Enter the count of molecules in each size fraction.
Enter the molecular weight of molecules in each size fraction (in Da or g/mol).
Number Average Molecular Weight (Mn)
—
Total Moles (ΣNi)—
Total Weight (ΣNiMi)—
PDI (Mw/Mn)—
Formula: Mn = Σ(Ni * Mi) / ΣNi
Where Ni is the number of molecules in a fraction and Mi is the molecular weight of that fraction.
Molecular Weight Distribution
Understanding Number Average Molecular Weight (Mn)
What is Number Average Molecular Weight (Mn)?
The Number Average Molecular Weight, commonly denoted as Mn, is a fundamental property in polymer science and other fields dealing with polydisperse materials. It represents the arithmetic mean of the molecular weights of all the molecules in a sample. Imagine a bag containing different sized marbles; the Mn would be the average weight of a single marble if you considered the total weight of all marbles divided by the total count of marbles. In polymer chemistry, Mn is crucial because it directly relates to the number of molecules present. Properties that depend on the *number* of particles, such as osmotic pressure, are directly related to Mn. For instance, a polymer sample with a higher Mn indicates that, on average, the polymer chains are longer, which can influence its mechanical strength, viscosity, and processing characteristics.
Who should use it? Chemists, material scientists, chemical engineers, and researchers working with polymers, macromolecules, colloids, or any mixture of substances with varying molecular sizes will find Mn calculation essential. It's particularly important when characterizing synthesized polymers, understanding their formation, or predicting their bulk properties. Misconceptions often arise about Mn being the *only* measure of molecular weight; however, it's just one of several averages (like weight average molecular weight, Mw), each providing different insights into the material's characteristics.
Number Average Molecular Weight (Mn) Formula and Mathematical Explanation
Calculating the Number Average Molecular Weight (Mn) involves summing the products of the number of molecules in each fraction and their respective molecular weights, and then dividing by the total number of molecules. The formula is derived from the definition of an average: the sum of values divided by the count of values.
The core formula for Mn is:
Mn = Σ(Ni × Mi) / ΣNi
Let's break down the components:
Ni: Represents the number of molecules (or moles) in a specific molecular weight fraction (i).
Mi: Represents the average molecular weight of the molecules in that specific fraction (i).
Σ: This is the summation symbol, indicating that we sum up the values for all the different molecular weight fractions present in the sample.
In essence, the numerator (Σ(Ni × Mi)) calculates the total weight of all molecules in the sample by considering how many molecules of each weight exist. The denominator (ΣNi) calculates the total count of molecules in the sample. Dividing the total weight by the total count gives us the average weight per molecule, which is the Number Average Molecular Weight (Mn).
Variables Table
Variable
Meaning
Unit
Typical Range
Ni
Number of molecules (or moles) in fraction i
Count (or mol)
Non-negative integer (or float)
Mi
Average molecular weight of fraction i
Daltons (Da) or g/mol
Can range from tens to millions, depending on the substance
Mn
Number Average Molecular Weight
Daltons (Da) or g/mol
Typically positive, reflects the average size
ΣNi
Total number of molecules (or moles)
Count (or mol)
Sum of all Ni values
Σ(Ni × Mi)
Total weight of all molecules
Daltons (Da) or g/mol
Sum of products of Ni and Mi for all fractions
Practical Examples (Real-World Use Cases)
Understanding the Number Average Molecular Weight is key in many practical applications. Here are a couple of examples:
Example 1: Polystyrene Synthesis Characterization
A research lab synthesizes a batch of polystyrene. They use Gel Permeation Chromatography (GPC) to separate the polymer chains into different size fractions and determine the number of chains and their average molecular weight in each fraction.
Interpretation: The number average molecular weight of this polystyrene batch is approximately 33,333 Da. This value is important for predicting properties like its glass transition temperature and solubility. It tells us that, on average, there are about 33,333 units of weight per polymer chain.
Example 2: Polyethylene Glycol (PEG) Mixture
A pharmaceutical company is preparing a formulation that requires a specific average molecular weight of Polyethylene Glycol (PEG) for its drug delivery properties. They mix two PEG samples: PEG 4000 (average MW ~4000 Da) and PEG 20000 (average MW ~20000 Da). They need to determine the ratio to achieve a target Mn. Let's assume they mix 2 moles of PEG 4000 with 1 mole of PEG 20000.
