Chemistry: Numbers in Science

• Numbers play an important roll in chemistry
• Describing scientific phenomenon requires small units that represent certain quantities

Concepts of Numbers in Science

• Units of Measurement
• Quantities that are measured and Calculated
• Uncertainty in Measurement
• Significant Figures
• Dimensional Analysis

Units of Measurement-SI Units

• Système International d'Unités is the International System of Units
• It is fundamental units from which all others are derived
• A different base unit is used for each quantity
• Mass has units of Kilograms with an abbreviation of kg
• Length has units of Meters with an abbreviation of m
• Time has units of Seconds with an abbreviation of s or sec
• Temperature has units of Kelvin with an abbreviation of K
• Amount of Substance units are Moles with an abbreviation of mol
• Electric Current units are Amperes with an abbreviation of A or amp
• Luminous intensity has units of Candela with an abbreviation of cd.

Metric System Prefixes

• Prefixes convert the base units into units that are appropriate for common usage or appropriate measure
• peta (P) = 10^15
• tera (T) = 10^12
• giga (G) = 10^9
• mega (M) = 10^6
• kilo (k) = 10^3
• deci (d) = 10^-1
• centi (c) = 10^-2
• milli (m) = 10^-3
• micro (μ) = 10^-6
• nano (n) = 10^-9
• pico (p) = 10^-12
• femto (f) = 10^-15
• atto (a) = 10^-18
• zepto (z) = 10^-21

Non-SI Metric Units Commonly Used in Chemistry

• Length is measured in angstroms (Å) with the relation 1 Å = 0.1 nm = 10^-10 m
• Mass uses the atomic mass unit (u or amu), where 1 u = 1.66054 × 10^-27 kg
• mass can also be measured in metric tons (t) where 1 t = 10^3 kg
• Time is measured in minutes (min.) where 1 min. = 50 s
• Time can also be measured in hours (h) where 1 h = 60 min. = 3600 s
• Temperature is measured in degree Celsius (°C) where Tk = toc + 273.15
• Volume is measured in liters (L) where 1 L = 1000 cm^3

Useful Conversions

• Length can be measured in inches (in.) where 1 in. = 2.54 cm
• Length can be measured in yards (yd) where 1 yd = 0.9144 m
• Length can be measured in miles (mi) where 1 mi = 1.609 km
• Mass can be measured in pounds (lb) where 1 lb = 453.6 g
• Mass can also be measured in ounces (oz) where 1 oz = 28.35 g
• Volume can be measured in gallons (gal) where 1 gal = 3.785 L
• Volume can be measured in quarts (qt) where 1 qt = 946.4 mL
• Volume can be measured in fluid ounces (oz) where 1 oz = 29.6 mL

Decimal Multipliers

• Prefixes on SI base units are used when number is too large or too small for convenient usage
• Numerical values of multipliers can be interchanged with prefixes (1 mL = 10^-3 L).

Mass & Length

• These are basic units measured in science
• Mass is a measure of the amount of material in an object
• SI uses the kilogram as the base unit and the metric system uses the gram as the base unit
• Length is a measure of distance and the meter is the base unit

Mass

• The SI unit is kilogram (kg)
• Grams (g) are frequently used in laboratory as more realistic size
• 1kg = 1000g, 1g = 0.001 kg = 1/1000 kg
• Mass is measured by comparing weight of sample with weights of known standard masses
• An instrument used to measure weight is a balance

Length

• SI unit is meter (m)
• Meters (m) are too large for most laboratory measurements
• Commonly Centimeters are used where 1 cm = 10^-2 m = 0.01 m
• Millimeters are also commonly used where 1 mm = 10^-3 m = 0.001 m

Volume

• Volume is the dimension of length cubed
• SI unit for volume is m^3
• Most lab measurements use V in liters (L)
• 1 L = 1 dm³ (exactly)
• Chemistry glassware can be marked in L or mL (1 L = 1000 mL)
• 1 mL = 1 cm³

Conventions

• 10 mm = 1 cm
• 10 cm = 1 dm
• 10 dm = 1 m
• 1 cm³ = (1 x 10^-2 m)³ = 1 x 10^-6 m³
• 1 dm³ = (1 x 10^-1 m)³ = 1 x 10^-3 m³
• 1 mL = 1 cm³ and 1 L = 1000 mL = 1000 cm³ = 1 dm³
• Volume is not a base until of SI, it is derived from length
• The most commonly used metric units for volume are the liter (L) and the milliliter (mL)
• A liter is a 1 decimeter cube (dm³) long on each side
• A milliliter is a 1 centimeter cube (cm³) long on each side, also called 1 centimeter cube (cm × cm × cm = cm³)

