Density Lab Notes: Water, Aluminum, and Foil Thickness

Water Density: Mass and Volume Measurements

• Objective: experimentally determine the density of water by measuring its mass and volume with a graduated cylinder, noting how density depends on temperature.

• Key concepts:

  • Density definition: \rho = \frac{m}{V} where \rho is density, m is mass, and V is volume.
  • For liquids, use the mass of the water (not the mass of the empty container) and the volume displaced in the graduated cylinder.
  • Density of water changes with temperature: include a temperature reading to interpret density correctly.

• Materials used:

  • Clean, dry 100 mL graduated cylinder (glass tube with volume markings).
  • Distilled water.
  • Mass balance (tared).
  • Black cardstock to improve visibility of the meniscus.
  • Thermometer to read the water temperature.

• Procedure (water, step-by-step):

  • Zero/tare the balance to account for the empty graduated cylinder mass (set the starting mass to zero).
  • Record the initial mass of the empty graduated cylinder (M_empty).
  • Add about 20–25 mL of distilled water to the cylinder.
  • Allow bubbles to dissipate; read the volume by looking at the lower (bottom) meniscus at eye level and align with the markings:
  • The bottom of the curved surface of the water is used to determine the volume reading.
  • If the reading falls between markings, estimate to the first decimal place (e.g., 46.5 mL when between 46 and 47 mL).
  • Weigh the graduated cylinder with water (Mwithwater) and subtract Mempty to obtain the mass of the water (mwater).
  • Repeat for additional 20–25 mL increments to reach roughly 40–45 mL, and then around 60–70 mL, ensuring to read the volume carefully and subtract the dry cylinder mass to get water mass.
  • After each addition, ensure to settle out bubbles; use black cardstock behind the cylinder to improve readability.
  • For each reading, if the lower meniscus lies between two markings, you may report to the nearest 0.1 mL (e.g., 46.5 mL).
  • Final steps for water reading:
  • Record the mass of the cylinder plus water after each increment (Mwithwater) and compute mwater = Mwithwater − Mempty.
  • Finally, insert the thermometer into the water and read the temperature (T) to note how ρ_w is affected by temperature.

• Example readings and practical notes:

  • Volume readings often fall between marks; estimate to the first decimal (e.g., 46.5 mL).
  • When reading, be at eye level and use the lower meniscus for accuracy.
  • If bubbles obscure the reading, wait for them to settle and use a cardstock backdrop to improve contrast.
  • Typical sequence of volumes in the demo: ~20–25 mL (first), ~40–45 mL (second), and ~60–70 mL (third).

• Calculations to perform:

  • For each measurement: \rho{water} = \frac{m{water}}{V{water}} = \frac{M{with\,water} - M{empty}}{V{reading}}
  • Compare readings at different temperatures to illustrate the temperature dependence of density.

• Observations and practical considerations:

  • Temperature affects water density; warmer water is less dense than cooler water.
  • Bubbles can slightly increase the apparent volume reading if not settled; ensure readings reflect true liquid volume.
  • The mass of the empty cylinder must be subtracted to isolate the mass of the water.

Aluminum Density via Mass and Displacement

• Objective: determine the density of aluminum by measuring its mass and its volume via displacement.

• Rationale: Aluminum density can be obtained from mass and displaced water volume using the same fundamental relation \rho = \frac{m}{V}, but using displacement avoids calculating volume from dimensions.

• Materials used:

  • Aluminum cylinder piece (used instead of pellets to avoid air bubbles clinging to pellets).
  • Mass balance (tare available).
  • Graduated cylinder with water for displacement measurements.
  • Black cardstock to improve readability of volume readings.
  • Optional: thermometer to note ambient temperature (for reference).

