Average Atomic Mass Calculator

Input Parameters

Colorblind Mode

Calculation Results

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The Average Atomic Mass Calculator is a chemistry and materials science computation tool used to determine the weighted average atomic mass of an element by combining the individual masses of naturally occurring isotopes with their respective relative abundances expressed as fractions or percentages. It automates the weighted mean calculation required for accurate atomic weight determination, supporting applications in general chemistry, nuclear science, mass spectrometry, and isotopic analysis. As described in Chemistry: The Central Science by Theodore L. Brown and colleagues, the atomic mass of an element represents the weighted average of the masses of its naturally occurring isotopes. The calculator enables users to analyze isotope compositions by adding multiple isotope entries, converting abundance formats, normalizing data, and obtaining precise atomic mass values for applications such as student problem solving, laboratory analysis, isotope verification, and educational demonstrations. Its methodology follows the principles presented in Quantitative Chemical Analysis by Daniel C. Harris, which explains that atomic weights are derived by combining isotopic masses and abundances through weighted averaging.

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What is Average Atomic Mass Calculator?

An Average Atomic Mass Calculator is a sophisticated online tool that computes the weighted average atomic mass of an element based on its naturally occurring isotopes, incorporating their precise masses in atomic mass units (u) and relative abundances (either as fractions or percentages). This free average atomic mass calculator online free automates the weighted mean formula, essential for understanding elemental properties in chemistry, physics, and materials science. — A relevant chemistry reference is Chemistry: The Central Science by Theodore L. Brown and colleagues, which states, “The atomic mass of an element is the weighted average of the masses of the naturally occurring isotopes of that element.”

This isotope abundance calculator stands out for its dynamic interface, allowing users to add or remove isotope entries seamlessly while handling complex scenarios like percentage-to-fraction conversions and normalization. Whether you’re a student calculating the atomic mass of chlorine for exam prep, a researcher verifying isotopic compositions in mass spectrometry, or an educator demonstrating nuclear stability, this free online average atomic mass calculator delivers instant, accurate results. It excels in supporting high-CPC queries like “calculate average atomic mass from isotope data” or “best weighted average atomic mass tool for chemistry labs.” — The fundamental isotope abundance approach is also described in Quantitative Chemical Analysis by Daniel C. Harris, which explains, “Isotopic masses and abundances are combined through weighted averages to obtain the atomic weight of an element.”

What truly elevates this average atomic mass calculator is its array of special features: relevant visualizations including interactive bar charts for isotope contributions and pie charts for abundance distributions, a dedicated section for comments, analysis, and expert recommendations to provide contextual insights like stability implications or lab applications, step-by-step calculation breakdowns for educational depth, the ability for users to download or export results in CSV format for easy integration into spreadsheets or reports, and a groundbreaking colorblind view mode for improved accessibility—ensuring visually impaired users can navigate and interpret data with high-contrast patterns and shapes. These elements make it indispensable for topics like “isotope mass abundance calculator” and “elemental average atomic weight predictor.”

In industries from pharmaceuticals (isotope labeling) to environmental science (trace element analysis), having a reliable average atomic mass calculator from multiple isotopes is crucial for precise molar mass determinations and reaction stoichiometry. By processing real-world data inputs, it bridges theoretical concepts with practical computations, outperforming basic spreadsheets for “online tool to find average atomic mass of elements.”

Interpreting the Average Atomic Mass Results

The Average Atomic Mass Calculator determines the weighted average atomic mass of an element from the masses of its isotopes and their relative abundances. Unlike a simple arithmetic mean, the calculation gives each isotope a weight according to how frequently it occurs.

The fundamental calculation is:

\(\displaystyle \bar{m} = \sum_{i=1}^{n} m_i f_i\)

where:

(\bar{m}) = weighted average atomic mass
(m_i) = mass of isotope (i)
(f_i) = fractional abundance of isotope (i)

When abundances are entered as percentages:

\(\displaystyle \bar{m} = \frac{m_1P_1 + m_2P_2 + \cdots + m_nP_n}{100}\)

The resulting value, expressed in unified atomic mass units (u), represents the average mass of an atom of that element for the isotope composition represented by the supplied abundances. It is not necessarily equal to the mass of any individual isotope.

Normal or Expected Values

There is no universal “normal” average atomic mass. Each element has its own characteristic isotopic composition and therefore its own standard atomic weight.

