Name: atomic mass uncertainty in isotope samples
Uploaded: 2024-04-04T10:51:11.000Z
Duration: 1 min 10 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Chemistry

CHEM.2

CHEM.2.B

propagation-of-uncertainty-from-systematic-error

Propagation of Uncertainty from Systematic Error

The atomic mass of an element obtained from different sources changes slightly due to the variation in relative isotope concentration from one source to another. This is an example of uncertainty arising from a systematic error.
Here, the distribution of atomic masses is rectangular, unlike the Gaussian distribution observed in random errors. This means there is an approximately equal probability of finding values between the lowest and highest values.
The standard deviation or standard uncertainty calculated from the plot serves as a measure of uncertainty.
The individual uncertainties from different atoms that make up a molecule contribute to the net uncertainty in the molecular mass.
The propagation of systematic uncertainty in the molecular mass of a heteroatomic molecule is estimated by multiplying the number of identical atoms with their standard uncertainties, followed by summing up the squares of total standard uncertainties from all elements present in the molecule, and finally, finding the square root.

propagation of uncertainty from systematic error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample&#39;s relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular sample will give rise to sampling error or a type of systematic error.This means that there is an uncertainty in the atomic mass of oxygen. It is not a random error but specific to the system or a systematic error. The probability of finding a particular value is the same within the uncertainty range. The distribution plot of values has a rectangular shape. With a sharp cut in the frequency of occurrence at the minimum and maximum reported values, the values in between have roughly the same frequency of appearance. This differs from the Gaussian distribution of values with random errors, where the frequency of occurrence is high near the central or mean value and tapers down as the value deviates outwards from the mean.
The standard deviation or standard uncertainty obtained from the rectangular distribution plot can be used to measure uncertainty. The individual uncertainties from constituent atoms in a molecule propagate to the uncertainty of its molecular mass. The propagation of systematic errors does not follow the same rules as that of random errors. Because systematic errors are not reduced by repeated measurements, their propagation is much more additive. Consider the following example: In a molecule containing multiple identical atoms, the uncertainty in the mass of one atom is multiplied by the number of similar atoms to get the total contribution of uncertainty toward the molecular mass of those particular atoms. If heteroatoms are present in a molecule, the uncertainty in the molecular mass is the square root of the sum of the squares of total uncertainties from each type of atom present.

The atomic mass of an element obtained from different sources changes slightly due to the variation in relative isotope concentration from one source to ...

Education

JoVE Core

Analytical Chemistry

Unit 1

Chemical Applications of Statistical Analyses

chemical-applications-of-statistical-analyses

KEY TERMS AND CONCEPTS

atomic mass uncertainty in isotope samples

Atomic mass can vary because different samples contain different isotope ratios. For oxygen, the measured atomic mass is a weighted average of the isotopic masses in that sample. A single sample may not match the true atomic mass of oxygen found across all oxygen molecules on Earth. That creates sampling error, which is a type of systematic error.

This kind of uncertainty is not random. It is tied to the system being measured. Within the uncertainty range, the chance of finding a particular value is the same. The values form a rectangular distribution, with a sharp cutoff at the minimum and maximum reported values. The values in between appear at roughly the same frequency.

This pattern is different from a Gaussian distribution, which is typical of random error. In a Gaussian plot, values cluster near the mean and taper off as they move away from it. The rectangular distribution can be used to find the standard deviation, also called the standard uncertainty. That value describes the uncertainty in the atomic mass measurement.

The uncertainty in individual atoms also affects the uncertainty in a molecule’s mass. Systematic errors do not decrease with repeated measurements, so their propagation is more additive. For a molecule with several identical atoms, the uncertainty for one atom is multiplied by the number of those atoms. If a molecule contains different kinds of atoms, the total molecular mass uncertainty is found by taking the square root of the sum of the squares of the total uncertainties from each atom type.

