### Learning Objectives

By the end of this section, you will be able to:

- Calculate the density of a sugar solution.
- Evaluate lab sources of error and their effect on an experiment.

**Introduction**

The density of an object is defined as the ratio of its mass to its volume. We write this mathematically by using the equations:

**Equation 1**

[latex]\displaystyle\text{density}=\frac{\text{mass}}{\text{volume}};\text{ d}=\frac{\text{m}}{\text{V}}\\[/latex]

For an example of density, consider the following: Imagine a brick that is made of Styrofoam. Imagine a second brick that is made of lead. Note that even though the bricks take up the same amount of space – that is, they have the same volume – there is a major difference in their mass. We would say that the lead is denser, that is it has more mass in the same volume.

It is important to note that water has a density of 1.0 g/mL. Objects that have a density less than water, that is, less than 1.0 g/mL, will float on the surface of the water. Those that have a density greater than 1.0 g/mL will sink. Consider our two bricks again. The brick of Styrofoam will float if we toss it into water. The lead will quickly sink.

Modern ship manufacturers make use of density when designing the ships they build. They use materials that are denser than water but shape the materials so that they take up enough space to float. Although the ships weigh several thousand tons, that mass takes up a lot of space. Overall, the ship has a density less than water and therefore floats.

Two factors have an effect on the density of water:

1) Temperature will have a small effect on the density. For water, density increases as temperature decreases. See Table 1 for the density of water at different temperatures.

2) If more dense materials are dissolved in the water, the solution density will increase. We will see this effect in today’s lab when we measure the effect of dissolving sucrose on the density of water.

Table 1. Density of water at different temperatures |
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Temp (°C) | dH2O (g/mL) | Temp (°C) | dH2O (g/mL) |

18.0 | 0.99860 | 22.0 | 0.99777 |

18.5 | 0.99850 | 22.5 | 0.99765 |

19.0 | 0.99841 | 23.0 | 0.99754 |

19.5 | 0.99830 | 23.5 | 0.99742 |

20.0 | 0.99820 | 24.0 | 0.99730 |

20.5 | 0.99809 | 24.5 | 0.99716 |

21.0 | 0.99799 | 25.0 | 0.99704 |

21.5 | 0.99788 | 25.5 | 0.99690 |

In this experiment you will test your laboratory technique by calibrating a 10 mL graduated cylinder, making up an aqueous sucrose solution of a particular mass percent in solute, and measuring the density of the solution with the calibrated graduated cylinder. The density result will be evaluated by students for accuracy and precision. Since the correct density will depend on a correctly prepared sugar solution, *careful sample preparation will be critical.*

There are many ways of describing the concentration of a solution. The mass percent of solute in a solution is given by the symbol and the equation:

[latex]\displaystyle\text{Mass }\%=\frac{\text{Mass of Solute}}{\text{Total Mass of Solution}}\times100\\[/latex]

The advantage of this type of concentration unit is that it depends only on the mass, which is accurately measured with an analytical balance. It is not dependent on the temperature.

Note: Volumes **are** dependent on temperature. For example, a 10.000 mL volume of water will increase by 0.016 mL when the temperature is raised from 18°C to 25°C. Table 1 gives the density of water at different temperatures.

Another useful property is using the percent error to determine the amount a measurement is off from the theoretical value. The equation for finding percent error is:

[latex]\displaystyle\text{Percent Error}=\frac{\text{Experimental Value}-\text{Theoretical Value}}{\text{Theoretical Value}}\times100\\[/latex]

This allows us a more reasonable comparison of numbers than looking at the difference only because the magnitude of the theoretical value is considered. A table of the theoretical values of density for sucrose solutions of various (w/w)% is included in Table 2 below.

Table 2: Theoretical Density Values of Sucrose Solutions with Known Mass Percent |
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Mass % |
Density (g/mL) |
Mass % |
Density (g/mL) |

0.00 | 1.000 | 12.50 | 1.051 |

2.50 | 1.011 | 15.00 | 1.062 |

5.00 | 1.021 | 17.50 | 1.073 |

7.50 | 1.030 | 20.00 | 1.084 |

10.00 | 1.042 | 22.50 | 1.102 |

## Graphing Data

It is imperative students learn to properly organize and graph data. Students may wish to review graphing data and calculating the slope prior to coming to lab this week if it has been a few years since you have had a math course. A brief review is included here but may not be sufficient for some students.

Manual graphs should always:

- Be drawn on graph paper (included within the lab handout).
- Include data points (and possibly the labels as well).
- Have labels for the graph itself (named Y vs. X), the axes (with both name and units), and (if applicable) the legend.
- Be drawn large enough to visually see all components.
- Include axis scales that are appropriate (they may not start at 0, depending on the data).
- Contain a line of best-fit.

Graphs done in Microsoft Excel should always

- Include all of the components of manual graphs.
- Be in the “scatter” chart type unless otherwise specified.
- Include the equation for the line of best fit.