Measurement, Significant Figures, and Density in Chemistry

Measurement and Problem Solving

Physical Quantities and Units

In chemistry, measurements are used to quantify physical properties such as mass, volume, and length. Each measurement consists of a number and a unit, which together describe the magnitude and type of the quantity.

• Physical Quantity: A property of matter that can be measured (e.g., mass, volume, temperature).

• Unit: A standard of measurement for a physical quantity (e.g., grams for mass, liters for volume).

• Example: The mass of a sample might be recorded as 45.34 g, where 45.34 is the measured value and g is the unit.

Significant Figures

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated. They reflect the precision of a measurement and the limitations of the measuring instrument.

• Definition: All the digits in a measurement that are certain, plus one uncertain (estimated) digit.

• Purpose: To communicate the precision of a measurement and avoid overstating accuracy.

• Rules for Counting Significant Figures:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros are significant only if there is a decimal point.

• Example: 0.019 has two significant figures (1 and 9).

• Uncertain Digit: The last digit in a measurement is always uncertain and is estimated.

Precision and Accuracy

Precision and accuracy are two important concepts in measurement:

• Precision: How close repeated measurements are to each other.

• Accuracy: How close a measurement is to the true or accepted value.

• Range of Precision: The uncertainty in a measurement, often indicated by the last significant digit.

• Example: If a balance reads 39.0 g, the measurement is precise to the tenths place.

Recording Measurements and Using Significant Figures

When recording measurements:

• Always include all certain digits plus one estimated digit.

• Record the measurement with the correct number of significant figures.

• When using measuring devices, read from the bottom of the meniscus for liquids.

• When measuring mass with a container (e.g., weighing boat), subtract the mass of the empty container to find the mass of the sample.

Calculations with Significant Figures

• Multiplication and Division: The result should have as many significant figures as the measurement with the fewest significant figures.

• Addition and Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

• Rounding: Round off digits according to the rules for significant figures after completing calculations.

Density and Its Determination

Definition and Formula

Density is a physical property that relates the mass of a substance to its volume. It is commonly used to identify substances and assess purity.

• Definition: Density is the mass per unit volume of a substance.

• Formula:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Units: Common units are g/mL for liquids and g/cm3 for solids.

• Example: If a solid has a mass of 18.5 g and a volume of 4.0 mL, its density is $\frac{18.5\ \text{g}}{4.0\ \text{mL}} = 4.6\ \text{g/mL}$ (rounded to two significant figures).

Measuring Density in the Laboratory

• Measure the mass using a balance.

• Measure the volume using a graduated cylinder (for liquids) or by displacement (for irregular solids).

• For regular solids (rectangular, cylindrical), use geometric formulas to calculate volume.

• Always use proper significant figures in both mass and volume measurements.

Graphical Determination of Density

Density can also be determined graphically by plotting mass versus volume and finding the slope of the best-fit line.

• Plot mass (y-axis) against volume (x-axis).

• The slope of the line ($\Delta y / \Delta x$) gives the density.

• Ensure that data points are as accurate as possible and that the best-fit line is used.

Criteria for a Well-Made Graph

• Title that describes the graph.

• Labeled axes with units.

• Evenly spaced and appropriately scaled axes.

• Data points plotted accurately.

• Best-fit line or curve drawn if appropriate.

Accuracy, Precision, and Percent Error

Definitions

• Accuracy: Closeness of a measured value to the true value.

• Precision: Closeness of repeated measurements to each other.

• Percent Error: A measure of accuracy, calculated as:

$\text{Percent Error} = \left| \frac{\text{Experimental Value} - \text{True Value}}{\text{True Value}} \right| \times 100\%$

• Percent error can be calculated from experimental data or from a graph (using the best-fit line).

• Percent error is not a measure of precision; it measures accuracy.

Physical Properties and Classification of Matter

Physical Properties

• Properties that can be observed or measured without changing the substance's identity (e.g., density, color, melting point).

• Used to identify and classify substances.

Classification of Matter

• Matter can be classified based on physical and chemical properties.

• Physical properties are often used in laboratory settings to distinguish between substances.

Summary Table: Key Concepts in Measurement and Density

Concept	Definition	Example
Physical Quantity	Property that can be measured (e.g., mass, volume)	Mass of a sample: 45.34 g
Significant Figures	Digits that reflect the precision of a measurement	0.019 has 2 significant figures
Density	Mass per unit volume	Density = 18.5 g / 4.0 mL = 4.6 g/mL
Percent Error	Measure of accuracy	See formula above

Additional info: Some explanations and examples were expanded for clarity and completeness based on standard introductory chemistry curriculum.