the photon) to move to an orbital with higher energy. Any photon without this precise amount of energy will be ignored by the electron. The atom exists in the excited state for a very short amount of time. When an electron drops back down to the lower energy level, it emits a photon that has energy equal to the energy difference between the levels. The amount of energy is indicated by the frequency of the light that is given off and can be measured. Each element has a unique emission and absorption spectrum due to its unique electron configuration and specific electron energy jumps that are possible for that element. Being aware of the quantum mechanical model as the currently accepted model for the atom is important for science literacy as it explains and predicts subatomic interactions, but details should be reserved for more advanced study. Electron energy levels consist of sublevels (s, p, d and f), each with a characteristic number and shape of orbitals. Orbital diagrams and electron configuration can be constructed to show the location of the electrons in an atom using established rules. Valence electrons are responsible for most of the chemical properties of elements. In this course, electron configuration (extended and noble gas notation) and orbital diagrams can be shown for any element in the first three periods. Although the quantum mechanical model of the atom explains the most experimental evidence, other models can still be helpful. Thinking of atoms as indivisible spheres is useful in explaining many physical properties of substances, such as the state (solid, liquid or gas) of a substance at room temperature. Bohr's planetary model is useful to explain and predict periodic trends in the properties of elements.

Atomic structure

The Periodic Table

As early chemists discovered more elements, they realized that various elements could be grouped by their similar chemical behaviors. One such grouping ...

Periodic Classification of the Elements

The periodic table arranges atoms based on increasing atomic number so that elements with the same chemical properties recur periodically. When their ...

atomic-radii-and-effective-nuclear-charge

Atomic Radii and Effective Nuclear Charge

The elements in groups of the periodic table exhibit similar chemical behavior. This similarity occurs because the members of a group have the same number ...

Ionic Radii

Ionic radius is the measure used to describe the size of an ion. A cation always has fewer electrons and the same number of protons as the parent atom; it ...

Ionization Energy

The amount of energy required to remove the most loosely bound electron from a gaseous atom in its ground state is called its first ionization energy ...

Electron Affinity

The electron affinity (EA) is the energy change for adding an electron to a gaseous atom to form an anion (negative ion).



This process can be either ...

Alkali Metals

Group 1 elements are soft and shiny metallic solids. They are malleable, ductile, and good conductors of heat and electricity. The melting points of the ...

Halogens

Group 17 elements, known as halogens, are nonmetals. At room temperature, fluorine and chlorine are gases, bromine is a liquid, and iodine a solid. ...

Noble Gases


The elements in group 18 are noble gases (helium, neon, argon, krypton, xenon, and radon). They earned the name &ldquo;noble&rdquo; because they were ...

Periodic Table
• Properties
• Trends

Content elaboration: In the Physical Science course, the concept that elements are placed in order of increasing atomic number in the periodic table such that elements with similar properties are placed in the same column is introduced. How the periodic table is divided into groups, families, periods, metals, nonmetals and metalloids is also included and will be revisited here. In this course, with more information about the electron configuration of elements, similarities in the configuration of the valence electrons for a particular group can be observed. The electron configuration of an atom can be determined from the position on the periodic table. The repeating pattern in the electron configuration for elements on the periodic table explains many of the trends in the properties observed. Atomic theory is used to describe and explain trends in properties across periods or down columns including atomic radii, ionic radii, first ionization energies, electronegativities and whether the element is a solid or gas at room temperature. Additional ionization energies, electron affinities and periodic properties of the transition elements, and the lanthanide and actinide series are reserved for more advanced study.

Periodic Table

Types of Chemical Bonds

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Lewis Symbols and the Octet Rule

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Ionic Bonding and Electron Transfer

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The Born-Haber Cycle

Lattice Energy&#xA0;

An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a ...

trends-in-lattice-energy-ion-size-and-charge

Trends in Lattice Energy: Ion Size and Charge

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Covalent Bonding and Lewis Structures

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Electronegativity

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Electronegativity values of ...

bond-polarity-dipole-moment-and-percent-ionic-character

Bond Polarity, Dipole Moment, and Percent Ionic Character

Bond Polarity

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lewis-structures-of-molecular-compounds-and-polyatomic-ions

Lewis Structures of Molecular Compounds and Polyatomic Ions

To draw Lewis structures for complicated molecules and molecular ions, it is helpful to follow a step-by-step procedure as outlined:


	Determine the ...

