Density Determinations Class Notes

Techniques you will be learning.

1. Measurements and Uncertainty

All measurements are limited by some uncertainty due to the measuring process or the measuring instrument. There will always be a +/- range used to be sure the true value is included. By using measuring instruments capable of finer measurements we can reduce the uncertainty in our measurements, but we can never entirely eliminate it.

Examples: Using a balance that weighs to the thousandths place will give you a finer measurement and less uncertainty than a balance that weighs to the hundredth place, because you are certain of more digits in your measurement using the balance that weighs to the thousandths place.

Using a millimeter ruler to measure length will give a finer measurement with less uncertainty than when you use a centimeter ruler.

Using a graduated cylinder marked off in units of tenths will give you a finer measurement with less uncertainty than a graduated cylinder marked off in units of ones.

When you are recording your measurements, they are made up of digits that you are certain of and one uncertain digit at the far right. This uncertain digit is called the doubtful digit.

Example: You record a measurement of 121.98. You are certain of 121.9, the 8 is your doubtful digit.

2. Significant Figures of Digits

The concept of significant figures applies to measured quantities because of the uncertainty associated with every measurement. It does not apply to exact numbers such as counting numbers.

The number of significant digits in a measurement is the number of digits that you are certain of plus one that is doubtful. The doubtful digit is always the last digit shown.

Counting significant digits always begins with the first nonzero digit to the left and ends with the last digit shown - the doubtful digit.

EXAMPLE: 135.997      6 significant digits

Tail end zeros to the right of the decimal point are considered significant and must be counted even when that zero is the doubtful digit

EXAMPLE: 985.6210     7 significant digits

When there are only zeros to the left of the decimal point -DO NOT COUNT THESE ZEROS. They are merely place holders.

EXAMPLE: 000.154      3 significant digits

EXAMPLE: 00.2830      4 significant digits

When there are no nonzero digits to the left of the decimal point and also zeros between the decimal point and the first nonzero digit to the left of the decimal point - DO NOT COUNT THESE ZEROS. These zeros are merely place holders.

Examples: 0.5 One significant digit.           0.0010 Two significant digits.           0.006 One significant digit

Addition and Subtraction of Significant Digits

Write the measurements placing them so that the decimals line up. Do the math operation. Then look to see which value has the least number of decimal places to the RIGHT of the decimal. Do not count digits to the left of the decimal. Round so that your answer has the same number of digits to the right of the decimal point as the value with the LEAST number of decimal places.

  12.01

  14.0105      answer rounded to 14.01

Multiplication and Division of Significant Digits

Do the math operation, then round your answer to the same number of significant figures as the smallest number of significant figures in any factor.

2.03 x 10.90 = 22.12270     Answer rounded to 22.1

3. Reading the graduated cylinder

When a liquid is placed into a glass container it forms a meniscus a curved surface that is lower in the middle than at the edges. In order to read the graduated cylinder correctly, it must be placed on a stable surface such as the desk top of the work area. Never try to read the graduated cylinder while holding it in your hand. Your eye must be level with the bottom of the meniscus. Do not look down at or up at the bottom of the meniscus.

To determine the volume of liquid use the number that is directly at or below the bottom of the meniscus. Read to the unit the graduated cylinder is marked off in (your certain digits) THEN ESTIMATE ONE MORE PLACE (your doubtful digit). If the graduated cylinder is marked off in units of one, your volume reading must go to the tenths place in order to have your doubtful digit.

Examples: Using a 100 ml graduated cylinder (marked off in units of ones, the bottom of the meniscus is between 12 and 13, but closer to 12 than 13. You read it as 12.3 ml, with the 3 being your best estimation of where the bottom of the meniscus is. You are certain of the 12, the 3 is your doubtful digit.

If the graduated cylinder is marked off in units of tenths, then your volume reading must go to the hundredth place.

Remember: Read to the unit you are certain of, then estimate one more place.

4. Reading the Centimeter Ruler

The centimeter ruler is marked off in units of tenths so you must take measurements to the nearest tenth of a centimeter and then estimate the hundredth place ( your doubtful digit). All centimeter ruler measurements will have two places to the right of the decimal point.

Example: If you measure a line and find it measures exactly 12 cm, then your measurement is 12.00 cm.

Example: If you measure a line and find it measures between 1.5 and 1.6 cm but it is closer to 1.6, you estimate the uncertain digit. Your measurement could be 1.68 cm. The 8 is the doubtful digit and is estimated.

Remember: Read to the unit you are certain of, then estimate one more place.

5. Weighing by Difference

Chemicals are never placed directly on the pan of the lab balance. Instead, the mass is determined by a process known as weighing by difference. A suitable container such as a beaker is weighed empty on the balance. The desired chemical is added to the container, and the container plus the chemical is reweighed. By subtracting the mass of the empty container from the mass of the container plus the chemical, you find the mass of the chemical.

6. Full Calculator Readout vs Value Corrected for Significant Digits

You would think that the value of a derived quantity using a full calculator readout would be more reliable than a value corrected to the proper number of significant figures because the calculator readout gives you more digits in the answer, but this reasoning is incorrect. Using a full calculator readout gives a false sense of the measurement accuracy because the calculator displays as many digits as it can fit on the calculator screen. These additional digits are meaningless. The corrected values are more reliable because they take into consideration the uncertainty of the measurements. Your answer cannot be any more accurate than your least accurate measurement. Some calculators will display only 2 decimal places when more decimal places are needed to show the uncertainty in the measurement. Also calculators will often drop tail-end zeros from the derived answer even when these zeros are significant.

Density

Density is a physical property of a substance defined as mass per unit volume and determined by dividing the mass of the substance by its volume.

Density = mass/volume    ( D = m/v)

Units for density of a solid substance is grams/cm³. Units of density for liquids are grams/mL or grams/cc or grams/cm³.

Density of a Liquid

The density of a liquid is determined by using a clean graduated cylinder to hold the liquid. The mass of the liquid is calculated by using weighing by difference (ie [Mass of container + liquid] - mass of container = mass of liquid). The volume of the liquid is determined by reading the graduated cylinder at the bottom of the meniscus.

The graduated cylinder must be clean and dry before taking these measurements. If the graduated cylinder is dirty inside, you are getting a volume measurement that may not be correct, since the dirt also has a volume that may be included in the volume measurement. If the cylinder is wet inside, you will get a volume larger than the true volume due to the water being present in the bottom of the graduated cylinder. Always make sure the outside of the graduated cylinder is dry before placing it on the balance.

Density of a Regular Solid

A regular solid is something that we can measure easily with the centimeter ruler. We will be using cylinders of different metals. The mass of the regular solid is easily obtained by placing it on the balance. The volume of the cylinder is determined by measuring the height and diameter of the cylinder.

Use the following equation:

Density of an Irregular Solid

You cannot simply measure an irregular solid and calculate its volume without using calculus. An example of an irregular solid is a screw. Without knowing how to determine the area where the threads are, you cannot simply measure its length and diameter. We will be using a technique called liquid displacement. By determining the amount of liquid displaced by a solid, you can determine its volume. The irregular shape must be completely covered by the liquid in order to get the correct volume of the solid. Mass of the irregular shape is determined by weighing by difference.

(mass of container + water + irregular solid) - (mass of container + water) = mass of irregular solid

The volume is determined by reading the volume of a specified amount of water, then placing the irregular solid in the water and reading the volume again. This is called liquid displacement.

(Volume of liquid + irregular solid) - volume of liquid = volume of irregular solid

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last updated: January 11, 2010