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Química Geral · Atoms, moles and stoichiometry

Medidas, unidades e algarismos significativos

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O que você vai aprender

  • Use SI base units, derived units and metric prefixes, and write numbers in standard scientific notation.
  • Count significant figures and round the results of multiplication, division, addition, subtraction and logarithms correctly.
  • Convert units by dimensional analysis, squaring or cubing the conversion factor for areas and volumes.
  • Calculate density, mass or volume from the other two and use density to compare or identify materials.
  • Convert between the Kelvin, Celsius and Fahrenheit scales and distinguish a temperature from a temperature change.
  • Distinguish accuracy from precision, systematic from random error, and calculate percent error.

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SI units and prefixes

The SI has seven base units: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for current, kelvin (K) for temperature, mole (mol) for amount of substance and candela (cd) for luminous intensity. The kilogram is the only base unit that already carries a prefix. Every other unit is derived from these: volume in m³, density in kg/m³, force in newtons (1 N = 1 kg m s⁻²), pressure in pascals (1 Pa = 1 N/m² = 1 kg m⁻¹ s⁻²) and energy in joules (1 J = 1 kg m² s⁻²). The litre is not an SI unit but is accepted alongside it: 1 L = 1 dm³ = 1000 cm³, so 1 mL = 1 cm³. A prefix multiplies the unit by a power of ten: mega (M) 10⁶, kilo (k) 10³, centi (c) 10⁻², milli (m) 10⁻³, micro (µ) 10⁻⁶, nano (n) 10⁻⁹, pico (p) 10⁻¹². The yellow sodium line at 589 nm is 589 × 10⁻⁹ m = 5.89 × 10⁻⁷ m = 0.589 µm.

Scientific notation and significant figures

Standard scientific notation writes a number as a coefficient from 1 up to (but not including) 10 times a power of ten: 0.0003070 g = 3.070 × 10⁻⁴ g. Significant figures are the digits of a measurement that carry meaning: every certain digit plus one estimated digit. Nonzero digits always count; zeros between nonzero digits count; leading zeros never count; trailing zeros count when the number has a decimal point. So 0.004050 has four (4, 0, 5, 0), and 1500 is ambiguous until it is written 1.5 × 10³ (two) or 1.50 × 10³ (three). Exact numbers, such as counted objects and defined factors (1 m = 100 cm, 1 in = 2.54 cm), have unlimited significant figures. Rounding rules: in multiplication and division keep the fewest significant figures of any measured factor (4.56 cm × 1.4 cm = 6.384 → 6.4 cm²); in addition and subtraction keep the fewest decimal places (102.5 g + 18.27 g = 120.77 → 120.8 g); for a logarithm, the number of decimal places equals the significant figures of the argument (pH of [H₃O⁺] = 2.5 × 10⁻⁴ M is 3.60). Carry extra digits through a multistep calculation and round only at the end.

Dimensional analysis, including squared and cubed units

Write each conversion as a fraction equal to 1, oriented so the unwanted unit cancels: 250.0 mL × (1 L / 1000 mL) = 0.2500 L. Chain factors for multistep problems: a 15.0 mg/kg dose for a 154 lb patient is 154 lb × (1 kg / 2.205 lb) × (15.0 mg / 1 kg) = 1.05 × 10³ mg. When a unit is squared or cubed, the factor is squared or cubed too, because each length dimension converts separately: 1 m² = (100 cm)² = 10⁴ cm², and 1 m³ = (100 cm)³ = 10⁶ cm³ = 1000 L. A density of glycerol (drawn), 1.261 g/cm³, is therefore 1.261 g/cm³ × (1 kg / 1000 g) × (100 cm / 1 m)³ = 1261 kg/m³; forgetting the cube would give 0.1261. Checking that the units cancel to the requested unit catches most setup errors.

Density

Density is mass per unit volume, d = m/V, usually in g/cm³ (= g/mL) for solids and liquids. Rearranged, m = d × V and V = m/d. Ethanol (drawn) has d = 0.789 g/mL at 20 °C, so 250.0 mL has a mass of 250.0 × 0.789 = 197 g. The volume of an irregular solid is the volume of water it displaces: a 44.5 g metal sample that raises the water level from 25.0 mL to 30.0 mL displaces 5.0 mL and has d = 44.5 g / 5.0 mL = 8.9 g/mL. An object floats in a liquid that is denser than it and sinks in one that is less dense: ice (0.917 g/cm³) floats on water but sinks in ethanol. Density depends on temperature, because most materials expand on warming while their mass stays the same; water is densest near 4 °C (0.99997 g/mL) and is 0.9982 g/mL at 20 °C.

Temperature scales

The kelvin scale starts at absolute zero and has degrees the same size as Celsius degrees: T(K) = T(°C) + 273.15. A Fahrenheit degree is 5/9 of a Celsius degree and water freezes at 32 °F: T(°F) = 1.8 × T(°C) + 32, and T(°C) = (T(°F) − 32)/1.8. Liquid nitrogen (drawn) boils at 77 K = −196 °C; ethanol boils at 78.4 °C = 1.8 × 78.4 + 32 = 173.1 °F; −40 °F and −40 °C are the same temperature. A temperature change converts without the offsets: a rise of 25.0 °C is a rise of 25.0 K and of 1.8 × 25.0 = 45.0 °F.

