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

Masa atomowa pierwiastka uzyskanego z różnych źródeł zmienia się nieznacznie ze względu na zmianę względnego stężenia izotopów z jednego źródła na drugie. Jest to przykład niepewności wynikającej z błędu systematycznego.
Tutaj rozkład mas atomowych jest prostokątny, w przeciwieństwie do rozkładu Gaussa obserwowanego w błędach losowych. Oznacza to, że istnieje w przybliżeniu równe prawdopodobieństwo znalezienia wartości między najniższą a najwyższą wartością.
Odchylenie standardowe lub niepewność standardowa obliczona na podstawie wykresu służy jako miara niepewności.
Poszczególne niepewności dotyczące różnych atomów tworzących cząsteczkę przyczyniają się do niepewności netto masy cząsteczkowej.
Propagację niepewności systematycznej w masie cząsteczkowej cząsteczki heteroatomowej szacuje się poprzez pomnożenie liczby identycznych atomów przez ich niepewności standardowe, a następnie zsumowanie kwadratów całkowitych niepewności standardowych ze wszystkich pierwiastków obecnych w cząsteczce, a na końcu znalezienie pierwiastka kwadratowego.

propagacja niepewności z błędu systematycznego

Masa atomowa pierwiastka zmienia się ze względu na względny stosunek jego izotopów. Względna proporcja izotopów tlenu w próbce wpływa na jej średnią masę atomową. Na przykład, jeśli mielibyśmy zmierzyć masę atomową tlenu z próbki, masa ta byłaby średnią ważoną mas izotopowych tlenu w tej próbce. Ponieważ jest mało prawdopodobne, aby pojedyncza próbka doskonale odzwierciedlała rzeczywistą masę atomową tlenu dla wszystkich cząsteczek tlenu na Ziemi, masa, którą otrzymamy z tej konkretnej próbki, będzie powodować błąd próbkowania lub rodzaj błędu systematycznego. Oznacza to, że nie ma jest niepewnością dotyczącą masy atomowej tlenu. Nie jest to błąd losowy, ale specyficzny dla systemu lub błąd systematyczny. Prawdopodobieństwo znalezienia określonej wartości jest takie samo w przedziale niepewności. Wykres rozkładu wartości ma kształt prostokąta. Przy ostrym zmniejszeniu częstotliwości występowania minimalnych i maksymalnych zgłaszanych wartości, wartości pomiędzy nimi mają mniej więcej tę samą częstotliwość występowania. Różni się to od rozkładu Gaussa wartości z błędami losowymi, gdzie częstotliwość występowania jest wysoka w pobliżu wartości środkowej lub średniej i maleje w miarę odchylania się wartości na zewnątrz od średniej.

Do pomiaru niepewności można zastosować odchylenie standardowe lub niepewność standardową uzyskaną z prostokątnego wykresu rozkładu. Indywidualne niepewności wynikające z atomów składowych cząsteczki rozprzestrzeniają się na niepewność dotyczącą jej masy cząsteczkowej. Propagacja błędów systematycznych nie podlega tym samym zasadom, co błędów losowych. Ponieważ powtarzane pomiary nie zmniejszają błędów systematycznych, ich propagacja jest znacznie bardziej addytywna. Rozważmy następujący przykład: w cząsteczce zawierającej wiele identycznych atomów niepewność dotyczącą masy jednego atomu mnoży się przez liczbę podobnych atomów, aby otrzymać całkowity udział niepewności w masie cząsteczkowej tych konkretnych atomów. Jeżeli w cząsteczce obecne są heteroatomy, niepewność masy cząsteczkowej jest pierwiastkiem kwadratowym z sumy kwadratów całkowitych niepewności każdego rodzaju obecnego atomu.

Propagacja niepewności z błędu systematycznego

Masa atomowa pierwiastka uzyskanego z różnych źródeł zmienia się nieznacznie ze względu na zmianę względnego stężenia izotopów z jednego źródła do ...

Education

JoVE Core

Analytical Chemistry

Chemia analityczna

Unit 1

Jednostka 1

Chemical Applications of Statistical Analyses

Zastosowania analiz statystycznych w chemii

chemical-applications-of-statistical-analyses

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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

Degrees of freedom describe how many values in a statistical calculation are free to vary. They represent the minimum number of independent numbers needed to determine a statistic. In a data set, some values may be independent while others depend on them. A simple example uses three unknown numbers with a mean of 10. The first two numbers can be chosen freely. The last number cannot be chosen on its own because it must keep the mean at 10. In that case, the data set has two degrees of freedom.<br...

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Mean, Median, Range, Precision & Accuracy

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types-of-errors-detection-and-minimization

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systematic error: methodological and sampling errors

systematic-error-methodological-and-sampling-errors

Reducing Bias in Measurement Errors

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random error

Random Error in Measurements

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standard deviation of calculated results

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uncertainty: overview

Measuring and Reporting Uncertainty

Uncertainty is part of analytical chemistry measurement. When repeated measurements are taken, random errors can cause the results to differ in small ways. These repeated values are compared with the estimated or expected value to judge how much uncertainty is present. Uncertainty is usually written after the estimated or expected value. It should also be reported with the correct number of significant figures, which are the digits needed to show a precise result. The possible variation from...

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...

propagation of uncertainty from systematic error

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

Confidence intervals show a range of values around a sample mean that may contain the true mean. They are written as a probability percentage. A 95% confidence interval means the statistician is 95% confident that the true mean lies within that range. The two ends of the range are called confidence limits. These limits are estimated from the sample mean, the standard deviation, and the statistical factor t, or t-score. The t-score depends on the number of degrees of freedom and the confidence...

significance testing: overview

Significance Testing in Analytical Chemistry

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identifying statistically significant differences: the f-test

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comparing experimental results: student's t-test

comparing-experimental-results-student-s-t-test

Student's t-Test for Significance Checks

Student's t-test is a statistical tool for checking whether two means are different in a meaningful way. It can compare a sample mean with a population mean, or it can compare the means from two data sets. The test uses the mean, standard deviation, and number of measurements, along with a chosen confidence interval. The test statistic is then compared with a table of critical values at that confidence level. If the test statistic is smaller than the critical value, the null hypothesis is...

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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Interpreting Correlation in Calibration Curves

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Using Regression to Measure Variable Links

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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

analytical chemistry

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Standard Deviation of Calculated Results

A plot of relative deviation from the mean and its frequency of occurrence appears as a Gaussian curve. This probability distribution curve of a ...

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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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