Inputs:
Fraction 1 (PEG 4000): Ni = 2 moles, Mi = 4,000 Da
Fraction 2 (PEG 20000): Ni = 1 mole, Mi = 20,000 Da
Calculation:
Total Moles (ΣNi) = 2 + 1 = 3 moles
Total Weight (ΣNiMi) = (2 × 4,000) + (1 × 20,000) = 8,000 + 20,000 = 28,000 Da
Mn = 28,000 Da / 3 moles = 9,333.33 Da
Interpretation: The resulting mixture has a Number Average Molecular Weight of approximately 9,333 Da. This Mn value influences the viscosity and permeability characteristics of the drug formulation, directly impacting drug release rates. This calculation helps verify if the mixture meets the required specifications.
How to Use This Number Average Molecular Weight Calculator
Our Number Average Molecular Weight calculator is designed for simplicity and accuracy. Follow these steps to get your results:
Enter Initial Fraction: The calculator starts with one row. Input the Number of Molecules (Ni) for the first fraction in the "Ni" field and the corresponding Molecular Weight (Mi) in the "Mi" field. Ensure units are consistent (e.g., Da or g/mol).
Add More Fractions: If your sample consists of multiple molecular weight distributions, click the "Add Molecule Fraction" button. New input fields for Ni and Mi will appear. Enter the values for each additional fraction. You can add as many fractions as needed.
Monitor Results: As you enter valid numbers, the Total Moles (ΣNi), Total Weight (ΣNiMi), and the primary Number Average Molecular Weight (Mn) will update automatically in real-time below the input fields. The PDI (Polydispersity Index) is also shown, though it requires the Weight Average Molecular Weight (Mw) for a complete calculation.
Understand the Chart: The dynamic chart visually represents the distribution of your polymer sample based on the fractions you entered. It shows how the number of molecules (Ni) varies across different molecular weights (Mi).
Reset: If you need to start over or clear the current inputs, click the "Reset" button. This will clear all input fields and reset the results.
Copy Results: Use the "Copy Results" button to copy the calculated Mn, intermediate values, and key assumptions to your clipboard for easy pasting into reports or notes.
Reading Results: The main result is your calculated Mn. The intermediate values provide context: ΣNi is the total count of molecules, and ΣNiMi is the total mass represented by these molecules. PDI (Mw/Mn) is a measure of the broadness of the molecular weight distribution. A PDI of 1 indicates a perfectly monodisperse sample (all chains are the same length), while higher values indicate broader distributions.
Decision Making: Use the calculated Mn to compare different batches of a material, predict physical properties, ensure consistency in manufacturing processes, or determine the suitability of a polymer for a specific application. For instance, a higher Mn might suggest better mechanical properties but potentially more difficult processing due to higher viscosity.
Key Factors That Affect Number Average Molecular Weight Results
Several factors influence the resulting Number Average Molecular Weight (Mn) of a polymer or polydisperse substance. Understanding these is crucial for controlling and predicting material properties:
Initiation, Propagation, and Termination Rates: In polymerization reactions, the relative rates of these fundamental steps directly impact the chain lengths produced. Faster termination relative to propagation leads to shorter chains and a lower Mn. Conversely, slow termination can result in longer chains and a higher Mn.
Monomer Concentration: Higher initial monomer concentrations generally lead to more chain growth events occurring simultaneously. This can influence the number of chains formed and their average length, affecting Mn.
Catalyst Type and Concentration: The catalyst plays a critical role in initiating and controlling polymerization. Different catalysts have varying efficiencies and mechanisms, leading to different chain lengths and Mn values. Higher catalyst concentrations can sometimes lead to more chains but potentially shorter ones, depending on the mechanism.
Chain Transfer Agents: These additives are used to deliberately control molecular weight. They interrupt growing polymer chains and initiate new ones, effectively shortening the average chain length and thus reducing Mn.
Reaction Temperature: Temperature affects reaction kinetics. Higher temperatures often increase reaction rates, including termination and chain transfer, which can lead to shorter chains and a lower Mn. However, the specific effect depends heavily on the polymerization mechanism.
Reaction Time: While Mn is an average, extended reaction times can allow for continued chain growth or potential degradation. In some systems, prolonged reaction times might slightly increase Mn, while in others, side reactions or degradation could eventually lower it.
Presence of Impurities: Certain impurities can act as inhibitors or chain transfer agents, significantly affecting the polymerization process and leading to lower Mn values.
Frequently Asked Questions (FAQ)
What is the difference between Mn and Mw?
Mn (Number Average Molecular Weight) is the total weight of all molecules divided by the total number of molecules. Mw (Weight Average Molecular Weight) gives more weight to larger molecules. Mw is always greater than or equal to Mn. Mn is sensitive to the presence of small molecules, while Mw is sensitive to large molecules.
What does a low PDI indicate?