Density

• Density is a physical property of a substance
• Density has units that are derived from the units for mass and volume
• The most common units for density are g/mL or g/cm³
• Density = mass / volume

Numbers in Science

• Exact numbers are counted or given by definition
• Inexact (or measured) numbers depend on how they were determined
• Scientific instruments have limitations

Uncertainty in Measurements

• Different measuring devices have different uses and different degrees of accuracy
• All measured numbers have some degree of inaccuracy

Concentrations

• Molarity (c) is the number of moles of a substance per liter of solution
• C = mol of substance/L of solution or μM = 10^-6 Μ
• Molality (m) is the Number of moles of a substance per kilogram of solvent (not total solution but solvent)
• Molality (m) changes with temperature because volume of solution increases with heating
• m = mol of solute/kg of solvent
• Weight percent (wt%) = mass of solute/mass of total solution or mixture X 100
• For example 95wt% ethanol implies 95g per 100g of solution is actually ethanol
• Volume percent (vol%) = volume of solute/volume of total solution or mixture X 100
• Density (p) = mass of solute/volume of total solution or mixture Unit = g/mL
• Specific gravity = Density of substance/Density of water at 4 C
• Density of water at 4°C is approximately 1g/mL, therefore specific gravity is similar to density though it has no unit

Trial Questions

• Parts per Million (ppm) measures the mean grams of a substance per million grams of total solution or mixture = mass of substance/mass of sample x 10^6
• Parts per Billion (ppb) measures the mean grams of a substance per billion grams of total solution or mixture = mass of substance/mass of sample x 10^9

Errors

• All measurements rely on techniques that have proven to be reliable from experience/experiment.
• Repeating a measurement tells about reproducibility (precision).
• Comparing many quantities gives confidence in the "truth" (accuracy)
• Uncertainty of every measurement is called experimental error

Errors Analysis

• Systematic errors (Determinate error) is error due to the instrument, method or analyst
• Badly worn out instruments, unlevelled balances, improper use and storage of reagents
• An instrument that is improperly calibrated or poor technique (e.g. carelessness with parallax) is also the source of systematic error
• Random errors (Indeterminate Error) Arise from limitations on our ability to make physical measurements and by chance from minor uncertainties which are always part of repetitive measurements

Uncertainty and Accuracy

• Accuracy refers to the proximity/nearness of a measurement to the "true value" of a quantity, it is True because somebody has measured that value
• Precision refers to the proximity of several measurements to each other. It is a measure of the reproducibility of a result. Measurements could be reproducible but wrong

Significant Figures

• Significant figures refers to digits that were measured
• It is the minimum number of digits needed to write a given value in scientific notation without loss of accuracy
• When rounding calculated numbers, pay attention to significant figures so the accuracy does not get overstated in the answers.

Rules for Significant Figures

• Non-zero numbers are significant
• Zeros between non-zero numbers are significant
• Trailing zeros count as significant if number has decimal point
• Final zeros on number without decimal point are NOT significant
• Final zeros to right of decimal point are significant
• Leading zeros, to left of first non-zero digit, are never counted as significant
• When addition or subtraction is performed, answers are rounded to the least significant decimal place
• Answer has same number of decimal places as quantity with fewest number of decimal places.
• When multiplication or division is performed, answers are rounded to the number of digits that corresponds to the least number of significant figures in any of the numbers used in the calculations
• Limitation is based on the number of digits contained in the number with the fewest significant figure

Logarithms

• Logarithm of n: n= 10ª means that log n = a and n = antilogarithm of a
• for example: log 100 = 2, where the Antilog 2 = 100
• Antilog has character (integer part) and mantissa (decimal part), ie: Log 339 = 2.530 2 = character 530 = mantissa
• Log 3.39 x 10-5 = -4.470

Calculations

• If z = x + y or z = x - y then the absolute uncertainty in z is given by σz = root(σx2+σy2)
• If z = x y or z = x/y then the percent uncertainty in z is given by %σ = root(%σx2+%σy2)
• Convert all absolute uncertainties to % relative uncertainties