Chemistry 100: Measurements and Significant Figures - Vocabulary Flashcards
Measurements in Chemistry
Topic: Chemistry 100 – Measurements and Significant Figures
Key ideas: what measurements are, how units and systems differ (English vs Metric), and how to read, record, and propagate uncertainty in measurements.
Measurements in Chemistry: Units and Systems
Measurement is the process of assigning a numeric value to a dimension, capacity, quantity, or extent of something.
Examples of measurable dimensions: mass, volume, length, time, temperature, pressure, concentration.
Systems of measurement:
English system: inch, foot, ounce, pound, quart, gallon.
Metric system: science standard; base units used in chemistry.
In science, metric units are preferred for clarity and consistency.
Metric System Basics: Base Units
Base length unit: meter (m)
1 meter ≈ 1.09 yards
1 ext{ meter} \,\approx\, 1.09\text{ yards}
Base mass unit: gram (g)
About 28 g ≈ 1 ounce (approximate conversion)
Base volume unit: liter (L)
1\text{ liter} \approx 1.06\text{ quarts}
Metric Prefixes and Conversions
Length:
1\text{ kilometer} = 1000\text{ meters} = 1 \times 10^3\text{ m}
1\text{ millimeter} = 1 \times 10^{-3}\text{ m}
Mass:
1\text{ kilogram} = 1000\text{ grams} = 1 \times 10^3\text{ g}
1\text{ milligram} = 1 \times 10^{-3}\text{ g}
Volume:
1\text{ kiloliter} = 1000\text{ liters} = 1 \times 10^3\text{ L}
1\text{ milliliter} = 1 \times 10^{-3}\text{ L}
Common Metric Volume Equivalences
1\text{ cm}^3 = 1\text{ cc} = 1\text{ mL}
Milliliter is commonly used in research laboratories.
Deciliter and cubic centimeter (cc) are commonly used in clinical laboratories:
1\text{ dL} = 100\text{ mL}
1\text{ cc} = 1\text{ mL}
Metric System Prefixes (Power-of-Ten Table)
giga: 10^9 (billion)
mega: 10^6 (million)
kilo: 10^3 (thousand)
deci: 10^{-1} (one-tenth)
centi: 10^{-2} (one-hundredth)
milli: 10^{-3} (one-thousandth)
micro: 10^{-6} (one-millionth)
nano: 10^{-9} (one-billionth)
pico: 10^{-12} (one-trillionth)
Learning Check – Fill in Blank (Base Units and Prefixes)
Learning Check text (as given):
"Relate the Base Units kg 10- 12 g nm 106 g cL 109 Hz 10- 9 m 10- 2 L pg Mg GHz 103 g"
Your Turn! Correct Conversions
Determine which statement is correct:
\text{Gm} = 10^9\ \text{m}
10^9\ \text{Gm} = \text{m}
\text{Gm} = 10^{-9}\ \text{m}
\text{none are correct}
Significant Figures: Basics
Significant figures (SF): digits in a number that carry information about precision.
Exact numbers: no uncertainty (defined values or counted quantities).
Inexact numbers: have uncertainty.
Uncertainty relates to accuracy and precision and to the limitations of measuring devices and observations.
Accuracy vs. Precision
Accuracy: how close a measurement is to the true value (a "hole-in-one").
Precision: how close repeated measurements are to each other (consistency).
With properly calibrated instruments, more significant figures generally indicate greater precision for a given measurement.
Significance indicates precision!
Sources of Error in Measurement
Inherent errors due to equipment or procedure:
Changing volumes due to thermal expansion or contraction (temperature changes).
Improperly calibrated equipment.
Procedural design allows variability or too few measurements.
Mistakes/blunders to avoid:
Spillage, incomplete procedures, misreading scales, using measuring devices incorrectly.
Do not use data that contain these errors.
Measurement Error and Best Practices
Measurements always have error.
Record all measured numbers, including the first estimated digit.
The estimated digit is an SF and carries uncertainty.
Conduct more than one trial and average to reduce error.
Calculate percent error (\%\text{ error} = \left|\text{experimental} - \text{true}/\text{true}\right| \times 100\%) when possible.