Atomic Mass Uncertainty in Isotope Samples

atomic-mass-uncertainty-in-isotope-samples

si units: 2019 redefinition

SI Units in Chemistry Measurements

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degrees of freedom

Degrees of Freedom in Statistical Calculations

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types of errors: detection and minimization

types-of-errors-detection-and-minimization

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systematic error: methodological and sampling errors

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Reducing Bias in Measurement Errors

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random error

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standard deviation of calculated results

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uncertainty: overview

Measuring and Reporting Uncertainty

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propagation of uncertainty from random error

propagation-of-uncertainty-from-random-error

How Uncertainty Changes in Calculations

Uncertainty changes as measurements move through calculations. In a multi-step experiment, each measurement adds some uncertainty. Because the steps happen one after another, the uncertainty from one step carries into the next step and affects the final result. The way uncertainty is combined depends on the math operation. For addition and subtraction, the final result is reported with absolute uncertainty. Absolute uncertainty is found as the square root of the sum of the absolute...

Atomic mass can vary because different samples contain different isotope ratios. For oxygen, the measured atomic mass is a weighted average of the isotopic masses in that sample. A single sample may not match the true atomic mass of oxygen found across all oxygen molecules on Earth. That creates sampling error, which is a type of systematic error. This kind of uncertainty is not random. It is tied to the system being measured. Within the uncertainty range, the chance of finding a particular...

uncertainty: confidence intervals

Confidence Intervals and Sample Size

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significance testing: overview

Significance Testing in Analytical Chemistry

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comparing experimental results: student's t-test

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Student's t-Test for Significance Checks

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detection of gross error: the q test

Using the Q Test to Flag Outlier Data

The Q test helps identify outlier data points that sit far from the rest of a data set. In high school lab work, this can help students decide whether an unusual value should stay in the data or be removed. Outliers often come from gross errors, or human mistakes, and may not reflect the real result being measured. Some points that look like outliers may still reflect a true difference in the phenomenon being studied. In those cases, a statistical test can help judge whether the value should...

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calibration curves: correlation coefficient

calibration-curves-correlation-coefficient

Interpreting Correlation in Calibration Curves

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correlation and regression

Using Regression to Measure Variable Links

Correlation and regression are used to study the relationship between two variables. Correlation shows how strongly the variables are associated. Regression helps describe that relationship with a line that can be used for prediction. In linear regression, the relationship is written as a correlation coefficient. This coefficient is shown as r and ranges from -1 to +1. A positive value means the variables move in the same direction. A negative value means they move in opposite directions. <br...

difference from background: limit of detection

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Limit of Detection and Quantification

The limit of detection, or LOD, is the smallest amount of analyte that can be distinguished from background noise. In measurement science, background noise is the natural signal in a sample that can interfere with the analyte signal. The LOD helps show whether a substance is present, but it does not usually give a reliable amount. The LOD is set at the concentration where the analyte signal is three times larger than the standard deviation of the blank signal. A blank is a sample without the...

quantifying and rejecting outliers: the grubbs test

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Grubbs Test for Spotting Data Outliers

The Grubbs test is a statistical way to spot a possible outlier in a data set. It is used when the data is assumed to follow a normal distribution. A Grubbs test helps decide whether one unusual numerical value should be treated as a true outlier. To run a two-tailed Grubbs test, first find the absolute difference between the questionable value and the mean. Then divide that difference by the sample standard deviation. The result is the Grubbs statistic, or G. Next, compare the calculated G...

what is anova?

ANOVA for Comparing Sample Means

ANOVA, or Analysis of Variance, is a statistical test used to compare the means of three or more samples. It was developed by Ronald Fisher in 1918. In high school statistics, ANOVA helps determine whether group averages are equal or whether at least one group is different. Before ANOVA is used, the samples should meet three key assumptions. The data should come from normally distributed samples. The samples should also be randomly and independently selected. In addition, the populations...

key terms and concepts

Explore how isotope ratios affect atomic mass, why systematic uncertainty follows a rectangular distribution, and how atomic uncertainties add into molecular mass.

Atomic Mass Uncertainty in Isotope Samples | JoVE K12

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CHEM.2.A

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SI Units: 2019 Redefinition

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Standard Deviation of Calculated Results

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CHEM.2.C

Scientific and engineering practices. The student analyzes and interprets data to derive meaning, identify features and patterns, and discover relationships or correlations to develop evidence-based arguments or evaluate designs. The student is expected to:

Atomic Mass Uncertainty in Isotope Samples

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