A valid calculation should satisfy:

Every isotope mass must be physically meaningful.
Isotopic abundances should be non-negative.
Fractional abundances should sum to 1.000.
Percentage abundances should sum to 100%.
The calculated average must fall within the range bounded by the lightest and heaviest isotope masses included.

For example, if the included isotope masses range from 10 u to 12 u, the calculated weighted average cannot logically be below 10 u or above 12 u.

When naturally occurring isotope abundances are used correctly, the result should generally correspond closely to the accepted atomic weight reported for that element. Small differences can occur when the supplied isotope abundances differ from the reference composition.

High vs. Low Results

A higher average atomic mass means that the isotope composition is weighted more heavily toward the heavier isotopes.

For example, if an element has two isotopes:

Isotope A = 10 u
Isotope B = 12 u

increasing the abundance of the 12 u isotope will move the weighted average upward toward 12 u.

A lower average atomic mass means that the isotope composition is weighted more heavily toward the lighter isotopes.

Therefore, a high or low calculated value is not inherently good or bad. It primarily reflects the relative abundance distribution of the isotopes used in the calculation.

Practical Interpretation

Consider two isotopes:

Isotope 1: (10,u), abundance (75%)
Isotope 2: (12,u), abundance (25%)

The weighted average is:

\(\displaystyle \bar{m} = (10 \times 0.75) + (12 \times 0.25)\)

\(\displaystyle \bar{m} = 7.5 + 3.0 = 10.5\,\text{u}\)

The result of 10.5 u does not mean that the element contains atoms weighing exactly 10.5 u. Instead, it means that the isotope population represented by the stated abundances has a weighted mean atomic mass of 10.5 u.

If the abundance of the 12 u isotope increases, the average moves closer to 12 u. If its abundance decreases, the average moves closer to 10 u.

What the Result Indicates

The calculator’s output indicates:

The weighted mean mass of the isotope mixture.
How isotope abundance influences the element’s average atomic mass.
Whether the supplied isotope composition produces an expected atomic-weight value.
The quantitative relationship between isotope masses and their relative abundances.

This result is useful for:

Stoichiometric calculations in chemistry.
Determining molar masses.
Isotopic analysis.
Mass spectrometry interpretation.
Nuclear and materials science.
Verification of isotope-composition data.
Educational demonstrations of weighted averages.

The key point is that average atomic mass is a property of an isotope distribution, not the mass of a single atom.

When the Result Should Raise Concern

The result warrants checking when:

Isotopic abundances do not sum to 100% or 1.0: The weighted average will be incorrect unless the data are normalized appropriately.
The calculated average lies outside the minimum and maximum isotope masses entered: This indicates a mathematical or input error.
An isotope has a negative abundance: This is physically meaningless.
The calculated value differs substantially from an accepted atomic weight: Check whether the isotope abundances are appropriate for the same natural or reference material and whether all relevant isotopes have been included.
Abundances are entered using the wrong format: For example, entering 25 when the calculator expects a fraction of 0.25 can produce a major error.
The sample is isotopically enriched or depleted: The standard atomic weight of an element does not necessarily represent a deliberately modified isotope composition.
Too few isotopes are included: Omitting an isotope with meaningful abundance can shift the weighted average.
The input data come from a sample with unusual isotopic composition: Natural isotope abundances can vary by source, environment, geological history, or analytical context.

The Average Atomic Mass Calculator should therefore be interpreted as a weighted isotopic-composition calculation, not simply as an arithmetic averaging tool. Its output is meaningful only in relation to the isotope masses and abundances supplied. For standard chemistry applications, naturally occurring reference abundances should be used; for laboratory or mass-spectrometric work, the actual measured isotope composition may be more appropriate.

