Significant Digits, Measurement Uncertainty, and Precision

Qualitative and Quantitative Measurements

Qualitative Descriptions: Refer to the identity or chemical/physical form of a substance present in a sample.
Quantitative Measurements: Refer to experimental determinations of the exact amount or quantity of a substance present.
Role of Measurements: Precise measurements allow for the exact identification of substances and enable the establishment of scientific generalities.

Classification of Numbers in Chemistry

Exact Numbers:
- Defined values: Exact conversion factors or defined equivalences, such as $1\,\text{kg} = 1000\,\text{g}$ or $1\,\text{dozen} = 12\,\text{objects}$ .
- Counted quantities: Numbers obtained by direct counting of discrete objects, such as $28\,\text{students in a class}$ .
- Exact numbers possess an infinite number of significant figures ( $\infty\,\text{S.F.}$ ) and do not limit the number of significant figures in a calculated result.
Inexact Numbers:
- Quantities obtained by any measurement method other than counting.
- Examples include physical measurements such as length, mass, volume, time, and speed.
- All inexact numbers contain inherent uncertainty and must be reported in a manner that reflects this uncertainty.

Uncertainty and Estimation in Measurement

Significant Figures (S.F.): The meaningful digits in a reported measured number.
Uncertain Digit: The last digit recorded in a measured number, which is estimated.
Magnitude of Uncertainty: Unless explicitly stated otherwise, the uncertainty in a measurement is generally assumed to be $\pm 1$ in the last reported digit.
Estimation Procedure:
- When reading a measuring instrument, obtain certain digits from the scale markings and estimate one additional digit between the finest scale divisions.
- Coarser Ruler Example: Measuring an object with a ruler marked only in centimeters allows estimation to the tenths place (e.g., $2.5\,\text{cm}$ ). The digit $2$ is known with certainty, while the digit $5$ is an estimated uncertain digit.
- Finer Ruler Example: Measuring the same object with a ruler marked in millimeters allows estimation to the hundredths place (e.g., $2.45\,\text{cm}$ ). The digits $2.4$ are known with certainty, while the digit $5$ is the estimated uncertain digit.

Rules for Determining Significant Figures

Nonzero Digits: All nonzero digits are significant.
- $112.1$ has $4\,\text{significant figures}$ .
Interior Zeros (Captive Zeros): Zeros located between nonzero digits are significant.
- $305$ has $3\,\text{significant figures}$ .
- $50.08$ has $4\,\text{significant figures}$ .
Leading Zeros: Zeros to the left of the first nonzero digit are not significant; they serve only to locate the decimal point.
- $0.0023$ has $2\,\text{significant figures}$ .
- $0.000001$ has $1\,\text{significant figure}$ .
Trailing Zeros with a Decimal Point: Zeros to the right of the last nonzero digit are significant if an explicit decimal point is present in the number.
- $1.200$ has $4\,\text{significant figures}$ .
Trailing Zeros without a Decimal Point: Zeros to the right of the last nonzero digit in a number without a decimal point are ambiguous.
- The value $100$ may represent $1$ , $2$ , or $3\,\text{significant figures}$ .
- Resolution via Scientific Notation: Scientific notation must be used to specify the exact number of significant figures:
  - $1\,\text{S.F.} = 1 \times 10^2$
  - $2\,\text{S.F.} = 1.0 \times 10^2$
  - $3\,\text{S.F.} = 1.00 \times 10^2$