Significant Figures and Uncertainty
Estimated digits introduce uncertainty into measurements.
Only one estimated digit should be part of a recorded measurement.
Two main types of information:
Measurement magnitude
Measurement uncertainty
Significant figures: digits known with certainty plus one uncertain digit.
Instrument Precision: Graduated Cylinders (Examples)
Example comparison:
Using a less precise cylinder: volume ≈ 21.3 mL with 3 SF (3 significant digits).
Using a more precise cylinder: volume ≈ 21.32 mL with 4 SF.
Best estimate corresponds to the instrument’s precision (e.g., 2 hundredths in this example).
Significant Figures Guidelines
Nonzero digits are always significant.
Example: 593{,}673 contains 6 significant figures.
Zeros:
Zeros between nonzero digits are significant (e.g., 1{,}001 has 4 SF).
Leading zeros are not significant (e.g., 0.005071 has 4 SF).
Trailing zeros are significant if a decimal point is present (e.g., 101.700 has 6 SF).
Trailing zeros are generally not significant if there is no decimal point (e.g., 37{,}000 has 2 SF).
Rounding (Significant Figures)
Rule 1: If the first digit to be deleted is 4 or less, drop it and all following digits.
Example: 0.4327 \rightarrow 0.43 (to two SF).
Rule 2: If the first deleted digit is 5 or greater, that digit and following digits are dropped and the last retained digit is increased by 1.
Example: 25.7 \rightarrow 26 (to two SF).
Practice: Learning Check (Significant Figures)
Examples and answers (from slides):
1,000 → 1 sig fig
0.00235 → 4 sig figs
10{,}000 \pm 1{,}000 → 2 sig figs
0.50500 → 6 sig figs
12 → 2 sig figs
1.23 × 10^3 → 3 sig figs
Your Turn (More Sig Figs Practice)
Examples and expected results (as given):
546{,}000 \pm 105 → 3 sig figs
546{,}000 \pm 10003 → 4 sig figs
0.502 → 3 sig figs
123{,}502 → 6 sig figs
0.502 → 3 sig figs
1000 → 1 sig fig
Combining Significant Figures
Multiplication and division:
The result has as many SF as the measurement with the least SF in the calculation.
Addition and subtraction:
The result has the same number of digits to the right of the decimal point as the measurement with the fewest digits to the right of the decimal point.
Practice: Quick Calculations (Learning Check)
Compute:
3.003 \times 0.0029 = 0.0087
\frac{7240}{233} = 31.1
Significance in Addition/Subtraction
The result should be rounded to the least precise decimal place among the operands.
Example: 3.247 (three decimal places) with 41.36 (two decimal places) + 125.2 (one decimal place) = 169.8 (rounded to one decimal place)
Scientific Notation: Overview
Numerical system expressing numbers as a product with a power of 10.
Rules:
Move the decimal so the coefficient is < 10 (and >= 1).
The exponent equals the number of places the decimal was moved.
Exponent is positive if original number is ≥ 10; negative if < 1.
For numbers 1–9, the exponent is 0.
Scientific Notation: Examples
Example: 93,000,000 (maintain 2 SF) → 9.3 \times 10^{7}
Coefficient: 9.3
Exponential term: 10^7
Example: 0.0000037 → 3.7 \times 10^{-6}
Coefficient: 3.7
Exponential term: 10^{-6}
What the notation means:
The coefficient contains only the significant figures.
Exponents track the decimal shift.
Quick tip: To multiply exponential terms, add exponents; to divide, subtract exponents.
If you have a \times 10^{m} \times b \times 10^{n}, product is (ab) \times 10^{m+n}.
If you have \dfrac{a \times 10^{m}}{b \times 10^{n}}, quotient is (a/b) \times 10^{m-n}.
Measurements in Chemistry
Measurements constitute a fundamental aspect of quantitative chemistry, providing the empirical data necessary for analysis and hypothesis testing. This discipline encompasses the understanding of measurement methodologies, the differentiation between various unit systems (e.g., English versus Metric), and the systematic approach to reading, recording, and propagating measurement uncertainty.
