Calculate The Polydispersity Index for The Following Molecular Distribution

The polydispersity index (PDI) is a measure of the width of a molecular weight distribution in a sample. It provides information about the homogeneity of the sample and is commonly used in polymer science, colloid chemistry, and materials science.

What is the polydispersity index?

The polydispersity index (PDI) is a dimensionless number that quantifies the width of a molecular weight distribution. It is calculated as the ratio of the weight-average molecular weight (Mw) to the number-average molecular weight (Mn).

The polydispersity index is also known as the polydispersity ratio or the dispersity ratio.

PDI values range from 1 to infinity. A PDI of 1 indicates a perfectly monodisperse sample where all molecules have the same molecular weight. Higher PDI values indicate broader molecular weight distributions.

Applications of the polydispersity index

Polymer characterization
Colloid and nanoparticle analysis
Quality control in chemical synthesis
Material science research
Biotechnology applications

How to calculate the polydispersity index

The polydispersity index is calculated using the following formula:

PDI = Mw / Mn

Where:

PDI = Polydispersity index
Mw = Weight-average molecular weight
Mn = Number-average molecular weight

Calculating Mw and Mn

The weight-average and number-average molecular weights are calculated from the molecular weight distribution using the following formulas:

Mw = Σ (Mi × Wi) / Σ Wi Mn = Σ (Ni × Mi) / Σ Ni

Where:

Mi = Molecular weight of species i
Wi = Weight fraction of species i
Ni = Number of molecules of species i

In practice, these calculations are often performed using specialized software or instruments like gel permeation chromatography (GPC) or mass spectrometry.

Interpreting the polydispersity index

The polydispersity index provides valuable information about the homogeneity of a sample. Here's how to interpret different PDI values:

PDI Range	Interpretation
1.0 - 1.1	Narrow distribution, highly homogeneous sample
1.1 - 1.5	Moderately broad distribution
1.5 - 2.0	Broad distribution, significant heterogeneity
2.0+	Very broad distribution, highly heterogeneous sample

Practical implications

In polymer science, a narrow PDI (close to 1) is often desirable for applications requiring uniform properties
In colloid chemistry, higher PDI values may indicate aggregation or other particle interactions
In biotechnology, PDI can indicate the purity and consistency of protein preparations

Worked example

Let's calculate the polydispersity index for a sample with the following molecular weight distribution:

Molecular Weight (g/mol)	Weight Fraction	Number of Molecules
10,000	0.3	300
20,000	0.5	200
30,000	0.2	100

Step 1: Calculate Mw

Mw = (10,000 × 0.3 + 20,000 × 0.5 + 30,000 × 0.2) / (0.3 + 0.5 + 0.2) Mw = (3,000 + 10,000 + 6,000) / 1.0 Mw = 19,000 g/mol

Step 2: Calculate Mn

Mn = (10,000 × 300 + 20,000 × 200 + 30,000 × 100) / (300 + 200 + 100) Mn = (3,000,000 + 4,000,000 + 3,000,000) / 600 Mn = 10,000 g/mol

Step 3: Calculate PDI

PDI = Mw / Mn = 19,000 / 10,000 = 1.9

The calculated polydispersity index is 1.9, indicating a moderately broad molecular weight distribution.

Frequently asked questions

What is the difference between polydispersity index and molecular weight distribution?

The molecular weight distribution shows the complete range of molecular weights present in a sample, while the polydispersity index is a single number that summarizes the width of that distribution.

How is polydispersity index different from polydispersity?

Polydispersity refers to the presence of multiple molecular weights in a sample, while polydispersity index is the specific numerical measure of that distribution width.

What is a good polydispersity index value?

A good PDI value depends on the application. For many polymer applications, a PDI of 1.1 or lower is considered good, while values between 1.5 and 2.0 may still be acceptable depending on requirements.

Can polydispersity index be less than 1?

No, the polydispersity index cannot be less than 1. A PDI of 1 indicates a perfectly monodisperse sample where all molecules have the same molecular weight.