Mark-Houwink Equation & Polymer Intrinsic Viscosity Reference

IIT-JAM │ GATE │ CSIR-NET │ TIFR │ BARC │ Polymer Chemistry

Exam Focus: The Mark-Houwink equation is one of the most important relations for determining the viscosity-average molecular weight of a polymer from intrinsic-viscosity measurements.

1. Mark-Houwink Equation

The intrinsic viscosity of a polymer solution is related empirically to the viscosity-average molecular weight of the polymer by the Mark-Houwink-Sakurada equation.

$$ [\eta] = K M_v^a $$

where:

• [η] = intrinsic viscosity
• M_v = viscosity-average molecular weight
• K = Mark-Houwink constant
• a = Mark-Houwink exponent

The values of K and a depend on the polymer, solvent, temperature and experimental conditions.

Golden Rule: Mark-Houwink equation gives M_v, not directly M_n or M_w.

2. Logarithmic Form

Taking logarithm on both sides:

$$ \log[\eta]=\log K+a\log M_v $$

Therefore, a plot of log[η] versus log M_v gives a straight line with:

• Slope = a
• Intercept = log K

This form is especially useful in GATE, CSIR-NET and university numerical problems.

3. Physical Significance of the Exponent 'a'

Approximate value of a	Typical interpretation	Important point
≈ 0.5	Ideal/θ-solvent behaviour	Polymer behaves approximately as an unperturbed random coil.
0.5 – 0.8	Good-solvent behaviour	Polymer coil is expanded because of favourable polymer–solvent interactions.
≈ 1 or higher	More extended or stiff chain	Can occur for relatively rigid or highly extended polymer conformations.
≈ 1 – 2	Rod-like/rigid-chain behaviour	Large values of a are associated with more extended chain conformations.

Important: The value of a is not determined only by polymer shape. It also depends on polymer–solvent interactions, molecular weight range, temperature and chain architecture.

4. θ-Solvent Condition

At the θ-condition, the polymer chain behaves approximately as an ideal random coil. For many ordinary flexible polymers:

$$ a \approx 0.5 $$

Thus:

θ-solvent → a ≈ 0.5 → unperturbed/ideal coil

A commonly cited example is polystyrene in cyclohexane near its θ-temperature (approximately 34 °C).

5. Good Solvent

In a good solvent, polymer–solvent interactions are favourable and the polymer chain becomes more expanded than in a θ-solvent. Consequently, for many flexible-chain systems:

$$ a > 0.5 $$

Typical values are often in the approximate range:

0.5 < a < 0.8

although actual experimental values depend strongly on the particular polymer–solvent system.

6. Molecular-Weight Average Obtained

For a polydisperse polymer, intrinsic viscosity is related to the viscosity-average molecular weight:

$$ [\eta]=K M_v^a $$

For the commonly encountered case of approximately 0 < a < 1:

$$ M_n \leq M_v \leq M_w $$

If:

$$ a=1 $$

then the viscosity-average molecular weight becomes equal to the weight-average molecular weight:

$$ M_v=M_w $$

Exam caution: The inequality involving M_n, M_v and M_w should be stated with the appropriate assumptions regarding the exponent and molecular-weight distribution.

7. Determination of M_v

From:

$$ [\eta]=K M_v^a $$

we obtain:

$$ M_v=\left(\frac{[\eta]}{K}\right)^{1/a} $$

This equation is frequently used directly in numerical problems.

8. Example Numerical

Question:

If [η] = 1.2 dL g⁻¹, K = 1.0 × 10⁻⁴, and a = 0.7, calculate the viscosity-average molecular weight.

$$ M_v= \left( \frac{1.2}{1.0\times10^{-4}} \right)^{1/0.7} $$ $$ M_v=(12000)^{1/0.7} $$ $$ M_v\approx6.72\times10^5\;g\,mol^{-1} $$

Answer: M_v ≈ 6.7 × 10⁵ g mol⁻¹

Correction to the earlier calculation: The value 1.58 × 10⁵ g mol⁻¹ is incorrect. The correct value is approximately 6.72 × 10⁵ g mol⁻¹.

9. Flory–Fox Equation

The Flory–Fox equation connects intrinsic viscosity with polymer molecular dimensions. A commonly used form is:

$$ [\eta]=\Phi\frac{\langle r^2\rangle_0^{3/2}}{M} $$

where:

• Φ = Flory constant
• ⟨r²⟩₀ = unperturbed mean-square end-to-end distance
• M = molecular weight

This relation is important for understanding the connection between chain dimensions, molecular weight and intrinsic viscosity.

