All the measurements have a certain degree of uncertainty regardless of their precision and accuracy. This is because of two factors, the limitation of the measuring instrument (called systematic error) and the skill of the experimenter doing the measurements (called random error). Too often, we come across values that are very close to each other and their average values. In such cases, we can say that the measurement is correct and precise. Specifying the uncertainty formula chemistry is important as it will help us to study the overall effect on output.
Scientific Notation
As we are aware of the fact that atoms and molecules have extremely low masses, but they are present in large numbers.
Chemists deal with figures which are as small as 0.00000000000000000000000166 g (Mass of Hydrogen atoms) and other constants that have very large values, like Avagadros number, Planck’s constant, Speed of light, Charge of particles, etc. Now the question arises how to handle such small and large numbers?
To handle these large or small numbers, we use the following notation: x x 10\[^{y}\], which is, x times ten raised to the power of y. In this expression, y is an exponent having positive or negative values and x is that number that can vary from 1.000… and 9.999…
In the same way, scientific notation 823.912 can be written as 8.23912 x 10\[^{2}\]. In this, the decimal is moved to the left side by two places and if it is moved three places to the left then the power of 10 will be 3. In the same manner, 0.00065 can be written as 6.5 x 10\[^{-4}\]. In this, the decimal is moved four places towards the right, so, −4 is the exponent in the scientific notation.
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Uncertainty Calculation
Uncertainty in Multiplication and Division:
Applying the same rule as discussed above we can solve the given problem as:
(4.3 x 10\[^{7}\]) x (2.7 x 10\[^{3}\]) = (4.3 x 2.7)(10\[^{7+3}\])
= (4.3 x 2.7) (10\[^{10}\])
= 11.6 x 10\[^{10}\]
Similarly for division,
\[\frac{4.9 \times 10^{-4}}{3.2 \times 10^{-6}}\] = (4.9 ÷ 3.2)(10\[^{-4-(-6)}\]) = 1.531 x 10\[^{2}\]
Uncertainty in Addition and Subtraction:
While doing addition or subtraction first of all we have to place these numbers in such a way that they have the same exponents. Eg: If there are two numbers 7.32 x 10\[^{3}\] and 9.55 x 10\[^{2}\]
Now adding both 7.32 x 10\[^{3}\] + 9.55 x 10\[^{2}\] = (7.32 + (0.955 x 10)) x 10\[^{3}\] = 8.275 x 10\[^{3}\]
Similarly for subtraction,
7.32 x 10\[^{3}\] - 9.55 x 10\[^{2}\] = (7.32 - (0.955 x 10))10\[^{3}\] = 6.365 x \[^{3}\]
Percentage Uncertainty Formula:
The uncertainty of a measured value can be represented in a percentage notation or as a simple ratio. It is calculated as:
percent uncertainty = \[\frac{Uncertainity}{\text{Actual value}}\] x 100
Solved Examples
Example: A scale measures wrongly a value as 6 cm due to some marginal errors. If the actual measurement of the value is 10 cm then calculate the percentage error.
Solution: Given,
Approximate value/wrong value = 6 cm
Exact value = 10 cm
Percentage Error = (Approximate Value - Exact Value)/Exact Value) x 100
Percentage Error = (10 – 6)/10 × 100
= 40 %
FAQs on Uncertainty in Measurement
1. What is uncertainty in measurement and why is it important in Chemistry?
Uncertainty in measurement refers to the doubt that exists about the result of any measurement due to the limitations of the measuring instrument and the skill of the person using it. It is crucial in Chemistry because all experimental data has some degree of uncertainty, and understanding it is essential for judging the reliability of experimental results, such as those used to determine reaction yields or atomic masses.
2. What is the key difference between accuracy and precision in experimental measurements?
Accuracy and precision are two fundamental concepts in measurement. Accuracy refers to how close a measured value is to the true or accepted value. Precision refers to how close multiple measurements of the same quantity are to each other. A measurement can be precise without being accurate, but for reliable results, both are desired.
3. What are the main types of errors that cause uncertainty in measurements?
The two main types of experimental errors are:
- Systematic Errors: These are consistent errors that occur in the same direction, either always high or always low. They are typically caused by faulty equipment or flawed experimental design (e.g., an uncalibrated thermometer).
- Random Errors: These are unpredictable errors that occur due to uncontrollable fluctuations in conditions, such as changes in temperature or human error in reading a scale. These errors can be either positive or negative.
4. How are significant figures used to represent uncertainty in a calculated result?
Significant figures are the digits in a number that are known with certainty plus the first digit that is uncertain. They are a crucial way to express the precision of a measurement. When performing calculations, the result cannot be more precise than the least precise measurement used. For example, in multiplication or division, the final answer should have the same number of significant figures as the input number with the fewest significant figures.
5. Can a measurement be precise but not accurate? Please provide an example.
Yes, a measurement can be highly precise but not accurate. This typically happens when there is a systematic error. For example, if you use a weighing scale that is incorrectly calibrated to show 0.5 kg less than the actual weight, and you weigh a 10.0 kg object three times, you might get readings like 9.5 kg, 9.5 kg, and 9.5 kg. These measurements are precise (very close to each other) but inaccurate (far from the true value of 10.0 kg).
6. What is the main purpose of using scientific notation for chemical measurements?
Scientific notation is used to express very large or very small numbers conveniently. Its main purpose in Chemistry, besides convenience, is to remove ambiguity in the number of significant figures. For instance, a number like 800 is ambiguous (it could have 1, 2, or 3 significant figures). Writing it as 8.00 x 102 clearly indicates that it has three significant figures, conveying the precision of the measurement.
7. How do you determine the uncertainty when adding, subtracting, multiplying, or dividing measurements?
The rules for determining uncertainty in calculations depend on the mathematical operation:
- For Addition and Subtraction: The result should be rounded to the same number of decimal places as the measurement with the least number of decimal places.
- For Multiplication and Division: The result should be rounded to the same number of significant figures as the measurement with the least number of significant figures.
8. How is absolute error different from relative error in evaluating a measurement?
Absolute error is the direct difference between the measured value and the true value, expressed in the same units as the measurement (e.g., ±0.1 g). Relative error expresses this error as a fraction or percentage of the true value (e.g., 0.1 g / 5.0 g = 0.02 or 2%). Relative error is often more useful as it puts the magnitude of the error into context, allowing for a better comparison of precision between different measurements.
