Name: significant figures and rounding rules
Uploaded: 2020-09-03T13:38:25.000Z
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Chemistry

Structure And Properties Of Matter

Quantifying matter

uncertainty-in-measurement-significant-figures

Uncertainty in Measurement: Significant Figures

Todos os números numa medida científica são certos, exceto o último dígito. A certeza da medida depende de dois fatores:o número de dígitos na medida e a precisão do instrumento utilizado. Numa quantidade medida, todos os dígitos, incluindo o último dígito incerto, são chamados de dígitos significativos e podem ser determinados utilizando regras específicas.Quaisquer dígitos não zero e todos os zeros cativos que se situam entre dois dígitos não zero são significativos. Por exemplo, 28 tem dois dígitos significativos, enquanto 26, 25 tem quatro e 208 tem três. Os zeros à esquerda nunca são significativos, eles apenas indicam o ponto decimal.Por exemplo, 0, 00208 tem três dígitos significativos. Tais quantidades podem ser expressas utilizando representações exponenciais. Assim, 0, 00208 pode ser escrito como 2, 08 10-³.Os zeros à direita são significativos apenas em números decimais. 2200 tem dois zeros à direita e dois dígitos significativos, enquanto que 2200.0 e 2200.1 têm 5 dígitos significativos. Para quantidades sem pontos decimais, o significado dos zeros à direita é ambíguo.Assim, 2200 pode ser escrito como 2, 2 10³ com dois dígitos significativos ou 2, 20 10³ com três dígitos significativos. Os dígitos significativos também ajudam a alcançar a certeza nas operações matemáticas. Na adição ou subtração, o resultado deve ser arredondado para ter o mesmo número de casas decimais que a medida com o menor número de casas decimais.O arredondamento para baixo deve ser efetuado quando o último dígito é inferior a 5, e o arredondamento para cima deve ser efetuado quando é 5 ou superior. Às vezes são utilizados outros métodos de arredondamento quando o último dígito é 5. Por exemplo, a soma de 2, 052 e 1, 2 é arredondada para 3, 3.No entanto, na multiplicação ou divisão, o resultado deve ser arredondado para ter o mesmo número de dígitos significativos que a medida com os dígitos menos significativos. Assim, o produto de 2, 052 e 1, 2 é arredondado para 2, 5. Os cientistas repetem frequentemente as suas experiências para alcançarem precisão nas suas medições.O desvio padrão é a expressão estatística dessa precisão e mede a dispersão relativamente ao valor esperado. Se a precisão for elevada, o desvio padrão é pequeno, e vice-versa. Por exemplo, dois grupos mediram a espessura de um livro em centímetros.Encontraram a mesma média 10, 6 centímetros. No entanto, as medidas do primeiro grupo são mais precisas, tendo assim um desvio padrão inferior. O segundo grupo tem mais medidas dispersas e um desvio padrão mais elevado.

incertezas na medição: valores significativos

Todos os dígitos em uma medição, incluindo o último dígito incerto, são chamados de algarismos significativos ou dígitos significativos. Note que o zero pode ser um valor medido; por exemplo, se uma balança que mostre o peso à libra mais próxima indicar “140”, então 1 (centenas), 4 (dezenas), e 0 (unidades) são todos valores significativos (medidos).

Um resultado de medição é devidamente comunicado quando os seus dígitos significativos representam com exatidão a certeza do processo de medição. Abaixo encontra-se um conjunto de regras para determinar o número de algarismos significativos em uma medição:

<ol>
<li>Todos os dígitos não-zero são significativos. Começando com o primeiro dígito diferente de zero à esquerda, conte esse dígito e todos os dígitos restantes à direita. Este é o número de algarismos significativos na medição. Por exemplo, 843 tem três dígitos significativos, 843,12 tem 5 dígitos significativos. </li>
<li>Zeros cativos, que são zeros entre dois dígitos diferentes de zero, são significativos. Por exemplo, 808,101 tem dois zeros cativos e 6 algarismos significativos.</li>
<li>Os zeros à esquerda são zeros à esquerda do primeiro dígito diferente de zero. Os zeros à esquerda nunca são significativos; eles representam apenas a posição do ponto decimal. Por exemplo, os zeros à esquerda em 0,008081 não são significativos. Este número pode ser expresso usando notação exponencial como 8,081 × 10−3, então o número 8,081 contém todos os algarismos significativos, e 10−3 localiza o ponto decimal.</li>
<li>A importância dos zeros à direita, que são zeros no final de um número, depende da sua posição. Os zeros à direita antes (mas depois de um dígito diferente de zero) e depois do ponto decimal são significativos. No entanto, para números que não têm pontos decimais, os zeros posteriores podem ou não ser significativos. Esta ambiguidade pode ser resolvida com o uso da notação exponencial. Por exemplo, a medida 1300 pode ser escrita como 1,3 × 103 (dois algarismos significativos), 1,30 × 103 (três algarismos significativos, se o local das dezenas foi medido), ou 1,300 × 103 (quatro algarismos significativos, se o local das unidades também foi medido).</li>
</ol>

<h4>Algarismos Significativos em Cálculos</h4>

A incerteza nas medições pode ser evitada expondo os resultados do cálculo com o número correcto de algarismos significativos. Isto pode ser determinado pelas seguintes regras para arredondar números: 

