Exploring Chemical Analysis
Exploring Chemical Analysis
5th Edition
ISBN: 9781429275033
Author: Daniel C. Harris
Publisher: Macmillan Higher Education
Question
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Chapter 3, Problem 3.12P

(a)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for [12.41(±0.09)÷4.16(±0.01)]×7.0682(±0.0004)=?

Concept Introduction:

Uncertainty:

Uncertainty means state of not certain in predicting a value.  In a measured value, the last digit will have associated uncertainty. Uncertainty has two values, absolute uncertainty and relative uncertainty.

Absolute uncertainty:

  Expressed the marginal value associated with a measurement

Relative uncertainty:

Compares the size of absolute uncertainty with the size of its associated measurement

  Relative uncertainty = absolute uncertaintymagnitude of measurement

For a set measurements having uncertainty values as e1,e2 and e3, the uncertainty (e4) in addition and subtraction can be calculated as follows,

  e4=e12+e22+e32

Percent relative uncertainty:

  Percent relative uncertainty = 100 × relative uncertainty

(a)

Expert Solution
Check Mark

Answer to Problem 3.12P

The absolute and percent relative uncertainty with a reasonable number of significant figures is 21.06±(0.16) and 21.06±0.6% respectively.

Explanation of Solution

Given data:

  [12.41(±0.09)÷4.16(±0.01)]×7.0682(±0.0004)=?

Calculation of absolute and percent relative uncertainty:

On solving the given division, we get

  12.41(±0.09)÷4.16(±0.01)2.98(±e)

For division of [12.41(±0.09)]÷[4.16(±0.01)]=?, convert absolute uncertainty to percent relative uncertainty.

For 12.41(±0.09), percent relative uncertainty is (±0.09/12.41)×100=±0.7%

For 4.16(±0.01), percent relative uncertainty is (±0.01/4.16)×100=±0.2%

Therefore,

   12.41(±0.7%)÷4.16(±0.2%)2.98(±e%)Since, %e2=0.72 + 0.22%e  =0.53=0.73    =0.7% (rounded to correct significant figure).

Convert relative uncertainty into absolute uncertainty as 2.98×0.7100=0.02.

Therefore,

The absolute uncertainty for the above division part obtained as 2.98(±0.02).

For the multiplication of [2.98(±0.02)]×(7.0682(±0.0004)) is calculated as follows,

  2.98(±0.02)×7.0682(±0.0004)21.06(±e)

For multiplication of [2.98(±0.02)]×(7.0682(±0.0004))=? , convert absolute uncertainty to percent relative uncertainty.

For 2.98(±0.02), percent relative uncertainty is (±0.02/2.98)×100=±0.6%

For 7.0682(±0.0004), percent relative uncertainty is (±0.0004/7.0682)×100=±0.005%

The uncertainty (e) is calculated as follows,

  %e2=0.0052 + 0.62    =0.360025=0.6%%e  =0.6% (rounded to correct significant figure).

Therefore, the relative uncertainty is given as 21.06±0.6%

Convert the relative uncertainty to absolute uncertainty as 21.06×0.6100=0.13

Therefore, the absolute uncertainty is given as 21.06±(0.16).

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is found as 21.06±(0.16) and 21.06±0.6% respectively.

(b)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for [3.26(±0.10)×8.47(±0.05)]0.18(±0.06)=?

Concept Introduction:

Refer part (a).

(b)

Expert Solution
Check Mark

Answer to Problem 3.12P

The absolute and percent relative uncertainty with a reasonable number of significant figures is (27.42±0.85) and 27.42±3.1%. respectively.

Explanation of Solution

Given data:

  [3.26(±0.10)×8.47(±0.05)]0.18(±0.06)=?

Calculation of absolute and percent relative uncertainty:

On solving the given multiplication, we get

    3.26 (±0.10)×8.47(±0.05)27.6122(±e)

For multiplication of [3.26 (±0.10)]×[8.47(±0.05)]=? , convert absolute uncertainty to percent relative uncertainty.

For 3.26 (±0.10), percent relative uncertainty is (±0.10/3.26)×100=±3.0%

For 8.47(±0.05), percent relative uncertainty is (±0.05/8.47)×100=±0.6%

The uncertainty (e) is calculated as follows,

  %e2=3.02 + 0.62    =9.36=3.1%%e  =3.1% (rounded to correct significant figure).

Therefore, the relative uncertainty is given as 27.6±3.1%

Convert 27.6±3.1% into absolute uncertainty as 27.6×3.1100=±0.85

For subtraction of [27.6(±0.85)]-0.18(±0.06)=? is calculated as follows,

  27.6(±0.85)-0.18(±0.06)27.42(e)Since, e2=0.852 + 0.062e  =0.7261=0.85    =0.8 (rounded to correct significant figure).

Convert absolute uncertainty (27.42±0.85) of into relative uncertainty as 0.8627.42×100=3.1%.

Therefore,

The absolute uncertainty is given as (27.42±0.85)

The percent relative uncertainty is 27.42±3.1%.

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is founds as (27.42±0.85) and 27.42±3.1%. respectively.

(c)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for 6.843(±0.008)×104÷[2.09(±0.04)1.63(±0.01)]=?.

Concept Introduction:

Refer part (a).

(c)

Expert Solution
Check Mark

Answer to Problem 3.12P

The absolute and percent relative uncertainty with a reasonable number of significant figures is (14.9±1.34)×104 and (14.9±9%)×104 respectively.

Explanation of Solution

Given data:

  6.843(±0.008)×104÷[2.09(±0.04)1.63(±0.01)]=?.

Calculation of absolute and percent relative uncertainty:

On solving the given subtraction, we get

  2.09(±0.04)1.63(±0.01)0.46±e

The uncertainty (e) is calculated as follows,

  e2=0.042 + 0.012    =0.0017=0.04e  =0.04 (rounded to correct significant figure).

Therefore, the absolute uncertainty of [2.09(±0.04)1.63(±0.01)] is given as 0.46±0.04.

For division, convert absolute uncertainty to percent relative uncertainty.

[6.843(±0.008)×104]÷[0.46±0.04.]=?

For 6.843(±0.008)×104, percent relative uncertainty is (±0.008/6.843)×100=±0.1%.

For 0.46±0.04., percent relative uncertainty is (±0.04/0.46)×100=±9%.

On solving the given division, we get

  6.843(±0.1)×104÷ 0.46(±9).14.9(±e%)×104

The uncertainty (e) is calculated as below,

  %e2=0.12 + 92    e=81.01=9%=9%(rounded to correct significant figure).

Convert relative uncertainty into absolute uncertainty as 14.9×9100=1.34

Therefore,

The absolute uncertainty is given as (14.9±1.34)×104.

The percent relative uncertainty is (14.9±9%)×104.

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is calculated as (14.9±1.34)×104 and (14.9±9%)×104 respectively.

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