Lab 1 - Density Lab

Fundamental Quantities and Metric Prefixes

• Mass: Physical property that quantifies the amount of matter in an object, measured in grams ($g$).
• Length: Distance between two points in space, measured in meters ($m$).
• Volume: Quantity of space an object occupies, measured in liters ($L$) or cubic centimeters ($cm^3$), where $1\,mL = 1\,cm^3$
• Density: Mass per unit volume, constant for a given substance at a specified temperature, measured in $g/cm^3$, $g/mL$, or $kg/m^3$.

Metric Prefixes

• Giga ($G$): $10^9$
• Mega ($M$): $10^6$
• Kilo ($k$): $10^3$
• Hecto ($h$): $10^2$
• Deka ($da$): $10^1$
• Deci ($d$): $10^{-1}$
• Centi ($c$): $10^{-2}$
• Milli ($m$): $10^{-3}$
• Micro ($\mu$): $10^{-6}$
• Nano ($n$): $10^{-9}$
• Pico ($p$): $10^{-12}$

Density Calculations and Unit Conversions

Unit conversion requires multiplying or dividing units such that unwanted units cancel out.

$\text{Density of Gold} = 18.3\,g/mL$

$18.3\,g/mL \times \left(\frac{1\,mL}{1\,cm^3}\right) \times \left(\frac{1 \times 10^6\,cm^3}{1\,m^3}\right) \times \left(\frac{1\,kg}{1000\,g}\right) = 18300\,kg/m^3$

Liquid Measurement and Displacement Techniques

Water Displacement Method

Measures the volume of irregular solid objects by determining the volume of liquid displaced:

$\text{Volume of Object} = \text{Final Volume} - \text{Initial Volume}$

• Example: An initial water volume of $3.00\,mL$ rising to a final volume of $3.50\,mL$ yields an object volume of $0.50\,mL$

Buoyancy and Density Behaviors

• Buoyancy: Tendency of a substance to float or rise in a liquid due to fluid displacement.
• Displacement: Volume of fluid displaced by an intervening substance.
• Mass vs. Weight: Mass is a constant measure of matter, whereas weight is relative to gravitational force.
• Liquid Layers: When immiscible liquids meet, the denser liquid sinks and the less dense liquid floats.

Reading Graduated Cylinders

• Position the graduated cylinder on a flat horizontal surface.
• Align eyes at eye level with the liquid.
• Read the volume measurement at the bottom of the meniscus.

Periodic Trends in Density

• Atomic Volume: Atomic radii increase down a group/family and decrease across a period.
• Atomic Mass: Mass increases down a column.
• Density Trend: Density generally increases down a column as mass increases faster than volume. Trends across a period vary and require experimental determination.

Experimental Procedures

1. Buoyancy of Sucrose Solutions: Compare relative densities of $6\%$, $12\%$, and $18\%$ sucrose solutions (red, blue, yellow) by layering them in test tubes using a disposable pipette.
2. Density of Sucrose Solutions: Measure mass and volume of approximately $5.0\,mL$, $7.0\,mL$, and $9.0\,mL$ increments of $diH_2O$ and sucrose solutions using a $10\,mL$ graduated cylinder to determine density.
3. Density of Solid Aluminum: Determine mass of Al bar(s), immerse in $50.0\,mL$ of water inside a graduated cylinder, and calculate volume change.
4. Density of Transition Metals: Measure mass of metal sample, submerge in water ($30\,mL$ to $50\,mL$ in a $100\,mL$ cylinder or $5\,mL$ in a $10\,mL$ cylinder), and measure displaced volume.
5. Density of Silver Coins: Determine mass, diameter ($\text{diameter} = 2r$), and thickness ($h$) using calipers. Calculate geometric volume using:

$V = \pi r^2 h$

Compare geometric volume against water displacement results.

Statistical Analysis and Error Calculations

Percent Error

$\% \text{ Error} = \frac{|\text{Density}_{\text{calculated}} - \text{Density}_{\text{literature}}|}{\text{Density}_{\text{literature}}} \times 100\%$

Standard Deviation

$S_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$

• Example calculation for density measurements $0.50\,g/mL$, $0.45\,g/mL$, and $0.65\,g/mL$ ($n = 3$):

$\bar{x} = 0.53\,g/mL$

$\sum (x_i - \bar{x})^2 = (0.50 - 0.53)^2 + (0.45 - 0.53)^2 + (0.65 - 0.53)^2 = 0.0217$

$\frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{0.0217}{2} = 0.01085$

$S_x = \sqrt{0.01085} = 0.104\,g/mL$

$\text{Reported Mean} = 0.53 \pm 0.104\,g/mL$

Significant Figures Rules

Significant Figures Identification Examples

• $1.004$: 4 significant figures
• $980$: 2 significant figures
• $103.00$: 5 significant figures
• $800.$: 3 significant figures
• $1.90 \times 10^2$: 3