Scientific Notation

Structure: Written as a product of a decimal coefficient and an exponential term:
- Decimal part: A number usually between $1$ and $10$ .
- Exponential part: $10$ raised to an integer exponent $n$ .
- Example: $1.51 \times 10^{-3}\,\text{m}$ (Decimal part = $1.51$ , Exponential part = $10^{-3}$ , Exponent = $-3$ ).
Exponents:
- A positive exponent $n$ indicates that $1$ is multiplied by $10$ , $n$ times.
- A negative exponent $-n$ indicates that $1$ is divided by $10$ , $n$ times.
Conversion Steps:
- Shift the decimal point left or right to obtain a coefficient between $1$ and 10$.\n * Multiply by 10 raised to the power corresponding to the number of places the decimal point was moved.\n * Preserve all original significant figures in the decimal coefficient.\n\n# Arithmetic Operations with Significant Figures\n\n* **Addition and Subtraction**:\n * The calculated result cannot have more digits to the right of the decimal point than any of the original numbers (limited by the least precise decimal place).\n * Example 1: 102.50 + 0.231 = 102.731 $. Since$ 102.50 $has 2 decimal places and$ 0.231 $has 3 decimal places, the answer is rounded to 2 decimal places:$ 102.73\n * Example 2: 143.29 - 20.1 = 123.19 $. Since$ 143.29 $has 2 decimal places and$ 20.1 $has 1 decimal place, the answer is rounded to 1 decimal place:$ 123.2\n* **Multiplication and Division**:\n * The number of significant figures in the final product or quotient is equal to the number of significant figures in the original factor that has the smallest number of significant figures.\n* **Calculations Involving Exact Numbers**:\n * Exact numbers have an infinite number of significant figures and do not constrain the precision of the final answer.\n * Example: Calculating the total mass of three pennies, each having a mass of 2.5\,\text{g} $. The count of 3 pennies is an exact number, so the total mass is$ 3 \times 2.5\,\text{g} = 7.5\,\text{g} $(retaining 2 significant figures based on$ 2.5\,\text{g}).\n* **Multistep Calculations and Rounding Rules**:\n * To prevent cumulative rounding errors, do not round intermediate steps. Round only at the end of the calculation.\n * Retain at least one extra digit (guard digit) throughout intermediate calculations.\n * *Rounding Rule*: If the first digit to be dropped is < 5 $, round down (drop the digit). If the first digit to be dropped is$ \ge 5, round up (add 1 to the last retained digit).\n * *Comparison of Rounding Approaches*:\n * Rounding after each step: Step 1: 3.66 \times 8.45 = 30.9 $; Step 2:$ 30.9 \times 2.11 = 65.2\n * Rounding at the end: Step 1: 3.66 \times 8.45 = 30.93 $; Step 2:$ 30.93 \times 2.11 = 65.3\n* **Mixed Operations and Order of Operations**:\n * Standard order of operations must be maintained:\n 1. Calculations inside parentheses\n 2. Exponential calculations\n 3. Multiplication or division (left to right)\n 4. Addition or subtraction (left to right)\n * Because addition/subtraction rules (decimal places) differ from multiplication/division rules (total significant figures), determine the number of significant figures at each intermediate step before proceeding to the next operation, while retaining extra digits to avoid round-off errors.\n\n# Logarithms and Significant Figures\n\n* **Logarithm Rule**: The number of digits after the decimal point (the mantissa) in the result of \log(x) $must equal the number of significant figures in$ x.\n * Example: \log(3.5 \times 10^{-5}) = -4.46 $. Since$ 3.5 \times 10^{-5} $has 2 significant figures, the result must be reported with 2 digits after the decimal point ($ -4.46).\n* **Inverse Logarithm Rule**: For an inverse log of x $(expressed as$ 10^x $), the number of significant figures in the final answer must equal the number of digits after the decimal point in$ x.\n * Example: 10^{-3.421} = 3.79 \times 10^{-4} $. Since$ -3.421 $has 3 digits after the decimal point, the answer must contain 3 significant figures ($ 3.79 \times 10^{-4}).\n\n# Accuracy and Precision\n\n* **Accuracy**: The closeness of a measurement to the true or accepted value.