1
1
mL/g
100,000
g/mol
0
11.512925
1
1
1
mL/g
100,000
g/mol
0
11.512925
1
The Mark-Houwink Equation Calculator relates a polymer's intrinsic viscosity to its molecular weight through the empirical Mark-Houwink-Sakurada (MHS) equation. This relationship is foundational to polymer characterization by viscometry, one of the simplest and most accessible methods for determining molecular weight. The calculator computes either intrinsic viscosity from known molecular weight or molecular weight from measured viscosity.
The Mark-Houwink parameters K and a are specific to each polymer-solvent-temperature system and provide insight into chain conformation. The exponent a ranges from 0 for rigid spheres to 2.0 for rigid rods, with flexible random coils in good solvents typically giving a = 0.6–0.8. Extensive tabulations of K and a values exist in the Polymer Handbook and other reference works.
The Mark-Houwink-Sakurada equation is:
$$[\eta] = K \cdot M^a$$
Where $$[\eta]$$ is the intrinsic viscosity (mL/g or dL/g), $$M$$ is the molecular weight (typically Mv, the viscosity average), and $$K$$ and $$a$$ are empirical constants for a specific polymer-solvent-temperature system.
To solve for molecular weight when viscosity is known:
$$M = \left(\frac{[\eta]}{K}\right)^{1/a}$$
The exponent $$a$$ reveals chain conformation in solution:
$$a = 0: \text{ rigid sphere (Einstein)}$$
$$a = 0.5: \text{ theta condition (unperturbed coil)}$$
$$a = 0.6-0.8: \text{ flexible coil in good solvent}$$
$$a > 1.0: \text{ rigid rod}$$
The intrinsic viscosity is determined experimentally by measuring solution viscosity at several concentrations and extrapolating to zero concentration using the Huggins or Kraemer equations.
A higher intrinsic viscosity at a given molecular weight indicates a more expanded chain conformation and stronger polymer-solvent interactions. The exponent a provides direct insight into how the chain occupies space: coiled polymers in good solvents (a ≈ 0.7) are expanded, while theta solvents (a = 0.5) give unperturbed dimensions. Universal calibration in GPC exploits the Mark-Houwink equation by plotting [η]×M versus elution volume, which yields a single curve for all polymer architectures.
Inputs
Results
[η] = 1.74×10⁻⁵ × 100000^0.73 ≈ 0.347 mL/g (flexible coil conformation)
Inputs
Results
M = (0.5 / 1.74×10⁻⁵)^(1/0.73) ≈ 171,813 g/mol
The Mark-Houwink-Sakurada equation [η] = K×M^a relates a polymer's intrinsic viscosity to its molecular weight through two empirical constants K and a. It is the basis for molecular weight determination by viscometry and provides information about chain conformation in solution.
K and a values are tabulated in the Polymer Handbook (Brandrup, Immergut, Grulke), polymer databases, and published literature for specific polymer-solvent-temperature combinations. Common values: polystyrene in toluene at 25°C has K = 1.74×10⁻⁵ mL/g and a = 0.73; PMMA in chloroform has K = 4.8×10⁻⁵ and a = 0.80.
Viscometry gives the viscosity average molecular weight (Mv), which falls between Mn and Mw. For a = 1, Mv = Mw; for a < 1, Mn < Mv < Mw. In practice, Mv is close to Mw for most flexible polymer systems where a is in the range 0.6–0.8.
The exponent a reveals chain conformation: a ≈ 0 indicates compact spheres, a = 0.5 corresponds to theta solvent conditions (unperturbed coil), a = 0.6–0.8 indicates an expanded random coil in a good solvent, and a > 1.0 suggests rod-like or very stiff chains. Semiflexible biopolymers like DNA can have a ≈ 0.7–1.0 depending on ionic strength.
Intrinsic viscosity is measured using a capillary viscometer (Ubbelohde or Ostwald type). Solution viscosity is measured at 4–5 concentrations, specific viscosity (ηsp/c) and inherent viscosity (ln ηr/c) are plotted against concentration, and both are extrapolated to c = 0. The intercept gives [η].
K has units of mL/g (or dL/g, depending on convention) when M is in g/mol. Be careful with unit consistency: if [η] is in dL/g and M in g/mol, K must also be in dL/g. Literature values vary in their unit conventions, so always check the original source.
Yes, but K and a values depend on copolymer composition as well as solvent and temperature. There is no universal set of constants for copolymers; they must be determined experimentally for each composition. For block copolymers, the effective K and a reflect the combined hydrodynamic behavior of both blocks.
Universal calibration plots log([η]×M) versus elution volume. Since [η]×M is proportional to hydrodynamic volume, all polymers fall on the same curve regardless of chemistry or architecture. By measuring [η] with an online viscometer and using Mark-Houwink parameters, absolute molecular weights are obtained without polymer-specific standards.
Both K and a change with temperature because polymer-solvent interactions are temperature-dependent. At the theta temperature, a = 0.5 and the chain adopts unperturbed dimensions. Above theta in a good solvent, a increases to 0.6–0.8. Published K and a values must be used at the specified temperature.
The equation is valid over a specific molecular weight range determined experimentally. At very low M (oligomers), end-group effects become significant and the relationship breaks down. At very high M, branching or other structural features may cause deviation. Always check the stated valid range when using published K and a values.
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