20.Significant Figures(2).

Since our last post we are trying to get an overview of the concept of significant figures and we shall continue to do so in this post as well.

Let us begin by understanding how the concept of significant figures deals with precision and accuracy.Suppose we are measuring volume of a liquid in the laboratory.This can be done using –
1] A beaker with volumes measured on the side OR
2] A graduated measuring cylinder OR
3] A burette.
Which glassware would give us the most precise volume measurement? To predict that , we use the RULE OF THUMB.

The Rule Of Thumb –

Read the volume to 1/10th (or 0.1) of the smallest division on the glassware (This rule applies to any measurement).This means that the reading error is ±0.1 of the smallest division on the glassware.Let us use this rule for all the three glass wares –

1] Beaker –
• The smallest division on a beaker is 10mL.
• So we can read the volume to ±0.1 of 10mL = ±1mL. Thus , the reading error is ±1ml.
• Thus,if we measure the volume of the liquid as 45mL , it basically is 45±1mL.So, it could         be 44mL,45mL or 46 mL.
•No.of significant figures = 2.

2]Measuring Cylinder –
• Water molecules are more attracted to glass than to each other i.e the adhesive force(force between two unlike molecules –  water & glass in this case ) is stronger than the cohesive force ( force of attraction between two water molecules).So, in a graduated cylinder, the surface of the liquid is curved.We call that a ‘meniscus‘.Here, we read the meniscus correctly(lower meniscus).
• The smallest division on the graduated cylinder is 1 mL.So, the reading error is 0.1×1 mL = ±0.1mL.
•Thus, a 45ml volume measured in this case could be 45±0.1mL. = 44.9mL,45mL or 45.1mL.
• No.of significant  figures = 3. 

3]Burette –
• Smallest division on a burette is 0.1mL.
• Reading error = 0.1 ×0.1 = 0.01mL.
• Thus, the 45mL liquid measured could be 45±0.01mL i.e 44.99mL , 45.00 mL or 45.01mL.
• No.of significant figures =4 

 

The more the no.of significant figures, more is the precision.

↑Significant figures ↑precision 

Thus, if we need a precise volume of liquid, we shall use a burette than a beaker. Now we know why we use a burette for titrations! We need the exact endpoint for a titration – that last drop which is responsible for the color change in the flask. This is how significant figures concept helps us accuracy and precision.

Recording a measurement to the correct number of significant figures is important
because it tells others about how precisely you made your measurement.
For example, suppose you check your weight on a normal weighing scale and suppose you weigh 50kgs. On a digital balance, you can weigh 49.7kgs .Thus, a digital balance will record your weight more precisely.

The following are some more important terms which are related to accuracy and precision-

1) Confidence Limit (CL) ⇒

•  The confidence limit is the level of confidence we have about a value being  statistically  significant.

Confidence Limit (CL)=  100 × (1 − α) (%) , where , α ⇒ standard Deviation.

• Lesser the standard deviation, more is the confidence limit.This means that if the values of a data set are closer to the mean (less standard deviation), we can be more confident that our value lies in the expected range and is thus significant (more CL). If the value of CL is less it means that we are unsure about it being in the expected range of values.

2)LIMIT OF DETECTION (LOD) – 

• This is the lowest limit below which the sample cannot be measured.
• LOD is the minimum amount/concentration of a component that can be detected with a given degree of confidence.
• It is the lowest quantity of a substance that can be distinguished from the absence of that substance (blank value) within a stated confidence limit(about 1% generally).

3)LIMIT OF QUANTIFICATION  (LOQ) – 

• It is the concentration at which quantitative results can be reported with high degree of confidence.
• The quantification limit is the lowest amount of analyte in a sample which can be quantitatively determined with suitable precision and accuracy i.e we can measure it quantitatively.

To understand these concepts let us take an example .Let us assume that we are at the airport with lots of noise from the jets taking off.

If a person next to you,
∗ speaks softly → Voice < LOD ∴ We cannot hear his voice.
∗ speaks a bit loudly → LOQ > Voice > LOD ∴We hear his voice but cant decipher what he said.
∗ speaks loudly → Voice > LOQ  ∴ We can clearly hear him.
Although , during the course of time, the noise from the jet could also get reduced ! Thus, the quantities LOD and LOQ change.So, the detection limits (i.e LOD and LOQ)  are dependent on both the signal intensity (voice ) and the noise (jet noise).

4)Limit of linearity(LOL) –

• This is the maximum value of concentration of a component upto which the instrument produces  a linear response.Beyond LOL, response becomes non-linear.

 

5)Sensitivity –

• It is a measure of the ability of the method to discriminate between two small concentration differences in the analyte.
• It is measured in terms of the slope of the calibration curve.

Let us try  some problems on significant figures-

1)Give the answer to the correct no. of significant figures for (1.3×10³ ) (5.724 ×10⁴) = ?
Ans –

(1.3×10³ ) (5.724 ×10⁴) = 7.4412 × 10⁷. The number with the least significant figures is 1.3. So , the answer should have only 2 significant figures (We learnt this rule in earlier post).

∴ Answer = 7.4× 10⁷.

2) A student made a mistake when measuring the volume of a big container.He found the volume to be 65litres.However, the real value was 50 litres.What is the percent error?
Ans – 

% error =[ (Experimental value – True value)/True value ] × 100 = (65-50)/50× 100 =30%.

3)Compute the addition w.r.t significant figures- 17.0+0.9205+0.00848+18.24+185.
Ans-

17.0+0.9205+0.00848+18.24+185 = 221.16898.

The minimum no of decimal places in the original nos. is zero.So, the final answer should not contain any digit after the decimal.
∴ Answer = 221.

All these parameters and concepts are very pertinent to the study of analytical Chemistry.We use these concepts with different analysis methods . Imagine how important precision and accuracy are in the industries where high quality metalurgical work  is carried out, or how the detection limits are so often used in clinical chemistry to find out the amount of tumour cells or platelets in a blood sample.The applications of these topics are countless in almost all industries today .

With this post I end the topics of measurements,error,significant figures, accuracy and precision(Statistical Chemistry Topics in Analytical Chemistry). I shall begin writing about a fresh topic soon.Till then ,

Be a perpetual student of life and keep learning …

Good Day !

 

References and Further Reading –

1. http://www.chem.utoronto.ca/coursenotes/analsci/stats/ConfLevel.html
2. https://www.researchgate.net/post/How_to_calculate_limit_of_detection_limit_of_quantification_and_signal_to_noise_ratio

Image source –

1. Image 1 -http://www.alanpedia.com/physics_measurement_and_density/measurement_and_density.html
2. Image 2 – https://www.wikipremed.com/image_science_archive_68/021200_68/191700_Burette_68.jpg