\n* **Precision**: The closeness of a series of replicate measurements to one another.\n* **Relationships**:\n * Good accuracy and good precision: Measurements are tightly clustered together and centered on the true value.\n * Poor accuracy but good precision: Measurements are tightly clustered together but systematically shifted away from the true value.\n * Poor accuracy and poor precision: Measurements are widely scattered and off from the true value.\n* **Case Study — Aspirin Mass Measurement**:\n * True mass of aspirin tablet = 0.370\,\text{g}.\n * Student A: Replicate measurements are precise but not accurate.\n * Student B: Replicate measurements are neither precise nor accurate.\n * Student C: Replicate measurements are both precise and accurate.\n\n# Measurement Tools and Precision in Single Measurements\n\n* **Instrument Dependence**: Precision in a single measurement is determined directly by the measuring device used.\n* **Estimation Standard**: When taking a single measurement, always estimate to one digit beyond the smallest scale division present on the tool.\n* **Scale Division Examples**:\n * A ruler marked in whole centimeters yields a measurement reported to the tenths place (e.g., 4.1\,\text{cm}).\n * A ruler marked in tenths of a centimeter (millimeters) yields a measurement reported to the hundredths place (e.g., 4.09\,\text{cm}).\n\n# Review of Key Principles\n\n* Measurement precision is directly reflected in the number of significant digits reported.\n* Significant figure rules for addition and subtraction are based on decimal places, whereas rules for multiplication and division are based on total significant figures.\n* Significant digits and decimal-place digits are distinct concepts with no mandatory relationship.\n* Standard electronic calculators do not display the correct number of significant figures automatically.\n* All digits in the decimal coefficient of a number written in scientific notation are significant.\n\n# Questions & Practice Calculations\n\n* **Scientific Notation Conversion Practice**:\n * Convert 8400 $to scientific notation:$ 8.4 \times 10^3\n * Convert 0.000125 $to scientific notation:$ 1.25 \times 10^{-4}\n * Convert 465000000 $to scientific notation:$ 4.65 \times 10^8\n* **Calculations with Significant Figures**:\n * *Problem A*: Calculate \frac{0.21\,\text{g}}{79.3\,\text{g} - 65.22\,\text{g}} \times 3.818\n * Step 1 (Parentheses/Subtraction): 79.3\,\text{g} - 65.22\,\text{g} = 14.08\,\text{g} $. The calculation is limited to the tenths place (1 decimal place from$ 79.3), giving 3 significant figures.\n * Step 2 (Multiplication/Division): \frac{0.21\,\text{g}}{14.08\,\text{g}} \times 3.818 = 0.056948…\,\text{g}.\n * The factor 0.21\,\text{g} contains 2 significant figures, which limits the final answer to 2 significant figures.\n * Final Answer: 0.057\,\text{g} $(or$ 5.7 \times 10^{-2}\,\text{g}).\n * *Problem B*: Calculate \left(\frac{92.12\,\text{mL}}{0.912\,\text{g/mL}}\right) + 223.02\,\text{g}\n * Step 1 (Parentheses/Division): \frac{92.12\,\text{mL}}{0.912\,\text{g/mL}} = 101.00877…\,\text{g} $. The denominator$ 0.912 $contains 3 significant figures, limiting the quotient to 3 significant figures (uncertainty in the units place,$ 101).\n * Step 2 (Addition): 101.00877…\,\text{g} + 223.02\,\text{g} = 324.02877…\,\text{g}.\n * Since the intermediate quotient is uncertain at the units place (101), the sum must be rounded to the units place.\n * Final Answer: 324\,\text{g}.\n* **Bathroom Scale Scenario**:\n * *Scenario*: A scale registers 2\,\text{lb} $with no load. An object weighed repeatedly on this scale yields a reading of$ 117\,\text{lb} each time.\n * *Question A*: Are the measurements precise?\n * Answer: Yes. The measurements are reproducible, repeatedly yielding the exact same value of 117\,\text{lb}.\n * *Question B*: Are the measurements accurate?\n * Answer: No. The scale has a systematic zero offset error of 2\,\text{lb}.\n * *Question C*: What is the probable true weight of the object?\n * Answer: 117\,\text{lb} - 2\,\text{lb} = 115\,\text{lb}$$.