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 3.13P

(a)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for [9.23(±0.03)+4.21(±0.02)]3.26(±0.06)=?

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 3.13P

The absolute and percent relative uncertainty with a reasonable number of significant figures is (10.18±0.07) and  10.18±(0.7%) respectively.

Explanation of Solution

Given data:

  [9.23(±0.03)+4.21(±0.02)]3.26(±0.06)=?

Calculation of absolute and percent relative uncertainty:

  • The first term in the bracket [9.23(±0.03)+4.21(±0.02)] is solved as follows,

On solving the given subtraction, we get

    9.23(±0.03)+4.21(±0.02)13.44(±e)

The uncertainty (e) is calculated as follows,

  e2=0.032 + 0.022    =0.0013=0.0360e  =0.04 (rounded to correct significant figure).

The absolute uncertainty for the above addition part [9.23(±0.03)+4.21(±0.02)] obtained as 13.44±0.04.

  • On solving [13.44±0.04.]-[3.26(±0.06)]=? as follows,

    13.44(±0.04)3.26(±0.06)10.18(±e)

The uncertainty (e) is calculated as follows,

  e2=0.042 + 0.062    =0.0052=0.07211e  =0.07 (rounded to correct significant figure).

Therefore, the absolute uncertainty of [9.23(±0.03)+4.21(±0.02)] is given as (10.18±0.07).

Convert the absolute uncertainty to relative uncertainty as 0.0710.18×100=0.7 %

Therefore, the relative uncertainty is given as 10.18±(0.7%).

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is found as (10.18±0.07) and 10.18±(0.7%) respectively.

(b)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for [91.3(±1.0)×40.3(±0.2)]÷21.2(±0.2)=?

Concept Introduction:

Refer part (a).

(b)

Expert Solution
Check Mark

Answer to Problem 3.13P

The absolute and percent relative uncertainty with a reasonable number of significant figures is 174±3 and 174±2% respectively.

Explanation of Solution

Given data:

  [91.3(±1.0)×40.3(±0.2)]÷21.2(±0.2)=?

Calculation of absolute and percent relative uncertainty:

  • On solving the given multiplication part, we get

  91.3(±1.0)×40.3(±0.2)3680(±e)

For multiplication of 91.3(±1.0)×40.3(±0.2)=? , convert absolute uncertainty to percent relative uncertainty.

For 91.3(±1.0), percent relative uncertainty is (±1.0/91.3)×100=±1.1%

For 40.3(±0.2), percent relative uncertainty is (±0.2/40.3)×100=±0.5%

The uncertainty (e) is calculated as follows,

  %e2=1.12 + 0.52    =1.46=1.2%%e  =1.2% (rounded to correct significant figure).

Therefore, the relative uncertainty is given as 3680±1.2%

  • For division [3680±1.2%]÷21.2(±0.2)=?, convert absolute uncertainty to percent relative uncertainty.

For 21.2(±0.2), percent relative uncertainty is (±0.2/21.2)×100=±0.9%

Therefore,

  3679(±1.2%)÷21.2(±0.9%)174(%e)Since, %e2=1.22 + 0.92%e  =2.25=1.5    =2% (rounded to correct significant figure).

Convert relative uncertainty into absolute uncertainty as 174×2100=3

Therefore,

The absolute uncertainty is given as 174±3.

The percent relative uncertainty is 174±2%.

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is founds as 174±3 and 174±2% respectively.

(c)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for [4.97(±0.05)1.86(±0.01)]÷21.2(±0.2)=?.

Concept Introduction:

Refer part (a).

(c)

Expert Solution
Check Mark

Answer to Problem 3.13P

The absolute and percent relative uncertainty with a reasonable number of significant figures is (0.147±0.003) and (0.147±2%) respectively.

Explanation of Solution

Given data:

  [4.97(±0.05)1.86(±0.01)]÷21.2(±0.2)=?.

