By: Kirill Polyanskiy

**Introduction**

The purpose of this experiment is to test the accuracy and precision of the person doing the experiment as well as find the density of a metal rod. The accuracy and precision with which one performs experiments is a very important factor in their success as a scientist. In this laboratory, the accuracy and precision of my own laboratory techniques was tested by finding the mass, volume, and eventually density of a brass rod and comparing it to known values.

It is useful to know the density of an object because it allows us to identify the object’s composition. Back in the day, Archimedes was posed with a problem. The king did not know whether or not his crown was truly made of gold. Archimedes knew that gold had a mass/volume density ratio unlike any other, and if the crown had this ratio, then it was truly made of gold. Archimedes was perplexed though because he could not find the volume of the crown. It was not a perfect shape such as a cube or sphere, and because of this, it was difficult to find the volume of the crown. He remedied this problem by displacing the crown with water and seeing how much water flowed out of a filled container when the crown was allowed into it. Once he found the volume of the water, he knew it had to be the same as the volume of the crown because the volume of a liquid does not change with its container, only its shape. Upon taking the ratio, he found that the density of the crown was less than that of the pure gold sample he was given. The crown-maker had alloyed the crown with a less-worthy metal! [1]

The purpose of calibrating the graduated cylinder is to determine how accurate measurements can be made with it. Results from this analysis will tell how accurate the cylinder is and how many significant figures are usable in calculating the density of any substance using its volume measurement. The purpose of having standards and constants in this experiment to have a control. The purpose of having a control is to test for how much change occurred in the experiment; it serves as a basis for comparison. The control and/or standard used in this experiment is the actual density of the brass rod, and we can compare our calculated density of the brass rod with that of the standard to find out how close our techniques were to finding the correct measurement. Our measurements will inevitably have variation within them, and to find the amount of variation, we use a statistical process called standard deviation. Using standard deviation to find the average variation of measurement, we can find a mean line to symbolize our data and use that as the most accurate measurement because it symbolizes hundreds of trials within the time it took to make four or five. We can also assess the quality of our density measurements by comparing our measurements to that of the known measurements. This method is statistically relevant according to the lab manual.

**Materials and methods**

The material being analyzed in this laboratory is brass. The brass cylinder is about an inch and a half in length, 4mm in diameter with sawed-off edges. It is golden with stain spots in color, and smells metallic. It is smooth and warm to the touch. The brass rod resonates at a high pitch as compared to typical vocal ranges. In part A, the mass of the graduated cylinder was found to the nearest thousandth of a gram because that is how accurate the balance is. In part B, this cylinder and its most accurate reading were then used to find the density of a known metal by first finding the mass of the metal using the balance to its most accurate reading. The mass of three measurements and the average volume of the three measurements was found to increase the accuracy. In part C, The same technique as in part B was used, but it was applied to an unknown metal.

For part A, the accuracy of the graduated cylinder was calculated by finding the mass of the water through subtracting the mass of the empty cylinder from the mass of the full cylinder. The actual mass of the water was calculated by using its known density at 22.0 degrees Celsius, and it was compared to what was found from the cylinder to find the accuracy of the cylinder. In part B, the volume of the metal rod was calculated by subtracting the calculation of the water and the metal combined volume from the water only volume. The mass was found using the balance and the density values were made sure to be within 0.3 g/mL for precision.

**Results**

Part A: Table 1

Cylinder quality |
Mass (g) |
True Volume (mL) |
Percentage Difference |

Cylinder – Dry |
25.648 g |
0 mL |
0% |

Cylinder with 3.0 mL of water |
28.538 g |
2.890 mL |
3.7% |

Cylinder with 5.1 mL of water |
30.640 g |
4.992 mL |
2.1% |

Cylinder with 7.2 mL of water |
32.575 g |
6.927 mL |
3.8% |

Cylinder with 9.0 mL of water |
34.503 g |
8.885 mL |
1.3% |

Results: the mass of the cylinder and the water together increases linearly as more water is added to the cylinder. The volume of the graduated cylinder is off by about .11 mL for every measurement.

Part B:

Table 2: Measured Densities of a Brass Rod

Trial |
Mass (g) |
Final Volume (mL) |
Initial Volume (mL) |
Total Volume (mL) |
Density (g/mL) |

1 |
10.330 g |
7.2 mL |
6.0 mL |
1.2 mL |
8.608 g/mL |

2 |
10.354 g |
6.2 mL |
5.0 mL |
1.2 mL |
8.628 g/mL |

3 |
10.350 g |
8.2 mL |
7.0 mL |
1.2 mL |
8.625 g/mL |

4 |
10.343 g |
8.0 mL |
6.8 mL |
1.2 mL |
8.619 g/mL |

The final volume is the volume of the water with the brass rod inside it.

The initial volume is the volume of the water without the brass rod inside it.

The total volume is the final initial volume – the initial volume. It stands for the volume of the brass rod.

Results: The density of the brass rod was measured to be around 8.620125 g/mL. The actual density of a brass rod is 8.47 g/mL, about 0.15 g/mL away from the average.