Measurements in Chemistry: Units and Systems
A measurement is defined as the process of assigning a numeric value to a specific dimension, such as mass, volume, length, time, temperature, pressure, or concentration. While the English system employs units like inches, feet, ounces, pounds, quarts, and gallons, the metric system is universally adopted in scientific contexts due to its inherent clarity, consistency, and base-ten structure. This standardization facilitates global scientific communication and reproducibility.
Metric System Basics: Base Units
The fundamental base units in the metric system, crucial for chemical measurements, include:
Length: The meter (m), with an approximate equivalence of 1\text{ meter} \approx 1.09\text{ yards}.
Mass: The gram (g), with an approximate conversion of about 28\text{ g} \approx 1\text{ ounce}.
Volume: The liter (L), with an approximate equivalence of 1\text{ liter} \approx 1.06\text{ quarts}.
Metric Prefixes and Conversions
Metric prefixes enable scaling of base units to represent larger or smaller quantities by powers of ten:
Length:
1\text{ kilometer} = 1000\text{ meters} = 1 \times 10^3\text{ m}
1\text{ millimeter} = 1 \times 10^{-3}\text{ m}
Mass:
1\text{ kilogram} = 1000\text{ grams} = 1 \times 10^3\text{ g}
1\text{ milligram} = 1 \times 10^{-3}\text{ g}
Volume:
1\text{ kiloliter} = 1000\text{ liters} = 1 \times 10^3\text{ L}
1\text{ milliliter} = 1 \times 10^{-3}\text{ L}
Common Metric Volume Equivalences
Specific volume equivalences are frequently utilized in laboratory and clinical settings:
1\text{ cm}^3 = 1\text{ cc} = 1\text{ mL}
The milliliter (mL) is a standard unit in research laboratories.
The deciliter (dL) and cubic centimeter (cc) are commonly employed in clinical laboratories, with specific relationships:
1\text{ dL} = 100\text{ mL}
1\text{ cc} = 1\text{ mL}
Metric System Prefixes (Power-of-Ten Table)
A comprehensive list of metric prefixes and their corresponding power-of-ten multipliers:
giga: 10^9 (billion)
mega: 10^6 (million)
kilo: 10^3 (thousand)
deci: 10^{-1} (one-tenth)
centi: 10^{-2} (one-hundredth)
milli: 10^{-3} (one-thousandth)
micro: 10^{-6} (one-millionth)
nano: 10^{-9} (one-billionth)
pico: 10^{-12} (one-trillionth)
Significant Figures: Basics
Significant figures (SF) are the digits in a measured or calculated number that convey information regarding its precision. Exact numbers, arising from definitions or discrete counts, possess no inherent uncertainty. Conversely, inexact numbers, derived from empirical measurements, inherently carry uncertainty attributed to the limitations of measuring instrumentation and observational processes. This uncertainty directly affects the accuracy and precision of quantitative data.
Accuracy vs. Precision
Accuracy refers to the proximity of a measurement to the true or accepted value.
Precision denotes the degree of reproducibility among repeated measurements. With appropriately calibrated instruments, a greater number of significant figures in a measurement is indicative of enhanced precision. Thus, the significance of a measurement directly correlates with its precision.
Sources of Error in Measurement
Measurement errors can arise from various sources:
Inherent errors due to equipment or procedural design:
Variations in volume caused by thermal expansion or contraction, induced by temperature fluctuations.
Malfunctions or improper calibration of measuring devices.
Procedural designs that introduce variability or are based on insufficient replicate measurements.
Mistakes (blunders), which must be rigorously avoided, include:
Material spillage or incomplete execution of experimental protocols.
Incorrect interpretation of instrument scales or misuse of measuring apparatus.
Data containing such blunders should be excluded from analysis.
Measurement Error and Best Practices
All measurements are inherently subject to some degree of error. To mitigate and quantify this, best practices include:
Recording all observable digits, including the conventionally estimated final digit, which is considered significant and contributes to the total uncertainty.
Conducting multiple experimental trials and averaging the results to reduce random error.
Calculating the percent error (\text{% error} = \left|\frac{\text{experimental} - \text{true}}{\text{true}}\right| \times 100\%) when a true or accepted value is available, to assess accuracy.
