Accuracy in Measurement: A Study of Density

Austin Peay State University Department of Chemistry

Accuracy in Measurement: A Study of Density

CHEM 1011

Caution

  • Personal Protective Equipment: Always wear safety goggles, closed-toed shoes, and long pants in the lab. This is crucial as broken glass poses a potential danger.

Purpose

  • To gain familiarity with density.
  • To validate mathematical conversions with graphical analysis using Excel.
  • To report accuracy as a measure of percent error.

Introduction

Definition of Accuracy
  • Accuracy: Refers to how close experimental data is to theoretical values established by the scientific community.
  • Commonly quantified using percent error, which indicates how far the experimental data is from a theoretical perfect score (% error).
Percent Error Equation
  • The formula for calculating percent error is:
    \text{% error} = \left| \frac{\text{theoretical value} - \text{experimental value}}{\text{theoretical value}} \right| \times 100
Importance of Density
  • Density: Defined as a measure of a substance's mass divided by its volume. The formula for density is:
    d = \frac{m}{V}
  • The density of water remains constant unless temperature changes.

Goals of the Experiment

  • Gain experience with different types of glassware and their measuring capabilities.
  • Compare the density calculated from mass and volume measurements with the theoretical density of water.
Measuring Mass
  • Digital Balance: Ensure the scale reads zero before use by taring it.
  • Best practices recommend using a container (e.g., weigh boat) to avoid damaging the balance and ensure accurate mass recordings.

Measuring Volume: Water Displacement Technique

  • A method to measure the volume of irregular solids involves:
    1. Filling a graduated cylinder with a measured volume of water.
    2. Immersing the solid in water and recording the new water line (meniscus).
    3. The volume of the solid can be derived from the change in water level, presuming no dissolution or air pockets.

Key Physical Property - Density

  • Example: Density of copper is $8.96 \, \text{g/cm}^3$.
  • If a piece of copper has a volume of $3.95 \, \text{cm}^3$, its mass can be calculated as:
    3.95 \, \text{cm}^3 \times \frac{8.96 \, \text{g}}{1 \, \text{cm}^3} = 35.4 \, \text{g}
  • Conversely, if given a mass of $29.04 \, \text{g}$, its volume can be calculated using the density:
    29.04 \, \text{g} \times \frac{1 \, \text{cm}^3}{8.96 \, \text{g}} = 3.24 \, \text{cm}^3

Data Collection and Analysis

Graphical Analysis
  • Scatterplots and linear regression techniques will be used to analyze collected data.
  • The typical linear regression form is: y = mx + b
    • Where:
    • $y$: variable on the y-axis (mass in this study).
    • $x$: variable on the x-axis (volume in this study).
    • $m$: slope of the line (density in this context).
    • $b$: y-intercept (theoretical mass).
  • Coefficient of Determination ($R^2$): Indicates how well data points fit a linear model, with a perfect value being $1.000$. A value above $0.900$ is typically acceptable.

Experiment Description

Materials Needed
  • Equipment and Chemicals:
    • Balance
    • Solid aluminum
    • 25-mL graduated cylinder
    • 50-mL beaker
    • Thermometer
    • Calculator
    • Buret
    • 100-mL beaker
    • Microsoft Excel

Procedure

General Lab Procedure

  • Teams will occupy various stations to perform the experiments within the allotted time.

Station 1: Determining Density of Aluminum by Displacement

  1. Measure the mass of aluminum using the digital balance (record all digits).
  2. Fill a 25-mL graduated cylinder with approximately 15 mL of deionized water and record the meniscus level.
  3. Submerge the aluminum gently into the water, recording the new meniscus level.
  4. Dry the aluminum, dispose of the water, and reset the station.

Station 2: Reporting the Density of Water from a Beaker

  1. Weigh an empty 100 mL beaker and record its mass.
  2. Fill the beaker to about 1/4, 1/2, and full with tap water, measuring and recording mass and volume.
  3. Rinse the beaker after the measurements and ensure it is returned to its original state.

Station 3: Reporting the Density of Water from a Buret

  1. Fill the buret with deionized water, ensuring to displace any bubbles by letting some water flow out.
  2. Weigh an empty 50-mL beaker and record its mass.
  3. Measure and record the initial water level in the buret.
  4. Add approximately 2 mL and record the final mass and water level; repeat for totals of approximately 5 mL, 9 mL, and 12 mL.
  5. Pour water out and reset the station.

Calculations

Station 1: Density of Aluminum

  • Calculate the wood volume using the difference in water level before and after immersion:
    \text{Volume of Aluminum} = \text{Final Water Volume} - \text{Initial Water Volume}
  • Determine density:
    \text{Density of Aluminum} = \frac{\text{Mass of Aluminum}}{\text{Volume of Aluminum}}
  • Record and compare to the theoretical density of aluminum: find credible sources for this value.
  • Calculate percent error using:
    \text{% error} = \left| \frac{\text{Density Calculated} - \text{Density Theoretical}}{\text{Density Theoretical}} \right| \times 100

Station 2: Density of Water from Beaker

  • Calculate the mass of water:
    \text{Mass of Water} = \text{Mass of Beaker and Water} - \text{Mass of Empty Beaker}
  • Calculate density for each trial:
    \text{Density of Water} = \frac{\text{Mass of Water}}{\text{Volume of Water}}
  • Compare with theoretical density and determine percent error as done above.

Station 3: Density of Water from Buret

  • For each water addition, calculate precise volume and mass:
    • Volume:
      \text{Volume Added} = \text{Initial Buret Amount} - \text{New Buret Amount}
    • Mass:
      \text{Mass of Water} = \text{Mass of Beaker with Water} - \text{Mass of Empty Beaker}
  • Calculate density for each addition:
    \text{Density} = \frac{\text{Mass of Water}}{\text{Precise Volume}}
  • Find average percent errors and graphically determine density using Excel.

Graphical Analysis of the Buret Station

  1. Input data in Excel for both volume and mass for each measurement obtained.
  2. Create a scatterplot and add a linear trendline to derive the regression equation in the form: y=mx+b.
  3. Record the equation, slope (density), and $R^2$ value.
  4. Use this slope to calculate percent error against theoretical density and report the findings.

Data Sheets

Station 1: Density of Aluminum by Displacement

  • Records for:
    • Mass of Aluminum.
    • Volume of Water (initial & final).
    • Calculated Volume.
    • Theoretical and calculated densities.

Station 2: Density of Water from Beaker

  • Records for:
    • Mass of Empty Beaker.
    • Volume of Water for each fullness measure.
    • Mass of Water from beaker measurements.

Station 3: Buret Density Measurements

  • Records for:
    • Mass of the empty beaker.
    • Buret measurements before and after each addition.

Post-Lab Exercises

  1. Discuss how overall experiment accuracy is reported.
  2. Explain why percent errors are only reported as positive values.
  3. Compare manually determined versus graphically determined water density accuracy.
  4. Calculate mass of water for $6.318 \, mL$ using the regression equation.
  5. Calculate volume for a mass of $4.925 \, g$ using the regression equation.