Accuracy, precision and error

Accuracy is how close a result is to the true or accepted value; precision is how close repeated results are to one another. Weighings of 9.71, 9.70, 9.72 and 9.71 g for a 10.00 g standard are precise but not accurate. Systematic error shifts every result the same way (an untared weighing paper, a miscalibrated pipet) and spoils accuracy without changing the scatter; random error scatters results in both directions and spoils precision, and averaging more trials reduces its effect. Percent error = |measured − accepted| / accepted × 100 %: a measured aluminium density of 2.552 g/cm³ against the accepted 2.699 g/cm³ is 0.147 / 2.699 × 100 % = 5.45 % error. The tool sets the precision available: a volumetric pipet delivers 25.00 mL, a graduated cylinder marked every 1 mL is read to the nearest 0.1 mL, and a beaker only approximates volume.

What to carry forward

Carry units through every calculation and let them cancel; square or cube a length factor whenever the unit is squared or cubed. Report results with the significant figures the measurements justify, rounding only at the end. Density links mass and volume, kelvin and Celsius differ only by 273.15, and a good measurement is both accurate (close to the true value) and precise (reproducible).

Problemas de prática (40)

Praticar é grátis com uma conta.

Preveja o produto 1

  • Does ice float in water and in ethanol?Predict whether a solid floats or sinks by comparing its density with that of the liquid.

Classificar 5

Significant figures

  • Which number is exact?Recognise defined relationships as exact numbers that never limit significant figures.
  • Show three significant figures in 1500 mUse scientific notation to remove the ambiguity of trailing zeros in a whole number.

Temperature, accuracy and precision

  • Classify a set of weighingsClassify a data set as accurate and/or precise by comparing its spread and its mean with the true value.

Units, prefixes and scientific notation

  • Write a small mass in scientific notationConvert a decimal with leading and trailing zeros to standard scientific notation without losing significant figures.
  • Pick the SI base unitDistinguish an SI base unit from prefixed, derived and accepted non-SI units.

Comparar 4

Density

  • Rank densities given in different unitsConvert densities quoted in g/mL, kg/m³, kg/cm³ and kg/L to one unit and rank them.

Units, prefixes and scientific notation

  • Rank lengths written with different prefixesCompare lengths expressed with pico, nano, micro and milli prefixes by converting to a common unit.

Significant figures

  • Rank measurements by significant figuresRank measurements by their number of significant figures, applying the leading, captured and trailing zero rules.

Temperature, accuracy and precision

  • Rank temperatures on three scalesConvert Kelvin and Fahrenheit temperatures to one scale and rank them, including two that are close.

Calcular 23

Significant figures

  • Total mass of three portionsRound a sum to the decimal place of the least precise term rather than by counting significant figures.
  • Count significant figures in 0.004050 gApply the zero rules to count the significant figures of a decimal measurement.
  • Significant figures in a pH valueApply the decimal-place rule for logarithms when converting a concentration to pH.
  • Area of a rectangle with correct significant figuresRound a product to the significant figures of the least precise factor.
  • Significant figures in a two-step calculationTrack significant figures through a subtraction followed by a multiplication without rounding the intermediate.

Density

  • Density of a metal blockCalculate density from a measured mass and volume.
  • Density by water displacementFind the volume of an irregular solid by displacement and use it to calculate density.
  • Mass of a measured volume of ethanolUse density to convert a liquid volume into a mass.
  • Volume of a kilogram of mercuryCalculate a volume from mass and density after converting the mass to match the density's units.

Unit conversions

  • Square metres to square centimetresConvert an area by squaring the length conversion factor.
  • Cubic inches to litresConvert a volume between unit systems by cubing an exact length factor.
  • Dose from body mass in poundsChain several conversion factors, including a dose rate, to reach a mass in the requested unit.
  • Density of glycerol in SI unitsConvert a density from g/cm³ to kg/m³, converting the mass unit and cubing the length factor.
  • Mass of lead in a water tankCombine a cubed length conversion with a concentration and a prefix change to find a total mass.
  • Volume of a cubic tank in litresCalculate a volume from a length and convert cubic metres to litres.

Temperature, accuracy and precision

  • Boiling point of ethanol in °FConvert a Celsius temperature to Fahrenheit using both the scale factor and the offset in the right order.
  • Boiling point of liquid nitrogen in °CConvert a temperature from kelvins to degrees Celsius.
  • Percent error in a densityCalculate the percent error of a measurement relative to an accepted value.
  • A temperature change in K and °FConvert a temperature difference between scales without applying the zero-point offsets.

Units, prefixes and scientific notation

  • Multiply and divide in scientific notationCombine coefficients and exponents when multiplying and dividing numbers in scientific notation.
  • Caffeine concentration in µg/LConvert a concentration unit by changing the prefixes of both numerator and denominator.
  • The pascal in base unitsExpress a derived SI unit in base units by dimensional analysis of its defining equation.
  • The sodium line in micrometresConvert a length between two metric prefixes by way of the base unit.

Encontre o erro 2

Unit conversions

  • Find the error in an area conversionDiagnose a dimensional-analysis setup whose units fail to cancel when converting squared units.

Density

  • Fix an over-precise densityDiagnose a density reported with more significant figures than the measurements support.

Interprete as evidências 2

Temperature, accuracy and precision

  • Judge three students' pipet calibrationsInterpret three replicate data sets by computing their means and spreads and comparing them with the true value.

Significant figures

  • Read a graduated cylinderRecord a scale reading with all certain digits plus one estimated digit.

Trabalhe de trás para frente 1

  • Edge length of a gold cubeWork backwards from mass and density to the edge length of a cube.

Planeje uma síntese 1

  • Set up a millilitre-to-litre conversionChoose a conversion factor with the correct value and orientation so the original unit cancels.

Explique por quê 1

  • Effect of forgetting to tareExplain how a systematic error affects accuracy and precision differently.

Travou em medidas, unidades e algarismos significativos? Escaneie o problema.

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