Factors Affecting the Average Atomic Mass Results

The Average Atomic Mass Calculator determines an element’s weighted average atomic mass from the mass of each isotope and its relative abundance. Because the result depends directly on both quantities, even a small change in isotope abundance or isotope mass can change the final weighted average. The main factors are:

Input Sensitivity: The result is especially sensitive to the relative abundance of the isotopes. An isotope with a high abundance contributes more strongly to the final average than a rare isotope. Therefore, changing an abundance from 50.0% to 50.5%, for example, can shift the calculated atomic mass. The isotope masses themselves also matter, although their effect is proportional to their abundance. If abundances are entered as percentages, fractions, or mixed incorrectly, the result can be substantially wrong.
Environmental Conditions: The calculator performs a mathematical weighted average, but the isotopic composition of a real sample can vary with its source and environment. Geological origin, biological processes, industrial processing, evaporation, chemical reactions, and other fractionation processes can alter isotope abundances. Consequently, a sample-specific isotopic composition may differ from the standard natural abundance values normally used to represent an element.
Material and Isotopic Properties: Different elements have different numbers of naturally occurring isotopes, and their isotope masses and abundances determine the final atomic weight. Some samples may also contain radioactive isotopes, enriched isotopes, depleted isotopes, or non-natural isotope mixtures. In such cases, using standard terrestrial abundances may not represent the actual material being analyzed.
Human Factors: Incorrect isotope identification, transposed mass values, inaccurate abundance entries, or failure to distinguish percentage from decimal fraction can change the result. For example, entering 25 instead of 0.25 is a 100-fold difference if the calculator expects fractional abundance. Users must also ensure that the isotope abundances correspond to the same material or sample.
Measurement Quality: The reliability of the calculated average depends on the quality of the isotope masses and abundance data. Experimental isotope-abundance measurements can contain analytical uncertainty, while rounded textbook values may produce slightly different results from high-precision reference data. Mass spectrometry measurements can also vary with calibration, instrument precision, sample preparation, and isotope fractionation during analysis.
Operating Assumptions: The calculator generally assumes that the supplied isotope abundances are appropriate for the sample and that they represent the complete isotope distribution being analyzed. If abundances do not total 100% (or 1.00), the calculator may need to normalize them, depending on its operating design. Different normalization rules, rounding conventions, selected isotope masses, or reference abundance datasets can therefore produce slightly different average atomic masses.

In summary, two users entering slightly different values may obtain different average atomic masses because the calculation is a weighted average, not a simple arithmetic mean. Small changes in isotope abundance can shift the result, particularly when the isotopes have substantially different masses. Differences can also arise from sample-specific isotope composition, measurement uncertainty, rounding, normalization, and reference-data choices. Thus, a calculator can produce a mathematically precise result from its inputs while still differing slightly from a textbook value or a laboratory measurement when the underlying isotope abundances or assumptions differ.

Precision and Reliability of the Calculated Results

The Average Atomic Mass Calculator provides mathematically reliable weighted-average atomic masses when the isotope masses and their relative abundances are accurate and correctly entered. The underlying weighted-mean calculation is deterministic, so identical isotope data will produce the same result. However, the scientific accuracy of the final atomic mass depends primarily on the quality, reference basis, and representativeness of the isotope-abundance data.

Expected precision:
The calculator can produce atomic mass values to several decimal places, which is appropriate for chemistry calculations and educational isotope analysis. The displayed precision should not be interpreted as greater scientific accuracy than the input data support. If isotope masses or abundances are reported with limited significant figures, the final atomic mass should be rounded accordingly. For naturally occurring elements, the result is meaningful only when the supplied abundances correspond to the intended natural isotopic composition.

Numerical approximations:
Numerical approximations can arise when isotope abundances are rounded, entered as percentages, or normalized because their total does not equal exactly 100%. The calculator may normalize the abundances before calculating the weighted mean, but this does not correct inaccurate isotope measurements. Small changes in the abundance of a relatively heavy or light isotope can shift the calculated average, particularly when isotopic abundances differ substantially. For elements with significant natural isotopic variation, a single abundance set may therefore represent only a particular reference composition rather than every natural sample.

Floating-point limitations:
The calculation involves decimal multiplication and addition of isotope masses and fractional abundances. Floating-point arithmetic can introduce extremely small rounding differences, especially when several isotopes with many decimal places are included. These differences are normally insignificant compared with uncertainty in isotope masses and abundance measurements and should not be mistaken for chemically meaningful variation.

Situations where manual verification is advisable:
Manual verification is advisable when the calculated value is being used in high-precision analytical chemistry, isotope-ratio studies, mass-spectrometric interpretation, or published research. Users should confirm that isotope abundances sum appropriately, the correct isotope masses are being used, percentage and fractional abundance formats have not been confused, and the isotope data refer to the same reference composition. Independent calculation is particularly useful when the result differs unexpectedly from a published standard atomic weight.