• Procedure (aluminum cylinder, displacement method):

  • Zero the balance and measure the mass of the aluminum cylinder by itself (m_Al).
  • Prepare the displacement setup:
  • Record the initial volume of water in the graduated cylinder (V_i) at eye level with the backdrop behind for contrast.
  • Carefully insert the aluminum cylinder into the water by tilting it and sliding it down the side to avoid splashing and to reduce air bubbles.
  • Allow any air bubbles to escape and settle.
  • Read the final volume of water in the cylinder (V_f).
  • Compute displaced volume: ΔV = Vf − Vi. This ΔV equals the volume of the aluminum cylinder (V_Al).
  • Calculate aluminum density: \rho{Al} = \frac{m{Al}}{V{Al}} = \frac{m{Al}}{\Delta V}

• Key handling tips:

  • Tilt the cylinder during submersion to minimize splashing and to help bubbles escape.
  • Ensure the cylinder is fully submerged but not touching the bottom of the cylinder to avoid measurement artifacts.
  • Use a black backdrop to improve visibility of the meniscus for volume readings.
  • If any bubbles remain, allow them to dissipate before taking the final reading.

• Notes on readings:

  • Initial and final volume readings should be read at eye level, with attention to possible parallax error.
  • If the final volume lies between marks, estimate to the nearest 0.1 mL (or 0.1 cm^3 for the graduated cylinder units).

• Conceptual takeaway:

  • This displacement method directly yields the volume of a solid object in liquid, which, combined with its mass, gives density via the same fundamental principle, \rho = \frac{m}{V}.

Density by Mass–Volume Data: Using Cylinders in a Four-Point Span

• Objective: determine density by plotting mass vs. volume using four cylinders of the same type but arranged to form progressively longer “cylinders.”

• Rationale: With known diameter, the total volume for a stack is V = π (d/2)^2 L, where L is the total length (stack length) and d is the cylinder diameter.

• Modification to the lab protocol (due to availability of four different-size cylinders):

  • Use four identical cylinders of the unknown metal.
  • Treat the single cylinder as a small cylinder (1 unit), two side-by-side cylinders as a medium cylinder (2 units), three side-by-side as a large cylinder (3 units), and all four as an extra-large cylinder (4 units).
  • For each configuration, measure and record:
  • Mass: m1, m2, m3, m4 corresponding to 1, 2, 3, and 4 cylinders, respectively.
  • Length: L1, L2, L3, L4, measured by placing the cylinders side by side along the ruler and reading the total length from end-to-end. Ensure alignment to minimize parallax errors.
  • Diameter: d (assumed uniform across all cylinders; measured once for use in volume calculations).
  • Important note: when stacking, ensure cylinders are lined up evenly to obtain an accurate total length; avoid parallax by reading directly above the measurement line.

• Calculations for each configuration:

  • Cross-sectional area: A = \pi \left(\frac{d}{2}\right)^2
  • Volume for each configuration: Vi = A \cdot Li = \pi \left(\frac{d}{2}\right)^2 L_i
  • Density estimate for each configuration: \rhoi = \frac{mi}{V_i}
  • Data to plot: plot mass mi (y-axis) versus volume Vi (x-axis) to obtain a line of best fit, whose slope approximates the density \rho for the material.

• Measurements and reading tips:

  • Use a single diameter measurement for all configurations if cylinders are visually similar in diameter.
  • When taking length readings, align the stack along the ruler and ensure the measurement starts and ends at the outermost edges.
  • For lengths, read in centimeters; remember that 1 cm = 10 mm, and the ruler markings may include mm subdivisions.
  • For length readings that are between marks, estimate to the nearest 0.1 cm (e.g., 3.25 cm).
  • For masses, ensure the balance is tared to zero with the configuration removed before placing the cylinders.
  • Watch for parallax error and read at eye level; align the stack to minimize viewing angle issues.

• Conceptual takeaway:

  • Plotting mass vs. volume for multiple configurations allows confirmation of a linear relationship with slope equal to the material density; the line of best fit provides an averaged density value across the tested configurations.