Resonance

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Formal Charges

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Exceptions to the Octet Rule

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	...

Bond Energies and Bond Lengths

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VSEPR Theory and the Basic Shapes

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vsepr-theory-and-the-effect-of-lone-pairs

VSEPR Theory and the Effect of Lone Pairs

Effect of Lone Pairs of Electrons&nbsp;on Molecule Geometry

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Predicting Molecular Geometry

VSEPR Theory for Determination of Electron Pair Geometries

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Molecular Shape and Polarity

Dipole Moment of a Molecule

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Valence Bond Theory

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Hybridization of Atomic Orbitals I

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Hybridization of Atomic Orbitals II

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Molecular Orbital Theory I

Overview of Molecular Orbital Theory

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Molecular Orbital Theory II

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Bonding in Metals

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Chemical bonding
• lonic
• Polar/covalent

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Chemical bonding

Chemical Formulas

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ionic-compounds-formulas-and-nomenclature

Ionic Compounds: Formulas and Nomenclature

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Molecular Compounds: Formulas and Nomenclature

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Organic Compounds

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Classification of Elements and Compounds

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Molecular Models

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Representing compounds
• Formula writing
• Nomenclature
• Models and shapes (Lewis structures, ball and stick, molecular geometries)

Content elaboration: Using the periodic table, formulas of ionic compounds containing specific elements can be predicted. This can include ionic compounds made up of elements from groups 1, 2, 17, hydrogen, oxygen and polyatomic ions (given the formula and charge of the polyatomic ion). Given the formula, a compound can be named using conventional systems that include Greek prefixes and Roman numerals where appropriate. Given the name of an ionic or covalent substance, formulas can be written. Many different models can be used to represent compounds including chemical formulas, Lewis structures, and ball and stick models. These models can be used to visualize atoms and molecules and to predict the properties of substances. Each type of representation provides unique information about the compound. Different representations are better suited for particular substances. Lewis structures can be drawn to represent covalent compounds using a simple set of rules and can be combined with valence shell electron pair repulsion (VSEPR) theory to predict the three-dimensional electron pair and molecular geometry of compounds. Lewis structures and molecular geometries will only be constructed for the following combination of elements: hydrogen, carbon, nitrogen, oxygen, phosphorus, sulfur and the halogens.

Representing compounds

Measurement: Standard Units

Every measurement provides three kinds of information: the size or magnitude of the measurement (a number), a standard of comparison for the measurement ...

Measurement: Derived Units

The International System of Units or SI system, by international agreement, has fixed measurement units for seven fundamental properties: length, mass, ...

Uncertainty in Measurement: Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy ...

Uncertainty in Measurement: Reading Instruments

Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such ...

All the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a ...

Quantifying matter

Content elaboration: In earlier grades, properties of materials were quantified with measurements that were always associated with some error. In this course, scientific protocols for quantifying the properties of matter accurately and precisely are studied. Using the International System of Units (SI), significant digits or figures, scientific notation, error analysis and dimensional analysis are vital to scientific communication. There are three domains of magnitude in size and time: the macroscopic (human) domain, the cosmic domain and the submicroscopic (atomic and subatomic) domain. Measurements in the cosmic domain and submicroscopic domains require complex instruments and/or procedures. Matter can be quantified in a way that macroscopic properties such as mass can reflect the number of particles present. Elemental samples are a mixture of several isotopes with different masses. The atomic mass of an element is calculated given the mass and relative abundance of each isotope of the element as it exists in nature. Because the mass of an atom is very small, the mole is used to translate between the atomic and macroscopic levels. A mole is equal to the number of atoms in exactly 12 grams of the isotope carbon-12. The mass of one mole of a substance is equal to its molar mass in grams. The molar mass for a substance can be used in conjunction with Avogadro's number and the density of a substance to convert between mass, moles, volume and number of particles of a sample.

Significant Figures and Rounding Rules

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