When laboratory measurements remain necessary:
Laboratory measurements remain necessary when the actual isotopic composition of a specific sample must be determined rather than its expected or reference atomic mass. Techniques such as mass spectrometry and isotope-ratio mass spectrometry can directly measure isotope abundances and reveal natural or process-induced isotopic variations. Certified reference materials and authoritative atomic-weight data should be used when traceable high-accuracy values are required. Thus, the calculator accurately performs the weighted-average mathematics described in Chemistry: The Central Science by Theodore L. Brown and colleagues and Quantitative Chemical Analysis by Daniel C. Harris, but it cannot independently validate whether the supplied isotope masses and abundances accurately represent the physical sample.

Making Sense of Unexpected Average Atomic Mass Results

Unexpected results from an Average Atomic Mass Calculator generally arise from incorrect isotope masses, improperly entered abundances, inconsistent percentage-to-fraction conversions, or failure to normalize isotope abundances correctly. Because average atomic mass is a weighted mean, the result depends directly on both the mass of each isotope and its relative abundance.

Why is the result negative?
A negative average atomic mass is physically impossible. Atomic masses are positive quantities, so a negative result usually indicates an invalid isotope mass, negative abundance, incorrect sign in an input field, or a calculation/data-entry error. Relative abundances must also be non-negative. If a negative value appears, the isotope entries and units should be checked before interpreting the result.
Why is it zero?
An average atomic mass of zero normally indicates that no valid isotope data have been entered, isotope abundances are zero, or the calculation has been supplied with invalid or incomplete inputs. A real element cannot have an average atomic mass of zero because every isotope has positive mass. A zero result should therefore be treated as an input or calculation issue, not as a physically meaningful atomic weight.
Why is it extremely large?
An unusually large average atomic mass usually results from entering an incorrect isotope mass, confusing mass number (A) with atomic mass, using the wrong units, or entering an abundance incorrectly. For example, an isotope mass of 235 u is reasonable for a nuclide near mass number 235, whereas accidentally entering 235,000 u would distort the weighted average dramatically. If the abundances are properly normalized, the calculated average must lie between the smallest and largest isotope masses included in the calculation.
Why does changing one value have a dramatic effect?
Average atomic mass is calculated as a weighted sum:
Average Atomic Mass = Σ(mᵢ × fᵢ)
where mᵢ is the mass of isotope i and fᵢ is its fractional abundance. Consequently, changing the abundance of a dominant isotope can substantially shift the final atomic mass, whereas changing the abundance of a very rare isotope usually has a much smaller effect. A large difference between isotope masses also increases sensitivity: transferring even a small fraction of abundance from a lighter isotope to a substantially heavier isotope can noticeably increase the weighted average.

Before interpreting an unexpected result, verify the isotope masses, abundance percentages or fractions, isotope identities, units, and normalization of abundances. When abundances are entered as percentages, they should collectively represent the intended composition—typically totaling 100% for a complete isotopic distribution. The calculated value should also fall within the mass range of the isotopes included. Finally, remember that an average atomic mass based on user-supplied isotope abundances represents that particular isotopic composition; it may differ from a standard periodic-table atomic weight when the sample has a non-natural, altered, or isotope-enriched composition.

Why this Average Atomic Mass Calculator is Different than its Competitors?

Handles Real Isotope Data Instead of Fixed Atomic Weights
Calculates atomic mass directly from individual isotope masses and natural abundances, matching the actual scientific definition of average atomic weight.
Uses Accurate Weighted Average Methodology
Applies the same mathematical approach used in chemistry and analytical science to combine isotope contributions into a single representative atomic mass.
Supports Flexible Isotope Inputs
Allows users to add multiple isotopes, adjust abundances, and analyze elements with simple or complex isotope distributions.
Automatically Manages Abundance Conversions
Converts percentage abundances into fractional values and normalizes isotope contributions to prevent common calculation mistakes.
Provides Transparent Scientific Calculations
Shows how each isotope contributes to the final atomic mass, making results easier to verify and understand.
Useful Across Multiple Scientific Fields
Serves chemistry students, laboratory analysts, researchers, and educators working with isotope-based calculations.
Improves Accuracy in Downstream Calculations
Provides reliable atomic mass values for molar mass determination, chemical equations, and quantitative laboratory analysis.
Combines Simplicity with Research-Level Capability
Offers an easy interface for beginners while supporting the detailed isotope analysis requirements of advanced scientific applications.

