# Measured Quantities and Derived Quantities Class Notes

### 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 of of digits to the right of the decimal point .

12.01

+  2.0005

#### 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

REMEMBER THE 2 DIFFERENT RULES FOR SIGNIFICANT FIQURES

When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

When measurements are multiplied or divided, the answer can contain no more significant figures than least accurate measurement. (ie least number of significant digits).

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.

If you are using a 25 ml graduated cylinder, it is marked off in units of tenths. The liquid is between 12 and 12.1, but very, very close to 12. You read it as 12.01 ml. You are certain of 12.0, the 1 is your doubtful digit.

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.

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If it is reproduced, written permission must be granted by the author: D. Spurlock

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