Question

Question 3-10 points One of the most important functions of the membrane in living cells is its ability to selectively pass either sodium ions (Na) or potassium ions (K) through channels in the membrane. These ions have the same charge +le). The electric force exerted on them by the membrane will be similar. In addition, they are about the same size (RNa~0.12 nm, RK 0.15 nm). So how can channels in the membrane distinguish between them? One mechanism that has been proposed to account for the fact that the membrane channels treat these two ions differently is the suggestion that the ions attract water molecules, providing a sphere of water that magnifies the small size difference between the ions. This "water cloud" is illustrated in the figure at the right. A full calculation of the size of an ion's "water cloud" is very difficult. (For a start, it would require us to use quantum mechanics to include a repulsion force that keeps the atoms from getting too close to each other). However, we can get an idea of what is happening by exploring the electric force between the ion and one water molecule as shown in the figure at the right. We label the ion A, the oxygen in the water B, and the hydrogen atoms in the water C and D (a) Lets make a simple physical model - assuming that the ion and the two hydrogern atoms have a charge te, the oxygen has a charge -2e, and each atom can be treated as a point charge. Sketch, with reasonable accuracy, the layout of the atoms shown in the figure above, adding arrows to indicate the direction and relative magnitude of the force that the ion exerts on each of the three atoms in the water molecule, B, C, and D. (2 Points) (b) Create a free body diagram for the water molecule and use it to help calculate the net force that the ion (A) exerts on the water molecule in picoNewtons (10 N). The edges of the boxes in the grid above are Inm long. (Remember that forces are vectors). (8 Points)

Answer 1

a)

b)

Distance between ion and oxygen is $3\,nm=3*10^{-9}\,m$

Force between ion and oxygen molecule is $F_{BA}=\frac{1}{4\pi\epsilon_0}\frac{e*2e}{(3*10^{-9})^2}=8.99*10^9*\frac{2*(1.6*10^{-19})^2}{(3*10^{-9})^2}=51.1*10^{-12}\,N$ is directed towards right.

Distance between ion and hydrogen  is $r=\sqrt{4^2+1^2}=\sqrt{17}\,nm=\sqrt{17}*10^{-9}\,m$

Force vector on hydrogen atoms is making an angle $\theta=\tan^{-1}\left ( \frac{1\,n}{4\,n} \right )=18.44\degree$ with the horizontal.

Y-components of force on two hydrogen atoms due to ion cancel each other.

X-components of force on two hydrogen atoms due to ion will add up to $F_{CAx}+F_{DAx}=2 \frac{1}{4\pi\epsilon_0}\frac{e*e}{(\sqrt{17})^2}\cos\theta$

$F_{CAx}+F_{DAx}=2 *8.99*10^9*\frac{(1.6*10^{-19})^2}{(\sqrt{17}*10^{-9})^2}\cos18.44\degree=25.7*10^{-12}\,N$ is directed towards left.

Net force on the water molecule exerted by ion is

$F_{BA}-F_{CAx}-F_{DAx}=51.1*10^{-12}-25.7*10^{-12}=25.4*10^{-12}\,N$ is directed towards the ion.