<ol>
<li>Ao adicionar ou subtrair números, arredondar o resultado para o mesmo número de casas decimais que o número com o menor número de casas decimais.</li>
<li>Ao multiplicar ou dividir números, arredondar o resultado para o mesmo número de dígitos que o número com o menor número de algarismos significativos.</li>
<li>Se o dígito a ser eliminado (o imediatamente à direita do dígito a ser mantido) for menor que 5, “arredondar para baixo” e não alterar o dígito a ser mantido.</li>
<li>Se o dígito a ser eliminado (o imediatamente à direita do dígito a ser mantido) for 5 ou superior, “arredondar para cima” e aumentar o dígito mantido em 1. Também podem ser utilizados métodos alternativos de arredondamento se o dígito a ser eliminado for 5. O dígito mantido é arredondado para cima ou para baixo, o que render um valor uniforme. </li>
</ol>

Uma nota importante é que o arredondamento de algarismos significativos deve ser feito preferencialmente no final de um cálculo com várias etapas para evitar a acumulação de erros em cada etapa devido ao arredondamento. Assim, algarismos significativos e arredondamentos facilitam a representação correta da certeza dos valores medidos reportados.

Este texto é adaptado de <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>

Incertezas na Medição: Valores Significativos

Todos os dígitos em uma medição, incluindo o último dígito incerto, são chamados de algarismos significativos ou dígitos significativos. Observe que zero pode ser um ...

Education

JoVE Core

Química

Introduction: Matter and Measurement

Introdução: Matéria e Medida

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

Scientific laws and scientific theories describe science in two different ways. A law is a concise verbal or mathematical statement that summarizes many experimental observations. It describes or predicts aspects of the natural world that stay the same under the same conditions. A scientific theory is different because it explains why natural events happen. It is a unifying principle that gives a well-supported and testable explanation for a broad set of observations. Strong theories are the...

the scientific method

How Scientists Test a Hypothesis

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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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States of matter help describe how matter behaves in chemistry. Matter is anything that has mass and takes up space. It is found everywhere, including air, water, soil, mountains, and even our bodies. The three common states on Earth are solid, liquid, and gas. A solid has a definite shape and a definite volume. Its particles, such as atoms or molecules, are packed very closely together and held in place by strong forces, so they only vibrate slightly. This tight arrangement makes solids...

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classifying matter by composition

Matter: Elements, Compounds, and Mixtures

Matter can be grouped by composition into pure substances and mixtures. A pure substance has a constant composition throughout and uniform properties. For example, any sample of sucrose has the same composition and the same physical properties, such as melting point, color, and sweetness, no matter where it is isolated from. A mixture contains two or more kinds of matter in varying amounts. Its parts can be separated by physical changes, such as evaporation. The components keep their own...

measurement: derived units

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SI units and derived units help describe physical properties in chemistry. The International System of Units, or SI system, uses agreed-upon base units for seven fundamental properties: length, mass, time, temperature, electric current, amount of substance, and luminosity. Units made by combining these base units are called SI-derived units. Speed is one example. If distance is measured in meters and time in seconds, the derived unit for speed is meters per second, or m/s. Miles per hour is...

dimensional analysis

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uncertainty in measurement: accuracy and precision

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Measurement Error: Reading Results Correctly

Measurement error affects how scientists judge the quality of their results. Repeated measurements help show whether results are precise and accurate. A measurement is precise when repeated values are very similar. A measurement is accurate when it is very close to the true or accepted value. Precise values agree with each other, while accurate values agree with the true value. A pharmaceutical quality control chemist can use this idea to check machines that dispense 500 mL of cough syrup. The...

uncertainty in measurement: reading instruments

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Measurement Uncertainty and Significant Figures

Measurement uncertainty and significant figures show how carefully a value can be read from an instrument. Counting is different. If the number of objects does not change during counting, the result is exact. Defined quantities are exact too. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms. Most other measurements are not exact because the measuring process has limits. The result depends on the device used and on the person...

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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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

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The Quantum-Mechanical Model of an Atom

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The Pauli Exclusion Principle

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The Energies of Atomic Orbitals

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Atomic structure

The Periodic Table

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Periodic Classification of the Elements

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Atomic Radii and Effective Nuclear Charge

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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
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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

Chemical bonds are complex interactions between two or more atoms or ions, which reduce the potential energy of the molecule. Gilbert N. Lewis developed a ...

Ionic Bonding and Electron Transfer

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The Born-Haber Cycle

Lattice Energy&#xA0;

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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 ...

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Bond Polarity, Dipole Moment, and Percent Ionic Character

Bond Polarity

The absolute value of the difference in electronegativity (&Delta;EN) of two bonded atoms provides a rough measure of the polarity to be ...

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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

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Predicting Molecular Geometry

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Molecular Shape and Polarity

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Valence Bond Theory

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Hybridization of Atomic Orbitals II

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Molecular Orbital Theory I

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Chemical bonding
• lonic
• Polar/covalent

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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
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• 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 ...

Dimensional Analysis

Dimensional analysis, also known as the factor label method, is a versatile approach for mathematical operations. The main principle behind this approach ...

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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