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**1.** Given that the Debye length is about 10 nm in aqueous 0.001 mol
dm^{-3} KCl at room temperature, calculate its value
in aqueous 0.01 mol dm^{-3} solutions of (i) KNO_{
3}, (ii) CaCl_{2}, (iii) Na_{
2}SO_{4}, (iv) CuSO_{ 4}
, (v) LaCl_{3}.

The solubilities of a uni-univalent salt in an aqueous solution of sodium chloride (which had no ion in common with the salt) at 298 K were

NaCl/mol dm^{-3} | 0 | 0.001 | 0.005 | 0.01 | 0.02 |

Salt/mol dm^{-3} | 4.90 x 10^{-4} | 5.00 x 10^{-4} | 5.22 x 10^{-4} | 5.40 x 10^{-4} | 5.65 x 10^{-4} |

Calculate the solubility product of the salt and its activity coefficient in a saturated solution in water.

The degree of dissociation α of ethanoic acid, CH_{3}CO_{2}H, in aqueous solution at 298 K varies with concentration *c* in the following manner:

c/mol dm^{-3} | 2.801 x 10^{-5} | 1.532 x 10^{-4} | 2.414 x 10^{-3} | 5.912 x 10^{-3} |

α | 0.5393 | 0.2875 | 0.0829 | 0.054 |

Calculate the (thermodynamic) dissociation constant of ethanoic acid. How far do the data verify the Debye-Huckel theory?

**4.** The radius of gyration and second virial coefficient of a sample
of polystyrene dissolved in cyclohexane were measured at a series
of temperatures with the following results

T/K | 305.7 | 307.2 | 311.2 | 318.2 | 328.2 |

R_{g}/nm |
47.9 | 51.8 | 57.6 | 62.5 | 66.5 |

B | -0.40 | -0.20 | 0.37 | 0.95 | 1.58 |

Estimate the unperturbed root mean square end-to-end distance (empirically) and the θ temperature in this solvent.

**5.** The intensity of X-rays scattered by a 50 mg cm^{-3}
protein solution falls by a factor of 10 when κ
( = (4πsinθ)/λ) where 2θ is the scattering angle and λ is
the wavelength) increases from 0.1 to 1.0 nm^{ −1}
. If the scattering obeys the law ln I = ln I_{
0} - κ^{ 2} R^{
2}/3, what is the radius of gyration of the protein?

**6.** Describe how the short range chemical structure of a polymer
is related to its conformation in solution and derive an equation
relating the mean square end-to-end distance of a random flight
polymer to the number of formula units in a statistical segment.

The radius of gyration for poly(styrene) in cyclohexane at 310 K was found
to be 6.2 nm for a molecular weight of 50k. Given that the length
of the formula unit is about 0.3 nm estimate the number of formula
units in a statistical segment (the formula of styrene is C_{8}H_{8}).

**7.** (a) Explain what is meant by the radius of gyration of a polymer.
How is it related to the end-to-end distance for a single chain
polymer and what is its significance?

When κR_{g}<<1 the intensity, I,
of neutron scattering from a polymer molecule obeys the law I =
I_{0}exp(-κ^{ 2}
R_{g}^{ 2}/3) where I_{
0} is the incident intensity, κ is related
to the angle of scattering, and R_{ g}
is the radius of gyration. Two samples of amorphous polystyrene, A
and B, each containing a small fraction of deuterium labelled polymer,
gave the scattering data in the table below.

κ/nm^{-1} |
0.1 | 0.2 | 0.3 | 0.4 |

I_{A}/arbitrary units |
0.776 | 0.365 | 0.103 | 0.018 |

I_{B}/arbitrary units |
0.474 | 0.023 | 1.48x10^{-4} |
- |

Determine the radii of gyration of each of the polymers.

The molecular weights (M_{n}) of the two samples
are 100k (sample A) and 400k (sample B). Comment on the relation
between the two values you obtain.

**8.** The θ condition for a polymer in solution may be
achieved by varying either the temperature of a single solvent or
the composition of a mixed solvent. Comment on the factors involved.

Measurement of the osmotic pressure of a poly(isobutylene) sample in (a) cyclohexane and (b) benzene at 298 K gave the following results:

c/g dm^{-3} |
1 | 2 | 3 | 4 |

Π/Pa(a) | 5.17 | 13.95 | 27.26 | 45.68 |

Π/Pa(b) | 3.32 | 7.04 | 12.16 | 15.70 |

Determine the molecular weight and the second virial coefficient in each solvent. Comment on the results.

**9. **Measurement of the osmotic pressure Π (Pa)
of a poly(styrene) sample in mixtures of dichloroethane (A) and
cyclohexane (B) at 285 K in the A:B ratios given gave the following
results:

c/g dm^{-3} |
1 | 2 | 3 | 5 |

Π/Pa(65:35) | 2.52 | 6.92 | 13.20 | 31.40 |

Π/Pa(35:65) | 2.23 | 5.78 | 10.62 | 24.25 |

Π/Pa(6:94) | 1.76 | 3.90 | 6.39 | 12.50 |

Determine the molecular weight of the polystyrene and the second virial coefficient in each mixture of solvents. Use your values for the second virial coefficient to decide which of the two solvents is the better solvent for poly(styrene).