Mark-Houwink Viscosity Calculator

Calculate intrinsic viscosity and polymer molecular weight from Mark-Houwink equation.

Intrinsic viscosity [η] (mL/g) (mL/g)

Polymer-solvent system

K (mL/g × (mol/g)^a) — custom

a exponent (0.5-0.8) — custom

Please check your inputs and try again.

What Intrinsic Viscosity Reveals

Dissolved polymer chains occupy far more volume than their mass suggests, and they raise solution viscosity in proportion to that occupied volume. Intrinsic viscosity measures this at infinite dilution, and the Mark–Houwink relation converts it to molecular weight.

[η] = K M^a M_v = ([η]/K)^1/a

K and a are empirical constants specific to a polymer–solvent–temperature combination. They cannot be transferred between systems, which is why tabulated values always specify all three.

a value	Chain conformation	Solvent quality
0.5	Random coil, ideal	Theta solvent — segments neither attract nor repel
0.6–0.8	Expanded coil	Good solvent — chain swells
1.0	Semi-rigid rod	Stiff backbone, e.g. DNA
1.8–2.0	Rigid rod	Fully extended
0	Hard sphere	Compact, e.g. globular protein

The exponent is therefore a conformational probe. A value near 0.5 means the chain adopts its unperturbed dimensions; higher values mean the solvent is good enough that the chain expands to avoid itself.

Why M_v Is Its Own Average

Viscometry gives a viscosity-average molecular weight that sits between M_n and M_w, usually closer to M_w. It equals M_w exactly only when a = 1. This makes viscometry inexpensive and quick, but it cannot report a distribution — only gel permeation chromatography does that.

Worked Examples

Example 1: PS in toluene: [η]=150 mL/g

Mv=(150/1.72e-2)^(1/0.693)

Result: Mv≈486,000 g/mol

Viscometry gives quick MW estimate

Example 2: DNA sizing: [η]=500mL/g in 1M NaCl

Mv=(500/1.45e-1)^(1/0.571)

Result: Mv estimate for DNA fragment

Used with gel electrophoresis calibration

Example 3: Reading solvent quality

Same polymer gives a = 0.50 in one solvent, 0.75 in another

Result: Theta versus good solvent

In a theta solvent the chain adopts ideal random-coil dimensions. In a good solvent it expands, raising both intrinsic viscosity and the exponent.

Example 4: Why DNA gives a high exponent

a ≈ 1.0 for DNA in salt solution

Result: Semi-rigid rod behaviour

DNA's persistence length is far greater than a flexible synthetic polymer's, so it behaves as a stiff chain rather than a random coil.

Common Mistakes

⚠️

Using K and a from a different solvent

The constants are specific to polymer, solvent and temperature together. Transferring them between systems can give errors of tens of percent.

⚠️

Treating M_v as equal to M_w

The viscosity average lies between M_n and M_w and equals M_w only when a = 1. For typical a values it is somewhat lower.

⚠️

Measuring at a single concentration

Intrinsic viscosity is defined at infinite dilution. Several concentrations must be measured and extrapolated to zero.

⚠️

Expecting a distribution from viscometry

The method gives one average value only. Polydispersity requires gel permeation chromatography or light scattering.

Frequently Asked Questions

How is [η] measured?

Dilute solution viscometry. Measure relative viscosity η_rel = η_solution/η_solvent using Ubbelohde or Ostwald viscometer. Extrapolate η_sp/c vs c to c→0 gives [η]. Simple, inexpensive, gives viscosity-average MW.

a exponent meaning?

a=0.5: theta solvent (ideal chain, random coil). a=0.6-0.8: good solvent (expanded coil). a>0.9: rod-like chains (DNA, some polyelectrolytes). a<0.5: compact sphere (hyperbranched, dendrimers). Gives chain conformation information.

What does the Mark–Houwink exponent tell you?

Chain conformation and solvent quality. Around 0.5 indicates ideal random coil in a theta solvent; 0.6–0.8 indicates an expanded coil in a good solvent; near 1 or above indicates rod-like behaviour.

What is a theta solvent?

One in which polymer–solvent and polymer–polymer interactions balance, so the chain adopts unperturbed ideal dimensions. The exponent is then 0.5.

How does M_v relate to M_n and M_w?

It lies between them, usually nearer M_w, and equals M_w only when a = 1.

Why must K and a match the solvent?

Because they encode how the chain expands in that specific environment at that temperature. Using values from another system gives systematically wrong molecular weights.

Formula Explorer connections

Interpretation: This formula connects molecular properties, material structure or environmental transport to a macroscopic behavior or exposure estimate. Assumption: Use parameters measured for the same material, solvent, temperature and environment. Empirical correlations may not transfer outside their calibration range.

Glass Transition Temperature Calculator →Atmospheric Lifetime Calculator →BET Surface Area Calculator →Chemistry Formula Explorer →