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A short class with Sarah and her students. The other ways are under More.
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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.
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.
Lastly, 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.
When you add up precision, sum the errors with care, Combine them all together, you'll be fully aware.
Imagine a chef combining ingredients, measuring flour as 2 ± 0.1 kg and sugar as 3 ± 0.2 kg. Each time they bake, they take into account the precise amounts, ensuring delicious outcomes. They know to add the uncertainties for the best mixes!
For addition, think 'Sum it up with care', for multiplication, 'Add the percents to share'!
A.P.E. - Add for Plus, Percent for Times, Exponent for Powers (Addition, Percentage, Exponents).
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What is absolute uncertainty?
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How do you find total uncertainty in addition?
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What is the formula for percentage uncertainty in multiplication?
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Using powers, how do uncertainties change?
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The uncertainty of a measurement expressed in the same units as the measurement itself.
The absolute uncertainty expressed as a percentage of the measured value.
The process of determining the uncertainty of a derived quantity from the uncertainties of its measured values.
Consistent errors that result from flaws in the measurement system.
Unpredictable errors that cause fluctuations in measurements without a discernible pattern.
This section explains how to propagate uncertainties in calculations derived from experimental data.
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.
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.
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.
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.
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.