Average Value: 8.620125 g/mL

Standard Deviation: 7.548916667*10^-5 g/mL

Range:

Minimum: 8.61143655

Maximum: 8.62881345

Page 7

Part C:

Table 3: Measured densities of an unknown substance

Trial |
Mass (g) |
Final Volume (mL) |
Initial Volume (mL) |
Total Volume (mL) |
Density (g/mL) |

1 |
11.870 g |
6.5 mL |
5.2 mL |
1.3 mL |
9.131 g/mL |

2 |
12.759 g |
6.0 mL |
4.6 mL |
1.4 mL |
9.114 g/mL |

3 |
11.921 g |
7.6 mL |
6.3 mL |
1.3 mL |
9.164 g/mL |

4 |
11.850 g |
7.0 mL |
7.0 mL |
1.3 mL |
9.115 g/mL |

Results: The density of the unknown substance averaged around

Average Value: 9.13225 g/mL

Standard Deviation: 6.6091*10^-4 g/mL

Range:

Maximum of range: 9.1579583

Minimum of range: 9.1065417

**Discussion**

The possible sources of error may have come from the graduated cylinder being miscalibrated. The graduated cylinder may also have had pieces broken off of it on the inside leading to a minute gain in volume. Broken in this sense is meant to be scratched, dented, or misshapen in any way. The tolerance of the Purex brand 10 mL graduated cylinder is ±0.20 mL. [2]. The standard deviation says that the technique used is precise but not accurate. The standard deviation is reasonable because it gives results to the thousandths place and it shows that the result is accurate to three decimal places. This kind of standard deviation is reasonable because it gives three significant digits minimum in the result of final density. A reasonable standard deviation would be ±0.3 g/mL because that is what the laboratory manual wrote it to be.

The calculated value for the density of the brass rod was 8.62 g/mL while the actual value of the brass rod is 8.47 g/mL. Because the standard deviation is 7.548916667*10^-5 g/mL, this means that the actual brass rod density is not within the standard deviation range of the calculated brass rod density. This makes the answer very precise because of the extremely small standard deviation, but it makes it inaccurate because the actual density of the brass rod is not within the range of the standard deviation of the calculated density of the brass rod.

The calculated density of the unknown is now unreliable because the measured density in part B was not within the standard deviation of the actual value of the density of the brass rod. It is also unreliable because the measurements taken were not precise as mentioned earlier. The true value of the density is not expected to be within the standard deviation of the calculated value because of previously mentioned errors in accuracy. The calculated density of the unknown is expected to be as far away from the true value as the calculations made the calculated value of the density of the brass rod far away from the true value of the density of the brass rod. The error according to part A says that the volume should be off by 0.11 mL in either direction. The error according to part B says that the density should be off by 7.548916667*10^-5 g/mL in either direction. Compounding this error with the previous 0.11 mL, the total error in either direction is 0.11 g/mL because the other error is too miniscule to count in significant figures. This contributes to my answer in putting it anywhere from 9.02 g/mL to 9.24 g/mL.

According to the density range received from applying the error, this substance is identified as antifriction metal although it is still probably wrong. Antifriction metal according to engineering toolbox, has a density range of 9.130 g/mL to 10.600 g/mL [3]. It could also be erbium at 9.07 g/mL or copper at 8.96 g/mL [4]. The most probable outcome of the rod is copper because it is an easily obtainable resource for chemistry laboratories.

Being a human, the experimenter inevitably made mistakes in this laboratory. The experimenter may have misread the graduated cylinder because its markings may not have been accurate. The experimenter may also have added mass to the cylinder by touching it and getting their hand oil on it. The experimenter could have corrected this by wearing gloves. The experimenter would improve this experiment by using exact 1cm by 1cm by 1cm cubes of material because it would make the volume a constant. This eliminates any sort of volume measuring and also eliminates any kind of error in this part of the lab.

There is a density experiment done at North Seattle University which measures the densities of water and of salt water. The measurement was performed by measuring the mass of the control water, then the mass of the salt water. The volume was stayed constant because they used the same volume for all the experiments. A control was the water the salt water density was compared to, and another control was the volume of liquid used in all trials of water and salt water.

Endnotes:

[1] Baldwin, James. “The Golden Crown (Introduction).” *The Golden Crown (Introduction)*. N.p., n.d. Web. 22 Oct. 2013. <http://www.math.nyu.edu/~crorres/Archimedes/Crown/Vitruvius.html>

[2] “Pyrex, Vista Cylinder, 10ml 1/ea (70022-10).” *Pyrex, Vista Cylinder, 10ml 1/ea (70022-10)*. Purex, n.d. Web. 23 Oct. 2013. <http://www.coleparmer.com/buy/product/81583-pyrex-vista-cylinder-10ml-1-ea-70022-10.html>

[3] “Metals and Alloys – Densities.” *Metals and Alloys – Densities*. Engineering Toolbox, n.d. Web. 23 Oct. 2013. <http://www.engineeringtoolbox.com/metal-alloys-densities-d_50.html>

[4] “Densities of Metals – Chempendix.” *Densities of Metals – Chempendix*. Sapling Learning, n.d. Web. 23 Oct. 2013. <https://www.math.nyu.edu/~crorres/Archimedes/Crown/Vitruvius.html /site/chempendix/densities-of-pure-metals>