10. Stockmayer–Fixman Equation

For many polymer solutions, the Stockmayer–Fixman treatment gives:

$$ \frac{[\eta]}{\sqrt{M}} = K_\theta + kM^{1/2} $$

Thus, a plot of:

$$ \frac{[\eta]}{\sqrt{M}} \quad\text{vs.}\quad \sqrt{M} $$

can be used to obtain the θ-limit constant/intercept under the appropriate assumptions.

11. Applications

• Determination of viscosity-average molecular weight M_v
• Characterization of polymer molecular size
• Comparison of polymer–solvent interactions
• Study of chain expansion in different solvents
• Identification of θ-like behaviour using a ≈ 0.5
• Polymer quality control and industrial characterization

12. Limitations

• K and a are specific to the polymer–solvent–temperature system.
• The equation is empirical and should be applied within the appropriate molecular-weight range for the calibrated system.
• Polyelectrolyte solutions can show unusual concentration dependence; ionic strength and added salt may be important.
• The equation gives M_v, not directly M_n.
• Values of K and a from one polymer–solvent system should not be transferred blindly to another system.
• Branching, chain stiffness and polymer architecture can affect the Mark-Houwink parameters.

13. Typical Literature Values — Use with Caution

The following values are representative examples often encountered in polymer literature. They are not universal constants; different literature sources may report different values because K and a depend on experimental conditions and conventions.

Polymer	Solvent	Temperature	Approximate a	Exam Note
Polystyrene	Toluene	25 °C	≈ 0.7	Good-solvent behaviour
Polystyrene	Cyclohexane	≈ 34 °C	≈ 0.5	θ-condition
PMMA	Acetone	25 °C	≈ 0.7	System-dependent
Polyethylene	Decalin	High temperature	≈ 0.6–0.7	Depends on experimental conditions

Important for Competitive Exams: Do not memorize numerical K-values unless a specific question or data table provides them. The most important concepts are: [η] = KM_v^a, a ≈ 0.5 at θ-condition, and M_v as the molecular-weight average obtained.

14. High-Yield Exam Questions

1. The exponent a in the Mark-Houwink equation is approximately 0.5 for a polymer in:

Answer: θ-solvent

2. For many flexible polymers in a good solvent, the value of a is generally:

Answer: Greater than 0.5, commonly around 0.5–0.8.

3. The Mark-Houwink equation is:

Answer: [η] = K M_v^a

4. The molecular weight obtained from the Mark-Houwink equation is:

Answer: Viscosity-average molecular weight (M_v)

5. If a = 1, then:

Answer: M_v = M_w

6. In the logarithmic form of the Mark-Houwink equation, the slope of log[η] versus log M is:

Answer: a

7. Assertion: In a good solvent, the intrinsic viscosity generally increases more strongly with molecular weight than in a θ-solvent.

Reason: For many flexible polymers, a in a good solvent is greater than approximately 0.5.

Answer: Both Assertion and Reason are true, and the Reason correctly explains the Assertion under the stated assumptions.

8. MPSET-type Question:

The intrinsic viscosity of a polymer solution is related to molecular weight M by K and a as constants. The correct relation is:

A. η = KM^a
B. η = KM^1/a
C. η = K^aM
D. η = K + aM

Answer: A. η = KM^a

Strictly, the quantity used in the Mark-Houwink equation is intrinsic viscosity [η], so the more precise expression is [η] = KM^a.

15. Quick Revision Chart

Concept	Key Point
Mark-Houwink equation	[η] = K Mva
Log form	log[η] = log K + a log Mv
Slope of log plot	a
Intercept	log K
θ-solvent	a ≈ 0.5 for many flexible polymers
Good solvent	Usually a > 0.5
Rigid/extended chain	Often larger a values
Molecular weight obtained	Mv
a = 1	Mv = Mw
Numerical formula	Mv = ([η]/K)1/a
Flory–Fox	Relates [η] to unperturbed chain dimensions
Stockmayer–Fixman	[η]/√M = Kθ + k√M

Golden Exam Rules:

1. [η] = KM_v^a
2. a ≈ 0.5 → θ-condition for many flexible polymer systems.
3. a > 0.5 → generally good-solvent behaviour for many flexible chains.
4. Mark-Houwink gives M_v.
5. M_v = ([η]/K)^1/a.
6. For the common case 0 < a < 1, M_n ≤ M_v ≤ M_w.
7. a = 1 → M_v = M_w.
8. K and a are not universal constants; they depend on polymer, solvent and temperature.