20.Significant Figures(2).

Since our last post, we have been trying to get an overview of the concept of significant figures and we shall continue the discussion in this post as well.

Let us begin by understanding how significant figures deal with precision and accuracy. We studied in the previous post that – the number of significant figures is the number of digits believed to be exactly correct while doing the experiment. So the greater the number of significant figures, the more precise the result. To understand this concept better we shall learn the ‘RULE OF THUMB’.

---

THE RULE OF THUMB

Suppose we are measuring the volume of a liquid in the laboratory. This can be done using –
1] a beaker 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 requires us to read the volume to 1/10th (or 0.1) of the smallest division on the glassware. Thus, the reading error is ±0.1 of the smallest division on the glassware.

Let us use this rule for all the three glasswares –

1] Beaker –

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). Thus, in a graduated cylinder, the surface of the liquid is curved. We call that a ‘meniscus‘. We always read the lower meniscus.
• The smallest division on the graduated cylinder is 1 mL. Thus, the reading error is 0.1×1 mL = ±0.1mL.
• A 45ml volume measured in this case could be 45±0.1mL = 44.9mL, 45mL, or 45.1mL.
• Number of significant figures = 3.

3]Burette –
• The 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 use a burette rather than a beaker. In titrations, the endpoint is marked by the last drop that produces a colour change. Thus, titrations are considered to be very accurate.

Recording a measurement with the correct number of significant figures is important
because it tells others how precisely the measurement was made.
e.g.- Suppose a person weighs 50kgs on a normal weighing scale. On a digital balance, he could weigh 49.7 kg. Thus, a digital balance records the weight more precisely.

---

IMPORTANT PARAMETERS RELATED TO ACCURACY AND PRECISION

1) Confidence Interval and Confidence Limit (CL)  ⇒

Confidence in statistics is a way to describe the probability. The greater the level of confidence, the probability of that value being significant is more. The confidence limit is our level of confidence about a statistically significant value.The formula for calculating the CL is as follows-

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

We have studied the concept of standard deviation in post-18. Standard deviation tells us how closely our values are clustered around or dispersed from the mean. Thus, lower the standard deviation, the greater 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). The value of CL being less indicates that we are unsure about it being in the expected range of values.

2) LIMIT OF DETECTION (LOD) – 

Analytical techniques have their limitations. They might be unable to estimate analytes (component to be measured) at concentrations near zero i.e. when the analyte concentration is low. In lower concentrations, the signal produced by the analyte cannot be distinguished from the noise. Thus, a signal that can be easily distinguished from the noise is required to make correct observations. Normally, the minimum signal that is 3 times the size of the noise.

The limit of detection (LOD) is defined as the lowest quantity of the analyte substance that produces a signal at least three times the average noise level of the detector.

LOD is the lowest limit below which the sample cannot be measured. It is the minimum amount/concentration of a component that can be detected with a certain degree of confidence. Intuitively, the LOD would be the lowest concentration obtained from the measurement of a sample (containing the analyte) that we would be able to tell apart from the concentration obtained from the measurement of a blank sample (a sample not containing the analyte).

This concept is especially useful during trace analysis, where the component/s are found in minor quantities. In such cases, the analyst needs the LOD to present the results confidently. This concept is employed while studying toxic materials in food/soil/water. If a value falls below the LOD there is a good chance of that value being a FALSE POSITIVE.Thus, LOD is the minimum amount/concentration of a component that can be detected with a given degree of confidence.

However, this concept is one of the most controversial in Analytical chemistry. Most labs protect themselves against FALSE NEGATIVE results by increasing the LOD. As the LOD is higher, the probability of not reporting the analyte when it is present reduces.

Though LOD represents the lowest limit where the analyte can be detected, it may not be the limit where the measurement can be quantified. This means that even if the analyte is detected at low concentrations, it may not be possible to correctly know how much of it is present. The next parameter comes to the rescue in such cases.

3) LIMIT OF QUANTIFICATION  (LOQ) – 

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. It is the concentration at which quantitative results can be reported with a high degree of confidence.

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.
However, it is possible that during the course of time, the noise from the jet could 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 the concentration of a component up to which the instrument produces a linear response. Beyond LOL, the response becomes non-linear, as shown in the figure below.

 

5) Sensitivity –

It is a measure of the ability of the method to discriminate between two small concentration differences in the analyte. The procedure is considered sensitive if it can more accurately tell two close concentrations apart.

Sensitivity is measured in terms of the slope of the calibration curve. The greater the slope, the more sensitive the method.

It is important to get acquainted with these parameters as they are used frequently in Analytical laboratories.

---

NUMERICAL PROBLEMS

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 (refer to the 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 65 litres. However, the real value was 50 litres. What is the percentage 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 the parameters and concepts we have studied till now are pertinent to the study of Analytical Chemistry. We use these concepts with different analysis methods. These parameters are important in industries where high-quality analysis. The applications of these topics are countless.

We conclude the discussion on the basics of Analytical Chemistry with this topic. We shall start studying a new topic from the next post onwards. 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