How does this Average Atomic Mass Calculator work?

The average atomic mass calculator’s purpose is to empower users to derive the standard atomic weight of any element from its isotopic profile, facilitating everything from classroom exercises to advanced analytical chemistry. It processes multiple isotope data points dynamically, ensuring robust handling of edge cases like varying abundance formats.

Key inputs across the interface include:

Mass (u) for Each Isotope: Numeric value in atomic mass units (e.g., 34.9689 for Cl-35), with at least one required per entry.
Abundance for Each Isotope: Value as a decimal fraction (0-1) or percentage (0-100), auto-detected and normalized by the tool.
Dynamic Controls: Buttons to add/remove isotope rows (minimum 1, supports up to dozens), colorblind toggle for accessibility, and export options.
Batch Features: Implicit support via CSV export/import for multi-element workflows.

These make the tool ideal for “free average atomic mass calculator with isotope inputs.”

Where to use this Average Atomic Mass Calculator?

Chemistry Education and Laboratory Learning
Calculate the average atomic mass of elements from isotope masses and abundances while learning the relationship between isotopic composition and periodic table atomic weights.
Mass Spectrometry Data Analysis
Verify experimentally measured isotope distributions, interpret mass spectra, and compare observed isotope abundances with theoretical average atomic masses.
Nuclear Chemistry and Isotope Studies
Analyze elements with multiple naturally occurring isotopes and understand how variations in isotope abundance influence reported atomic weights.
Analytical Chemistry Applications
Support calculations involving elemental composition, sample characterization, and quantitative analysis where accurate atomic weights are required.
Materials Science and Metallurgy Research
Evaluate isotopic contributions in materials, alloys, and engineered substances where atomic mass affects molecular calculations and material properties.
Chemical Formula and Stoichiometry Calculations
Provide accurate atomic mass values for molar mass calculations, reaction stoichiometry, and quantitative chemical analysis.
Academic Assignments and Exam Preparation
Help students solve isotope abundance problems, weighted average calculations, and periodic table-based chemistry exercises.
Scientific Research and Data Validation
Cross-check isotope datasets and verify calculated atomic weights against published reference values.

Average Atomic Mass Formula

The average atomic mass calculator employs the weighted average formula:

\(M_{avg} = \sum_{i=1}^{n} (m_i \times f_i)\)

Where f_i is the normalized fractional abundance: \(f_i = \frac{a_i}{\sum a_j}\) (a_i = raw abundance)

For percentage inputs:

\(f_i = \frac{a_i / 100}{\sum (a_j / 100)}\)

Where:

M_avg = average atomic mass (u)
m_i = mass of isotope i (u)
f_i = fractional abundance of isotope i (unitless)
a_i = raw abundance of isotope i (fraction or %)
n = number of isotopes

The tool computes this exactly, with high-precision rounding.

How to Calculate Average Atomic Mass (Step-by-Step)

Calculating average atomic mass is intuitive yet precise with this tool. Here’s the complete step-by-step process:

Prepare Inputs: Open the free average atomic mass calculator. Add isotope rows via the “+” button—start with at least one for elements like hydrogen.
Enter Data: For each row, input mass (e.g., 1.0078 u for H-1) and abundance (e.g., 0.99985 or 99.985%). The tool auto-detects formats.
Validate: Use the built-in checks; errors highlight invalid entries like negative masses.
Compute: Click “Calculate Average Mass” (or Ctrl+Enter). Watch the loader for seamless processing.
Review Breakdown: Examine the step-by-step section, e.g., “Isotope 1: 34.9689 × 0.7576 = 26.496 u.”
Analyze Insights: Dive into the dedicated comments, analysis, and recommendations—e.g., “Dominant isotope drives 75% of mass; ideal for stoichiometry.”
Visualize: Interact with bar charts (contributions) and pie charts (abundances) for intuitive understanding.
Export and Iterate: Download CSV for reports. Toggle colorblind view for accessibility, then tweak inputs for “what-if” scenarios like rare isotopes.

This workflow supports searches for “step-by-step average atomic mass calculation online.”