• Practical notes from the demonstration:

  • The instructor noted that using pellets can introduce air bubbles and complicate density calculations; using a single solid cylinder helps avoid this issue.
  • Consistency in diameter across configurations is assumed to simplify volume calculations.
  • The phrase "Ivy League universities" appears in the transcript as a casual remark, not a methodological requirement; focus remains on the experimental approach.

Thickness of Aluminum Foil from Known Aluminum Density

• Objective: use the density of aluminum obtained earlier to determine the thickness of a piece of aluminum foil.

• Principle: For a sheet of foil, volume V = A × t, where A is the area (length × width) and t is the thickness. Since density ρ = m / V, thickness can be solved as:

  • t = \frac{m}{\rho \cdot A} where A = \text{length} \times \text{width}.

• Measurements needed:

  • Mass of foil piece: m_foil (measure with the balance after folding the foil to fit the balance pan).
  • Length and width of the foil piece (in cm) to compute area A.
  • Aluminum density ρ obtained from the previous density experiments (ρ_Al).

• Procedure:

  • Zero the balance and place the folded foil on the balance; record mass m_foil.
  • Measure the foil’s length (L) and width (W) with a ruler; use the same units (cm).
  • Compute area: A = L \times W in cm^2.
  • Use the previously determined aluminum density ρ_Al (in g/cm^3).
  • Compute thickness: t = \frac{m{foil}}{\rho{Al} \cdot A}, yielding thickness in cm (convert to micrometers if desired: 1 cm = 10,000 μm).

• Important considerations:

  • Ensure the foil piece lies flat and fully occupies the measurement area without gaps; folding should minimize air pockets affecting mass.
  • The accuracy of t depends on the accuracy of mfoil, ρAl, and A. Any error in L or W propagates into A and thus into t.
  • Parallax error should be minimized when reading L and W; read at eye level and align the ruler accordingly.

• Conceptual takeaway:

  • This approach demonstrates how a material’s density can be used to infer another geometric property (thickness) of a thin sheet when the area is known, illustrating the interdependence of mass, volume, area, and thickness.

• Connections to broader topics:

  • Demonstrates the relationship between macroscopic measurements and intrinsic material properties (density).
  • Highlights how measurement techniques (displacement vs. dimensional measurements) can be combined to determine a quantity that is not directly measured (thickness).

Cross-cutting Themes and Practical lab Skills

• Measurement accuracy and uncertainty:

  • Always tare/balance zero before measurements and subtract the mass of the dry container when calculating the mass of the contained substance.
  • Read volumes at eye level and at the bottom of the meniscus for accurate liquid volumes.
  • If readings fall between marks, estimate to the next decimal place, and clearly indicate that estimation in your data.

• Visibility and readability aids:

  • Use black cardstock behind the cylinder or the meniscus to improve visibility of volume readings.
  • If lighting causes glare, adjust the setup or orientation until the meniscus can be read clearly.

• Parallax error and measurement technique:

  • Align the eye with the markings to avoid parallax errors when reading scales.
  • When stacking cylinders, ensure alignment to avoid reading errors due to misalignment.

• Safety and handling:

  • Tilt the aluminum cylinder to minimize splashing and avoid bottom contact with the glass container to prevent cracking.
  • Allow bubbles to dissipate before final readings to improve accuracy.

• Real-world relevance:

  • Understanding density is fundamental to material science, quality control, and many engineering applications.
  • The concept that density is a mass-to-volume ratio with units of g/cm^3 (or kg/m^3) is central to material selection and design.

• Quick recap of key equations:

  • Density: \rho = \frac{m}{V}
  • Cylinder volume: V = \pi \left(\frac{d}{2}\right)^2 L
  • Foil thickness using known density: t = \frac{m}{\rho \cdot A}, \quad A = L \cdot W
  • Displacement volume: \Delta V = Vf - Vi = V_{object}

• Final note:

  • The transcript emphasizes careful measurement, error minimization, and linking experimental results to fundamental physical properties through explicit calculations and verifications. The measured densities serve as a basis for further inferences (e.g., foil thickness) and illustrate how different measurement techniques complement each other in a lab setting.