Calculation of absolute and percent relative uncertainty:

  • On solving the given subtraction part, we get

  4.97(±0.05)1.86(±0.01)3.11(±e)

The uncertainty (e) is calculated as follows,

  e2=0.052 + 0.012    =0.0026=0.05099e  =0.05 (rounded to correct significant figure).

Therefore, the absolute uncertainty of [4.97(±0.05)1.86(±0.01)] is given as 3.11±0.05

  • For division of [3.11±0.05]÷[21.2(±0.2)]=?, convert absolute uncertainty to percent relative uncertainty.

For 3.11±0.05, percent relative uncertainty is (±0.05/3.11)×100=±2%.

For 21.2(±0.2), percent relative uncertainty is (±0.2/21.2)×100=±1%.

On solving the given division, we get

  3.11(±2%)÷ 21.2(±1%)0.147(±e%)

The uncertainty (e) is calculated as below,

  %e2=22 + 12    e=5=2.23%=2%(rounded to correct significant figure).

Convert relative uncertainty into absolute uncertainty as 0.147×2100=0.002940.003

Therefore,

The absolute uncertainty is given as (0.147±0.003).

The percent relative uncertainty is (0.147±2%).

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is calculated as (0.147±0.003) and (0.147±2%) respectively.

(d)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for 2.0164(±0.0008)+1.233(±0.002)+4.61(±0.01)=?

Concept Introduction:

Refer part (a).

(d)

Expert Solution
Check Mark

Answer to Problem 3.13P

The absolute and percent relative uncertainty with a reasonable number of significant figures is (10.18±0.07) and  10.18±(0.7%) respectively.

Explanation of Solution

Given data:

  2.0164(±0.0008)+1.233(±0.002)+4.61(±0.01)=?

Calculation of absolute and percent relative uncertainty:

On solving the given addition, we get

  2.0164(±0.0008)+1.233(±0.002)4.61(±0.01)_7.86(±e)

The uncertainty (e) is calculated as follows,

  e2=0.00082 + 0.00220.012    =0.00010464=0.01023e  =0.01 (rounded to correct significant figure).

Therefore, the absolute uncertainty of given data is (7.86±0.01).

Convert the absolute uncertainty to relative uncertainty as 0.017.86×100=0.1%

Therefore, the relative uncertainty is given as 7.86±(0.1%).

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is found as (7.86±0.01) and 7.86±(0.1%) respectively.

(e)

Interpretation Introduction

Interpretation:

The absolute and percent relative uncertainty with a reasonable number of significant figures has to be found out for 2.0164(±0.0008)×103+1.233(±0.002)×102+4.61(±0.01)×101=?

Concept Introduction:

Refer part (a).

(e)

Expert Solution
Check Mark

Answer to Problem 3.13P

The absolute and percent relative uncertainty with a reasonable number of significant figures is 2185.8(±0.8) and 2185.8(±0.04%) respectively.

Explanation of Solution

Given data:

  2.0164(±0.0008)×103+1.233(±0.002)×102+4.61(±0.01)×101=?

Calculation of absolute and percent relative uncertainty:

By reducing scientific notion into 2016.4(±0.8)+123.3(±0.2)+46.1(±0.1)=?

On solving the given addition, we get

  2016.4(±0.8)+123.3(±0.2)46.1(±0.1)_2185.8(±e)

The uncertainty (e) is calculated as follows,

  e2=(0.82 +0.22+ 0.12)    =0.69=0.830e  =0.8 (rounded to correct significant figure).

Therefore, the absolute uncertainty of given data is 2185.8(±0.8).

Convert the absolute uncertainty to relative uncertainty as 0.82185.8×100=0.0365%=0.04%

Therefore, the relative uncertainty is given as 2185.8(±0.04%).

Conclusion

The absolute and percent relative uncertainty with a reasonable number of significant figures is found as 2185.8(±0.8) and 2185.8(±0.04%) respectively.

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