A low PDI (Polydispersity Index), ideally close to 1.0, indicates a narrow molecular weight distribution. This means most polymer chains in the sample have similar lengths. This is often desirable for applications requiring predictable material properties.
Can Mn be used alone to characterize a polymer?
No, Mn provides only one aspect of molecular weight characterization. Mw and PDI are also crucial for a complete understanding of a polymer's properties and behavior. Different applications may prioritize different averages.
Are the units for Ni and Mi important?
Yes, consistency is key. While Ni represents a count, Mi is typically in Daltons (Da) or grams per mole (g/mol). Ensure you use the same unit for Mi across all fractions for accurate calculation. The final Mn will have the same unit as Mi.
What is the typical range for molecular weight?
Molecular weights can vary enormously. Small molecules like water have MWs around 18 Da. Polymers can range from a few thousand Daltons (e.g., short oligomers) to millions of Daltons (e.g., very high molecular weight polymers like ultra-high-molecular-weight polyethylene).
How does Mn affect polymer solubility?
Generally, lower Mn polymers tend to be more soluble because they have a higher concentration of chain ends relative to their total mass, which can interact more favorably with solvents. Higher Mn polymers might be less soluble or require more aggressive solvents.
Can this calculator handle mixtures of different substances?
Yes, as long as you know the number of molecules (or moles) and their average molecular weights for each component in the mixture, the calculator can determine the overall number average molecular weight.
What happens if I enter zero for Ni or Mi?
Entering zero for Ni (number of molecules) means that fraction contributes nothing to the total count or total weight. Entering zero for Mi (molecular weight) is physically unrealistic and will lead to an incorrect calculation. The calculator includes basic validation to prevent negative inputs, but extreme valid inputs should be carefully considered for their physical meaning.
var moleculeFractions = [];
var chartInstance = null;
function addInputRow() {
var calculatorDiv = document.getElementById('calculator');
var currentRowCount = document.querySelectorAll('.molecule-fraction-input').length;
var newRowId = 'fraction-' + (currentRowCount + 1);
var newRowDiv = document.createElement('div');
newRowDiv.className = 'input-group molecule-fraction-input';
newRowDiv.id = newRowId;
newRowDiv.innerHTML = `
`;
calculatorDiv.insertBefore(newRowDiv, calculatorDiv.querySelector('.button-group'));
// Re-attach listeners or ensure initial calculation handles new inputs
calculateMn();
}
function removeInputRow(rowId) {
var rowToRemove = document.getElementById(rowId);
if (rowToRemove) {
rowToRemove.remove();
// Re-index labels if needed (optional for simplicity here)
updateFractionLabels();
calculateMn();
}
}
function updateFractionLabels() {
var rows = document.querySelectorAll('.molecule-fraction-input');
for (var i = 0; i < rows.length; i++) {
var label = rows[i].querySelector('label');
if (label) {
label.textContent = `Fraction ${i + 1}`;
}
}
}
function resetCalculator() {
document.getElementById('numChains').value = '';
document.getElementById('molWeight').value = '';
document.getElementById('numChainsError').style.display = 'none';
document.getElementById('molWeightError').style.display = 'none';
var existingRows = document.querySelectorAll('.molecule-fraction-input');
existingRows.forEach(function(row) {
row.remove();
});
document.getElementById('mainResultValue').textContent = '–';
document.getElementById('totalMoles').textContent = '–';
document.getElementById('totalWeight').textContent = '–';
document.getElementById('pdi').textContent = '–';
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
// Re-initialize canvas context if needed for a fresh chart
initChart();
}
function validateInput(element, errorElement, min, max) {
var value = parseFloat(element.value);
var errorDiv = document.getElementById(errorElement);
errorDiv.style.display = 'none'; // Hide error by default
if (isNaN(value)) {
if (element.value === '') {
// Empty is okay for reset/initial state, but not for calculation
return true; // Allow empty for now, calculateMn will handle logic
} else {
errorDiv.textContent = 'Please enter a valid number.';
errorDiv.style.display = 'block';
return false;
}
}
if (value max) {
errorDiv.textContent = `Value cannot exceed ${max}.`;
errorDiv.style.display = 'block';
return false;
}
return true; // Input is valid
}
function calculateMn() {
var totalMoles = 0;
var totalWeight = 0;
var niValues = [];
var miValues = [];
var labels = [];
