Water at has relative permittivity and Bjerrum length

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch Charges in salt water and the assembly of amphiphiles

Problem

Water at 298 K298\,\mathrm{K} has relative permittivity εr=78.4\varepsilon_r=78.4 and Bjerrum length ℓB=0.715 nm\ell_B=0.715\,\mathrm{nm}, obtained from ℓB=e2/4πε0εrkBT\ell_B = e^2/4\pi\varepsilon_0\varepsilon_r k_BT. Use this reference value to answer each part. ((a)) A patch of protein interior has εr=2\varepsilon_r=2 and is at body temperature 310 K310\,\mathrm{K}. Scale the reference value to find the Bjerrum length there. ((b)) A buffer has monovalent ionic strength I=20 mMI=20\,\mathrm{mM} at 298 K298\,\mathrm{K} in water. Using κ−1=0.304/I nm\kappa^{-1}=0.304/\sqrt{I}\,\mathrm{nm} (with II in mol L−1^{-1}), find the Debye length. ((c)) An anionic membrane patch carries surface charge density σ=−0.03 C m−2\sigma=-0.03\,\mathrm{C\,m^{-2}} in 150 mM150\,\mathrm{mM} salt at 298 K298\,\mathrm{K}. The Grahame equation reads σ=8c∞ε0εrkBT sinh⁡(eψ0/2kBT)\sigma=\sqrt{8c_\infty\varepsilon_0\varepsilon_r k_BT}\,\sinh(e\psi_0/2k_BT), and at this salt the prefactor 8c∞ε0εrkBT=0.045 C m−2\sqrt{8c_\infty\varepsilon_0\varepsilon_r k_BT}=0.045\,\mathrm{C\,m^{-2}}. Find the surface potential ψ0\psi_0 (state its sign). ((d)) A single-stranded polynucleotide carries one charge every b=0.34 nmb=0.34\,\mathrm{nm} (twice the spacing of double-stranded DNA's phosphates). Using ξM=ℓB/b\xi_M=\ell_B/b and the fraction neutralised 1−1/ξM1-1/\xi_M, find the fraction of its charge that Manning condensation neutralises.

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