The magnitude of the electric field at a distance $r$ from a point charge $Q$ is equal to $Q/4\pi {\epsilon }_0{r}^2.$ How close to a water molecule (of polarizability volume $1.48\times 10^{-30}{\mathrm{m}}^3$ ) must a proton approach before the dipole moment it induces has a magnitude equal to that of the permanent dipole moment of the molecule $(1.85\mathrm{D})?$

Key Concepts: Electric Field, Point Charge, Polarizability, Dipole Moment Explanation: We can use the formula for the induced dipole moment in a molecule due to an external electric field. We know that the induced dipole moment should have a magnitude of 1.85 D. We can equate this to the expression for dipole moment induced in a molecule due to an electric field, which is $p_{induced}=\alpha E$, where $\alpha$ is the polarizability of the molecule and $E$ is the electric field, and solve for $E$. Step by Step Solution: Step 1. Calculate the induced dipole moment of the water molecule using the given polarizability volume: $\alpha =1.48\times 10^{-30}{\mathrm{m}}^3$. Recall that $\alpha =3{ϵ}_{0}V$, where $V$ is the volume of the molecule and ${ϵ}_0$ is the permittivity of free space. Therefore, $\alpha =3{ϵ}_0(4/3)\pi (0.0965\mathrm{nm}{)}^3$ = 2.99 x 10^-30 Cm. Step 2. Substitute the induced dipole moment and polarizability into the dipole moment induced expression: $1.85D=2.99\times 10^{-30}Cm\times E$. Solve for $E$: $E=1.85D/(2.99\times 10^{-30}Cm)=6.18\times 10^{29}V/m$. Step 3. Use Coulomb's law to calculate the distance from the proton to the water molecule when it induces an electric field of this magnitude: $E=k_{e}Q/r^2$, where $k_e$ is Coulomb's constant. Rearranging for $r$: $r=\sqrt{k_{e}Q/E}$. Step 4. Substitute the values for $k_e$, $Q$, and $E$: $r=\sqrt{(9\times 10^{9}N{m}^2/C^2)(1.602\times 10^{-19}C)/(6.18\times 10^{29}V/m)}=2.93\times 10^{-}9m$. Final Answer: The proton must approach to a distance of $2.93\times 10^{-9}$ m from the water molecule to induce a dipole moment with a magnitude equal to the permanent dipole moment of the molecule.