var valid = true;
// Main inputs
var numChainsInput = document.getElementById('numChains');
var molWeightInput = document.getElementById('molWeight');
var numChainsValid = validateInput(numChainsInput, 'numChainsError', 0);
var molWeightValid = validateInput(molWeightInput, 'molWeightError', 0);
var initialNi = parseFloat(numChainsInput.value);
var initialMi = parseFloat(molWeightInput.value);
if (!isNaN(initialNi) && initialNi > 0 && !isNaN(initialMi) && initialMi > 0) {
totalMoles += initialNi;
totalWeight += initialNi * initialMi;
niValues.push(initialNi);
miValues.push(initialMi);
labels.push(`MW: ${initialMi.toLocaleString()}`);
}
// Additional rows
var fractionRows = document.querySelectorAll('.molecule-fraction-input');
fractionRows.forEach(function(row, index) {
var niInput = row.querySelector('.fraction-ni');
var miInput = row.querySelector('.fraction-mi');
var niErrorSpan = row.querySelector('.ni-error');
var miErrorSpan = row.querySelector('.mi-error');
var ni = parseFloat(niInput.value);
var mi = parseFloat(miInput.value);
niErrorSpan.textContent = ";
miErrorSpan.textContent = ";
var currentNiValid = true;
var currentMiValid = true;
if (isNaN(ni)) {
if (niInput.value !== ") {
niErrorSpan.textContent = 'Invalid number';
currentNiValid = false;
}
} else if (ni < 0) {
niErrorSpan.textContent = 'Cannot be negative';
currentNiValid = false;
}
if (isNaN(mi)) {
if (miInput.value !== '') {
miErrorSpan.textContent = 'Invalid number';
currentMiValid = false;
}
} else if (mi 0′;
currentMiValid = false;
}
if (currentNiValid && currentMiValid && !isNaN(ni) && ni > 0 && !isNaN(mi) && mi > 0) {
totalMoles += ni;
totalWeight += ni * mi;
niValues.push(ni);
miValues.push(mi);
labels.push(`MW: ${mi.toLocaleString()}`);
} else if (niInput.value !== " || miInput.value !== ") {
// Mark as invalid if there's any input but it's not fully valid
valid = false;
}
});
var mainResultValue = '–';
var pdiValue = '–';
if (totalMoles > 0 && totalWeight > 0) {
var mn = totalWeight / totalMoles;
mainResultValue = mn.toLocaleString(undefined, { minimumFractionDigits: 2, maximumFractionDigits: 2 });
}
// PDI calculation requires Mw, which is not calculated here.
// Placeholder or a note that PDI needs Mw.
// For simplicity, we'll just display '–' or a message.
// If Mw were available, PDI = Mw / Mn
document.getElementById('mainResultValue').textContent = mainResultValue;
document.getElementById('totalMoles').textContent = totalMoles > 0 ? totalMoles.toLocaleString() : '–';
document.getElementById('totalWeight').textContent = totalWeight > 0 ? totalWeight.toLocaleString() : '–';
document.getElementById('pdi').textContent = pdiValue;
updateChart(niValues, miValues, labels);
}
function initChart() {
var ctx = document.getElementById('molecularWeightChart').getContext('2d');
// Destroy previous chart instance if it exists
if (chartInstance) {
chartInstance.destroy();
}
chartInstance = new Chart(ctx, {
type: 'bar',
data: {
labels: [], // Will be updated by updateChart
datasets: [{
label: 'Number of Molecules (Ni)',
data: [], // Will be updated by updateChart
backgroundColor: 'rgba(0, 74, 153, 0.6)', // Primary color
borderColor: 'rgba(0, 74, 153, 1)',
borderWidth: 1
},
{
label: 'Molecular Weight (Mi)',
data: [], // This dataset represents Mi values for reference, not directly plotted as a separate series in a typical bar chart for distribution.
// A line chart overlay or separate plot might be better, but for simplicity with bar, we use it conceptually.
// Or, we can plot Ni vs Mi on the x-axis if Mi range is reasonable.
// Let's make this a line plot against the same x-axis labels for demonstration.
type: 'line', // Use line type for Mi
fill: false,
borderColor: 'rgba(40, 167, 69, 1)', // Success color
tension: 0.1,
yAxisID: 'y-axis-mi' // Assign to a secondary y-axis if needed
}]
},
options: {
responsive: true,
maintainAspectRatio: true,
scales: {
x: {
title: {
display: true,
text: 'Molecular Weight Fraction (Mi)'
}
},
y: {
title: {
display: true,
text: 'Number of Molecules (Ni)'
},
beginAtZero: true
},
y_axis_mi: { // Define the secondary y-axis
type: 'linear',
position: 'right',
title: {
display: true,
text: 'Molecular Weight (Mi)'
},
grid: {
drawOnChartArea: false, // only want the grid lines for one axis to show up
},
beginAtZero: true
}
},
plugins: {
legend: {
position: 'top',
},
title: {
display: true,
text: 'Molecular Weight Distribution'
}
}
}
});
}
function updateChart(niValues, miValues, labels) {
if (!chartInstance) {
initChart();
}
var datasetNi = {
label: 'Number of Molecules (Ni)',
data: niValues,
backgroundColor: 'rgba(0, 74, 153, 0.6)',
borderColor: 'rgba(0, 74, 153, 1)',
borderWidth: 1
};
// For the Mi line, we need to map it to the same x-axis points.
