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11.1.5. Propagating Uncertainties in Calculations
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Today, we're going to discuss how to propagate uncertainties, especially during addition and subtraction. Can anyone tell me why it's important to consider uncertainty in our measurements?
So we can understand how reliable our data is?
Exactly! Now, when we add or subtract quantities, what do you think we should do with their uncertainties?
I think we add the uncertainties together.
Correct! If C = A + B and A has an uncertainty of ΔA and B has an uncertainty of ΔB, we express this as ΔC = ΔA + ΔB. Let’s look at an example. If a beaker weighs 50.15 ± 0.01 g and we measure something inside it, say it weighs 52.34 ± 0.01 g, what is the mass of the contents and its uncertainty?
So, first, we subtract, right? 52.34 - 50.15 gives us 2.19 g. And the uncertainty would be ± (0.01 + 0.01), which is ± 0.02 g.
So the final answer would be 2.19 ± 0.02 g?
Exactly! Great job summarizing that.
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Now, let's talk about calculations where we multiply or divide numbers. Can anyone guess how we combine uncertainties in those cases?
I think we just add the uncertainties again?
Close, but instead of absolute uncertainties, we work with percentage uncertainties here. If C = A × B, the total percentage uncertainty in C is the sum of the percentages in A and B. Can someone help me understand how to apply this?
So if A has 1% uncertainty and B has 2% uncertainty, then C would have 3% uncertainty?
Exactly! And once you have the total percentage uncertainty, you can convert it back to absolute uncertainty for your final answer. Let's say we have moles measured as 0.0100 ± 0.0001 mol, with a 1% uncertainty, and a volume of 0.1000 ± 0.0001 L with a 0.1% uncertainty. How do we find the concentration?
We divide the moles by volume, which gives us 0.100 mol/L, and then add the percentage uncertainties, which would give us 1% + 0.1%.
Correct! Therefore, the concentration is reported as 0.100 ± 0.001 mol/L.
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Create a free accountLastly, let's discuss what happens when we raise a quantity to a power. If we have a volume measurement of 2.0 ± 0.1 L, how do we propagate uncertainty when we square this value?
So we have to calculate the percentage uncertainty first?
Exactly! The percentage uncertainty is 5% since 0.1/2.0 = 0.05. Now, if we square the volume, how does that affect the uncertainty?
The percentage uncertainty would be doubled because we raise to a power, right? So it would be 2 × 5%, making it 10%.
Spot on! Great job. This way, you can correctly report your final result with its uncertainty.
Overview
Short Summary
This section explains how to propagate uncertainties in calculations derived from experimental data.
Medium Summary
It details methods for combining uncertainties when performing arithmetic operations on quantities with associated uncertainties. This helps in communicating the precision of derived results effectively.
Detailed Summary
Propagating Uncertainties in Calculations
When carrying out calculations with experimental data, it's crucial to understand how to combine uncertainties associated with each measurement. This process, often referred to as uncertainty propagation, ensures that final results accurately reflect the degree of uncertainty inherent in the individual measurements.
Key Points:
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Addition and Subtraction: When quantities are added or subtracted, their absolute uncertainties are summed. This means that if you have measurements A and B, and both have associated uncertainties (ΔA and ΔB), the uncertainty in the result (C) is calculated as:
ΔC = ΔA + ΔB
For example, if the mass of a beaker is measured as 50.15 ± 0.01 g and the total mass of the beaker plus contents is 52.34 ± 0.01 g, the mass of the contents would be:
C = 52.34 - 50.15 = 2.19 g with an uncertainty of ΔC = 0.01 + 0.01 = 0.02 g.
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Multiplication and Division: For calculations involving multiplication or division, the percentage uncertainties of the measured quantities are added together. If C = A × B, the percentage uncertainty in C is given by:
Percentage Uncertainty in C = Percentage Uncertainty in A + Percentage Uncertainty in B
After determining the total percentage uncertainty, it should be converted back to absolute uncertainty for the final answer.
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Powers and Roots: When a quantity is raised to a power, the percentage uncertainty is multiplied by the absolute value of the exponent. For instance, if you want to calculate the square of a volume measurement:
Percentage Uncertainty in C = |n| × Percentage Uncertainty in A
Understanding these concepts is pivotal for accurately interpreting experimental results and effectively communicating findings in scientific reports. Accurate uncertainty propagation is particularly important in fields like chemistry, where measurements can significantly impact conclusions.
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Create a free account- For Powers (and Roots):
- If a quantity is raised to a power, C=A^n, then the percentage uncertainty in C is the absolute value of the power multiplied by the percentage uncertainty in A.
- Percentage Uncertainty in C=|n|×Percentage Uncertainty in A.
- Example: If a volume is 2.0 ± 0.1 L (5% uncertainty), and you calculate its square for some reason, the percentage uncertainty in the squared volume would be 2×5%=10%.
Detailed Explanation
In situations where measurements are raised to a power or taken to a root, the way we calculate uncertainty changes again. The percentage uncertainty of the result is determined by taking the absolute value of the power and multiplying it by the percentage uncertainty of the original measurement.
For instance, if we have a volume measurement of 2.0 liters that has an uncertainty of 0.1 liters, yielding a percentage uncertainty of 5%, and we are calculating the area (which is volume squared), we will multiply 5% by 2 (the exponent). This outcome tells us that our uncertainty in the result increases as we calculate higher powers.
Examples & Analogies
Think of it like measuring the height of a stack of books that’s two volumes tall. If one volume is known to have a height of ±1 cm, raising it to two volumes effectively doubles the uncertainty, because as you stack more books, any height uncertainty could affect the overall stack. Thus, you need to account for that in your final calculations to know exactly how uncertain that stack height is.
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Key Concepts
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
If a mass is measured at 10.0 ± 0.1 g, and another measurement is 20.0 ± 0.2 g, their sum is 30.0 ± 0.3 g.
For a division calculation: if A = 4.0 ± 0.1 and B = 2.0 ± 0.1, the percentage uncertainty in C = A/B would combine as 5% + 5% = 10%.
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Glossary
Absolute Uncertainty
The uncertainty of a measurement expressed in the same units as the measurement itself.
Percentage Uncertainty
The absolute uncertainty expressed as a percentage of the measured value.
Propagation of Uncertainty
The process of determining the uncertainty of a derived quantity from the uncertainties of its measured values.
Systematic Error
Consistent errors that result from flaws in the measurement system.
Random Error
Unpredictable errors that cause fluctuations in measurements without a discernible pattern.