T9 · Testing / StandardsPolymer ChemistryGATE 2027 Syllabus (XE5)

Mark-Houwink-Sakurada Equation (Intrinsic Viscosity to MW)

Empirical relation connecting limiting viscosity number (intrinsic viscosity [eta]) of a dilute polymer solution to viscosity-average molecular weight (Mv).

Mathematical Canonical Relation

[\eta] = K \cdot \overline{M}_v^a

Plain Formula: [eta] = K * (Mv)^a

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Mark-Houwink K (×10^-4 dL/g) (×10^-4 dL/g)

K_val

Range: 0.5 – 20 ×10^-4 dL/g

Exponent a (dimensionless)

a_val

Range: 0.5 – 0.9 dimensionless

Intrinsic Viscosity [eta] (dL/g)

eta_val

Range: 0.2 – 10 dL/g

Calculated Engineering Result:

114,080 g/mol

Engineering Verification:Conformation exponent a = 0.7 indicates good polymer-solvent interaction.

When & Where to Apply

Use when determining polymer molecular weight via capillary Ubbelohde viscometry in a specified solvent and temperature.

Common Engineering Pitfalls

Using K and a values for a different solvent or temperature
Confusing intrinsic viscosity [eta] (dL/g or mL/g) with dynamic melt viscosity (Pa·s)

Physical Assumptions & Validity Limits

Dilute solution regime where polymer coils do not overlap (c < c*)
Known Mark-Houwink constants K and a for the specific polymer-solvent-temperature system
Newtonian solvent behavior at zero shear limit

Variable Symbols, SI Units & Dimensional Specifications

Symbol	Variable Meaning	SI Unit	Typical Industrial Range
[\eta]	Intrinsic viscosity (limiting viscosity number)	dL/g	—
K	Mark-Houwink constant	dL/g	0.00016
a	Mark-Houwink exponent (conformation parameter)	dimensionless	0.70
\overline{M}_v	Viscosity-average molecular weight	g/mol	120000

Worked Industrial Numerical Example: High-Density Polyethylene Mv in Decalin at 135°C

Problem Statement:

For HDPE in decalin at 135°C, K = 6.2 × 10^-4 dL/g and a = 0.70. If measured [eta] = 2.15 dL/g, determine the viscosity-average molecular weight Mv.

Step-by-Step Numerical Solution:

Rearrange formula: Mv = ([eta] / K)^(1/a)
Compute [eta]/K: 2.15 / (6.2e-4) = 3467.74
Raise to power 1/0.70 (1.4286): (3467.74)^1.4286 = 115,200 g/mol

Final Answer: Mv = 115,200 g/mol

💡 Engineering Interpretation: The intrinsic viscosity of 2.15 dL/g corresponds to high molecular weight extrusion pipe grade HDPE suitable for PE-100 pressure pipe applications.

Authoritative Technical Reference & Source Provenance

Polymer Physics by Michael Rubinstein & Ralph Colby (Oxford University Press, 2003), 1st Edition, Page 38-42.

Connected Curriculum Lesson

Molecular Weight Averages & Dilute Solution Viscometry

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