// If using Mi as labels, this might be tricky. Revisit chart data structure.
// Let's simplify: Use Mi as labels on X-axis and plot Ni as bars and perhaps another metric if available.
// For this example, let's just plot Ni as bars and perhaps a line representing the Mi value at each bar.
chartInstance.data.labels = labels.map(function(label) { return label.replace('MW: ',"); }); // Use Mi values as labels, clean them up
chartInstance.data.datasets[0].data = niValues; // Ni as bars
// We can't easily plot Mi as a separate series against Mi labels.
// A common approach is to use Mi on the X-axis and Ni on the Y-axis.
// If we want to show Mi values, we could use a scatter plot or a line chart with Mi on X.
// For a bar chart, let's simulate the line for Mi values, which might look odd if Mi values are very different.
// Alternatively, we could show Ni distribution and then list relevant Mi values in the labels.
// Let's refine: use Mi values on the X-axis, Ni as bars, and perhaps a conceptual line for Mi value itself at that point.
// This requires careful consideration of axis scaling.
// For now, let's focus on displaying Ni distribution against Mi labels.
// The second dataset is conceptual for now – maybe a line plotting Ni trend or Mi value trend.
// Let's plot Mi values using the secondary axis.
chartInstance.data.datasets[1].data = miValues; // Mi values for the line plot
chartInstance.data.datasets[1].type = 'line'; // Ensure it's a line
chartInstance.data.datasets[1].yAxisID = 'y-axis-mi'; // Use the secondary y-axis
chartInstance.options.scales.x.title.text = 'Molecular Weight (Mi) in Da';
chartInstance.options.scales.y.title.text = 'Number of Molecules (Ni)';
chartInstance.options.scales.y_axis_mi.title.text = 'Molecular Weight (Mi) in Da';
chartInstance.update();
}
function copyResults() {
var mainResult = document.getElementById('mainResultValue').textContent;
var totalMoles = document.getElementById('totalMoles').textContent;
var totalWeight = document.getElementById('totalWeight').textContent;
var pdi = document.getElementById('pdi').textContent;
var assumptions = "Key Assumptions:\n";
var inputs = document.querySelectorAll('#calculator .input-group');
inputs.forEach(function(group, index) {
if (index === 0) { // Main input
var ni = document.getElementById('numChains').value;
var mi = document.getElementById('molWeight').value;
if (ni && mi) {
assumptions += `- Fraction 1: Ni = ${ni}, Mi = ${mi} Da\n`;
}
} else { // Additional rows
var niInput = group.querySelector('.fraction-ni');
var miInput = group.querySelector('.fraction-mi');
var ni = niInput ? niInput.value : ";
var mi = miInput ? miInput.value : ";
if (ni && mi) {
assumptions += `- Fraction ${index}: Ni = ${ni}, Mi = ${mi} Da\n`;
}
}
});
var resultText = `Number Average Molecular Weight (Mn) Calculator Results:\n\n`;
resultText += `Number Average Molecular Weight (Mn): ${mainResult}\n`;
resultText += `Total Moles (ΣNi): ${totalMoles}\n`;
resultText += `Total Weight (ΣNiMi): ${totalWeight}\n`;
resultText += `Polydispersity Index (PDI): ${pdi}\n\n`;
resultText += assumptions;
navigator.clipboard.writeText(resultText).then(function() {
// Optionally provide user feedback, e.g., change button text temporarily
var copyButton = event.target;
var originalText = copyButton.textContent;
copyButton.textContent = 'Copied!';
setTimeout(function() {
copyButton.textContent = originalText;
}, 2000);
}).catch(function(err) {
console.error('Failed to copy results: ', err);
// Handle error, maybe show a message to the user
});
}
function toggleFaq(element) {
var answer = element.nextElementSibling;
if (answer.style.display === 'block') {
answer.style.display = 'none';
} else {
answer.style.display = 'block';
}
}
// Initial setup when the page loads
document.addEventListener('DOMContentLoaded', function() {
initChart();
// Add a default row for demonstration, or keep it empty
// addInputRow();
calculateMn(); // Calculate on load with initial values if any
});