Significant Figures and Uncertainty
The presence of an estimated digit within a recorded measurement is the principal source of its intrinsic uncertainty. Consequently, only one estimated digit should be included in a reported measurement. Significant figures are thus defined as all digits in a measurement that are known with certainty, plus this single, estimated, and uncertain digit.
Instrument Precision: Graduated Cylinders (Examples)
The precision of a measurement is directly constrained by the capabilities of the measuring instrument. For instance, comparing graduated cylinders:
A less precise cylinder might yield a volume of approximately 21.3 mL, containing 3 significant figures.
A more precise cylinder could measure a volume of approximately 21.32 mL, indicating 4 significant figures.
The most reliable estimate of a quantity aligns with the inherent precision of the instrument used, e.g., to the hundredths place in the latter example.
Significant Figures Guidelines
Adherence to specific rules ensures proper identification and interpretation of significant figures:
Nonzero digits are always significant (e.g., 593{,}673 contains 6 SF).
Zeros:
Captive zeros (between nonzero digits) are significant (e.g., 1{,}001 has 4 SF).
Leading zeros (those preceding all nonzero digits) are not significant; they merely indicate the decimal place (e.g., 0.005071 has 4 SF).
Trailing zeros (at the end of the number) are significant if a decimal point is explicitly present (e.g., 101.700 has 6 SF).
Trailing zeros without an explicit decimal point are generally considered not significant, indicating ambiguity about their precision (e.g., 37{,}000 has 2 SF, implying the thousands place is uncertain).
Rounding (Significant Figures)
Rounding procedures are applied to ensure that calculated results retain the appropriate number of significant figures:
Rule 1: If the first digit to be removed is 4 or less, it and all subsequent digits are simply dropped. Example: 0.4327 rounded to two SF becomes 0.43.
Rule 2: If the first digit to be removed is 5 or greater, that digit and all subsequent digits are dropped, and the last digit retained is increased by 1. Example: 25.7 rounded to two SF becomes 26.
Combining Significant Figures
When performing arithmetic operations, the number of significant figures or decimal places in the result is governed by the precision of the input measurements:
Multiplication and Division: The product or quotient must be reported with the same number of significant figures as the measurement with the fewest significant figures in the calculation.
Addition and Subtraction: The sum or difference must be rounded such that it has the same number of digits to the right of the decimal point as the measurement with the fewest digits to the right of the decimal point.
Example: Summing 3.247 (three decimal places), 41.36 (two decimal places), and 125.2 (one decimal place) yields 169.8, which is rounded to one decimal place because 125.2 has the fewest decimal places.
Scientific Notation: Overview
Scientific notation is a standardized numerical system used to express numbers concisely, particularly very large or very small values, as a product of a coefficient and a power of 10. The rules for its application are:
The decimal point is moved such that the coefficient (the number multiplied by the power of 10) is greater than or equal to 1 but less than 10 (1 \le \text{coefficient} < 10).
The exponent of 10 corresponds to the number of places the decimal was moved. Its sign indicates the direction of movement: positive if the original number was 10 or greater (decimal moved left), and negative if the original number was less than 1 (decimal moved right).
For numbers between 1 and 9, the exponent is 0.
Scientific Notation: Examples
Illustrative examples of scientific notation:
For 93,000,000 (maintaining 2 SF), it is expressed as 9.3 \times 10^{7}. Here, 9.3 is the coefficient, and 10^7 is the exponential term.
For 0.0000037, it is expressed as 3.7 \times 10^{-6}. Here, 3.7 is the coefficient, and 10^{-6} is the exponential term.
Interpretation: In scientific notation, the coefficient distinctively contains only the significant figures of the number, while the exponent precisely tracks the magnitude and decimal shift.
Quick Tip for Exponential Arithmetic:
To multiply exponential terms (e.g., (a \times 10^{m}) \times (b \times 10^{n})), the product is (ab) \times 10^{m+n}.
To divide exponential terms (e.g., \frac{a \times 10^{m}}{b \times 10^{n}}), the quotient is (a/b) \times 10^{m-n}.