Examples

Example 1: Chlorine (Common Lab Element) Inputs: Isotope 1: Mass=34.9689 u, Abundance=75.76%; Isotope 2: Mass=36.9659 u, Abundance=24.24%. Steps: Normalize to fractions (0.7576, 0.2424); Weighted sum=35.453 u. Results: Average=35.45 u; Analysis: Matches periodic table; Recommendation: Use for HCl molar mass in titrations.

Example 2: Copper (Industrial Application) Inputs: Isotope 1: Mass=62.9296 u, Abundance=69.17%; Isotope 2: Mass=64.9278 u, Abundance=30.83%. Steps: Fractions (0.6917, 0.3083); Contributions yield 63.546 u. Results: Average=63.55 u; Comments: Reflects natural variability; Export CSV for alloy design simulations.

Average Atomic Mass Categories / Normal Range

Atomic masses categorize elements by mass ranges, influencing reactivity and applications. Reference table below:

Atomic Mass Range (u)	Category	Abundance Type	Examples	Typical Use Cases
1–20	Light Elements	High (near 100%)	H (1.008), He (4.003)	Fusion, astrophysics
20–50	Medium-Weight	Mixed isotopes	C (12.011), O (15.999)	Organic chemistry
50–100	Transition Metals	Balanced	Fe (55.845), Cu (63.546)	Metallurgy, catalysis
100–200	Heavy Elements	Dominant isotope	Ag (107.868), Pb (207.2)	Electronics, batteries
>200	Superheavy	Radioactive	U (238.029), Th (232.038)	Nuclear energy, medicine

Normal range for stable elements: 1–238 u. Average masses are abundance-weighted.

Limitations

This average atomic mass calculator assumes ideal isotopic data and ignores relativistic or quantum effects in ultra-precise measurements. It normalizes abundances but may slightly deviate for highly skewed distributions (>99% one isotope). Inputs are limited to numeric values; non-standard units (e.g., kg/mol) require conversion. Colorblind mode enhances contrast but doesn’t alter computational accuracy. For polyisotopic elements with >10 variants, manual entry is needed—batch CSV helps but is export-only.

Disclaimer

This average atomic mass calculator is designed exclusively for educational, research, and informational use. Outputs are computational approximations based on user-provided data and should not replace certified laboratory analyses, official periodic table values, or expert chemical engineering consultations. Users bear full responsibility for input accuracy and result applications; always verify with sources like IUPAC for professional or regulatory purposes. No warranties on precision for critical scenarios like drug synthesis or nuclear safety.

FAQs — Average Atomic Mass Calculator

Why is the average atomic mass on the periodic table usually not a whole number, and how is it different from the mass number of an isotope?

The average atomic mass is a weighted statistical value calculated from all naturally occurring isotopes of an element, while the mass number represents the total number of protons and neutrons in one specific isotope. Because most elements exist as mixtures of isotopes with different masses and abundances, their average atomic masses commonly appear as decimal values rather than integers.

How can an element have a different average atomic mass in different samples or geological sources?

Average atomic mass is based on isotope abundance, which is not always perfectly constant in nature. Processes such as radioactive decay, isotope fractionation, evaporation, geological history, and biological activity can alter isotopic ratios in certain environments. Therefore, some elements may exhibit sample-dependent atomic weights rather than a single universal value.

Why is isotope abundance more important than simply having many isotopes when calculating average atomic mass?

The contribution of each isotope depends on both its mass and its relative abundance. A rare isotope with a large atomic mass may have minimal influence on the final average, whereas a highly abundant isotope strongly dominates the calculation. Weighted averaging ensures that common isotopes contribute proportionally more to the final atomic mass value.

Can the average atomic mass calculator produce an accurate value if isotope abundances are incomplete, uncertain, or do not sum to 100%?

Yes, but only after proper data handling. If abundances are incomplete, the values must be normalized or corrected before calculation. Uncertain or inaccurate isotope percentages can directly affect the final atomic mass, meaning the reliability of the result depends strongly on the quality of the isotope composition data provided.

Why do some elements have no stable isotopes yet still have an accepted atomic mass value?

For elements composed entirely of radioactive isotopes, the reported atomic weight may be based on the mass of a representative isotope rather than a natural isotopic average. In such exceptional cases, conventional weighted averaging cannot be applied because no stable naturally occurring isotope mixture exists with a fixed abundance distribution.