Name__________________________



Name__________________________ |Section_____ Date___/___/____ | |Percent Error with sig fig and scientific notation Worksheet LHS Chemistry chapter 2

Percent Error Also called percent deviation and relative error

When scientists need to compare the results of two different measurements, the absolute

difference between the values is of very little use.  (Is being off by 10 cm a lot or a little?

It depends on whether you are measuring the length of a piece of paper or the distance to

Boston from Providence.)  What is used invariably is the percent error between the two

measurements.

If you are comparing your value to an accepted value, you first subtract the two values so

that the difference you get is a positive number; that is, subtract the smaller of the two

from the larger.  This is called taking the absolute value of the difference.  Then you

divide this result (the difference) by the accepted value to get a fraction, and finally

multiply by 100 to get the percent error.

So,     % error =   | your result - accepted value |   * 100

                                          accepted value

That's it.  That's all there is to this module.

A couple of points should be noted when using this equation to obtain a percent error.

1) When you do the subtracting that gives you the numerator of the fraction, note how

many significant figures remain after the subtraction and express your final answer to no

more than that number. 2)  If neither of the two values being compared is an "accepted value" , then use either

number in the denominator to get the fraction.  If one value is more reliable than the other, choose it for the denominator.

Example:  A student measures the volume of a 2.50 liter container to be 2.38 liters.

What is the percent error in the student's measurement?

Ans.      % error = (2.50 liters - 2.38 liters)  x   100

                                  2.50 liters

                       =  (.12 liters)     x    100

                            2.50 liters

                        =  .048     x   100

                        =  4.8% error

 

(Note only two sig figs left in the answer after the subtraction)

Some for practice:

What is the percent difference between a measurement of the density aluminum which

came out be 2.9 g/cm3 and the actual value of 2.2 g/cm3?

Compare a value of  7.428 with a value of  7.738

(Ans.:   4.17% or 4.01% depending on which you took as "actual")

         (Observed Value - True Value)

Percent Error = -------------------------------------------------- x 100

      True Value

|1.  Working in the laboratory, a student find the density of a piece of pure aluminum to be 2.85 g/cm3.  The accepted value for the density of aluminum is |

|2.699 g/cm3.  What is the student's percent error? |

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|2.  A student experimentally determines the specific heat of water to be 4.29 J/g x Co.  He then looks up the specific heat of water on a reference table |

|and finds that is is 4.18 J/g x Co.    What  is his percent error? |

|  |

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|3.  A student takes an object with an accepted mass of 200.00 grams and masses it on his own balance.  He records the mass of the object as 196.5 g.   What |

|is his percent error? |

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4) The literature value of the atomic mass of an isotope of nickel is 57.9 g/mol. If a laboratory experimenter determined the mass to be 55 g/mol, what is the percent error?

5) The mass of one mole of oxygen gas is determined in an experiment to be 30.41 g/mol. Calculate the percent error, given that the literature value for this mass is 32.0 g/mol.

6) At 20( C, the solubility of potassium chloride is actually 34.7 grams per 100 cm3 water. A laboratory experiment yielded 31.389 grams per 100 cm3 water at the value. What is the percent error?

7) The solubility product constant for silver oxide at 25(C is actually 1.51 x 10-8. An experimental value obtained in a lab was 1.57 x 10-8. What is the percent error?

More sig figs and some scientific notation

Whereas when counting objects, everyone would count the same number, and have an equally exact answer; when making measurements, there is no such thing as an EXACT measurement. Instead objects are measured to an accepted level of precision based on the limitations of the measuring apparatus. A digit is said to be a significant figure if it is either known with certainty or if it is the first estimated digit in a measurement.  

 

For example, suppose that there are three slips of paper on a desk. No matter which student counts them, they will all tell you the same answer, "There are three pieces of paper." However, if each student then measures the length of each slip of paper, there will most likely be a difference in their answers. One student might report the length as 8.20 cm, another as 8.19 cm and yet another as 8.22 cm. In each case, each answer has three significant figures. All three student agree that the slip of paper is greater than 8 cm long. All three would round off their answers to 8.2 cm. But they have each estimated a final digit. All three answers, within the limitations of their rulers, should be considered accept.

 

Let's look at some examples.

Refer to the following information for the next six questions.

[pic]

 

If each of the numbers (1, 2, 3, 4, 5) represent centimeters, then what is the reading for each of the specified locations? Note that in each of these measurements there should be two (2) certain digits and a third estimated digit.

A_______________ B_________________ C________________

D_______________ E_________________ F________________

Suppose you were now asked to state the TOTAL length of the ruler diagrammed? A reasonable value, to 3 significant figures, might be 5.28 cm. This measurement could also be stated as 52800 µm, 52.8 mm, 0.528 dm, 0.0528 m, 0.0000528 km. Because the value cannot become more accurate by converting it to other units, each of these new representations must also have only three significant figures. The question arises, when is zero a significant figure?

 

A zero is said to be significant if:

 

|(1) it is between two non-zero digits |3001 m, 30.001 m |4 SD, 5 SD |

|(2) it is at the end of a decimal expression |0.00310 km |3 SD |

|(3) it is required when expressing the number in scientific notation |3.10 x 106 m |3 SD |

 

Otherwise a zero is considered to be only a placeholder.

|150,000 m |all four zeros are placeholders |1.5 x 105 m |

|0.0015 km |all three zeros are placeholders |1.5 x 10-3 km |

|150. Gm |no zeros are placeholders |1.50 x 1011 m |

| |(note the deliberate inclusion of the decimal) | |

 

 

When multiplying or dividing two measurements, your answer should be rounded off so that it only has accurate as many significant digits as your least accurate original value. 

 

When adding or subtracting two measurements, first convert them to the same unit of measurement, then line up the decimals. Your final answer should be rounded off so that it only has as many decimal places as your least accurate original value.

Numerical constants (π, e, ½) do not have significant digits.

For example, the volume of sphere is calculated with the formula V = 4/3 πr3.

Using this formula, a sphere with a measured diameter of 24 cm would have a volume equal to

4/3 π(12)3 = 4/3 π(1728) = 2304π cm3

Since 12 only had two significant digits, your final value for the sphere's volume should only have 2 SD.

This means that a calculated value of V = 7238.229 cm3 should be expressed in final form as

7200 cm3 = 7.2 x 103 cm3

Scientific Notation. Express your value so that it has one digit to the left of the decimal and all other significant digits to the right of the decimal. It should then be multiplied by an appropriate power of 10.

(1) When the absolute value of the original number is greater than one, then moving the decimal point will require the resulting number to be multiplied by 10 raised to a positive exponent.

(2) When the absolute value of the original number is less than one, then moving the decimal point will require the resulting number to be multiplied by 10 raised to a negative exponent.

| | | |where the decimal was moved .... |

|0.066 g |6.6 x 10-2 g ||0.066| ................
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